113
Biol. Rev. (2007), 82, pp. 113–142.
doi:10.1111/j.1469-185X.2006.00006.x
Quantitative steps in the evolution of
metabolic organisation as specified by the
Dynamic Energy Budget theory
S. A. L. M. Kooijman and T. A. Troost
Department of Theoretical Biology Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam, The Netherlands
(Received 12 January 2006; revised 8 November 2006; accepted 14 November 2006)
ABSTRACT
The Dynamic Energy Budget (DEB) theory quantifies the metabolic organisation of organisms on the basis of
mechanistically inspired assumptions. We here sketch a scenario for how its various modules, such as
maintenance, storage dynamics, development, differentiation and life stages could have evolved since the
beginning of life. We argue that the combination of homeostasis and maintenance induced the development of
reserves and that subsequent increases in the maintenance costs came with increases of the reserve capacity. Life
evolved from a multiple reserves - single structure system (prokaryotes, many protoctists) to systems with multiple
reserves and two structures (plants) or single reserve and single structure (animals). This had profound
consequences for the possible effects of temperature on rates. We present an alternative explanation for what
became known as the down-regulation of maintenance at high growth rates in microorganisms; the density of the
limiting reserve increases with the growth rate, and reserves do not require maintenance while structure-specific
maintenance costs are independent of the growth rate. This is also the mechanism behind the variation of the
respiration rate with body size among species. The DEB theory specifies reserve dynamics on the basis of the
requirements of weak homeostasis and partitionability. We here present a new and simple mechanism for this
dynamics which accounts for the rejection of mobilised reserve by busy maintenance/growth machinery. This
module, like quite a few other modules of DEB theory, uses the theory of Synthesising Units; we review recent
progress in this field. The plasticity of membranes that evolved in early eukaryotes is a major step forward in
metabolic evolution; we discuss quantitative aspects of the efficiency of phagocytosis relative to the excretion of
digestive enzymes to illustrate its importance. Some processes of adaptation and gene expression can be
understood in terms of allocation linked to the relative workload of metabolic modules in (unicellular) prokaryotes
and organs in (multicellular) eukaryotes. We argue that the evolution of demand systems can only be understood
in the light of that of supply systems. We illustrate some important points with data from the literature.
Key words: dynamic energy budget, homeostasis, reserves, maintenance, supply and demand systems,
evolution.
CONTENTS
I. Introduction ......................................................................................................................................
II. Steps in metabolic evolution .............................................................................................................
(1) Variable biomass composition ....................................................................................................
(2) Strong homeostasis .....................................................................................................................
(3) Reserves .......................................................................................................................................
( a ) Partitionability and weak homeostasis ................................................................................
( b ) First-order dynamics ...........................................................................................................
( c ) Rejection of mobilised reserve ............................................................................................
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Address for correspondence: E-mail: [email protected]
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
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( d ) Structural homeostasis .........................................................................................................
( e ) Excretion ..............................................................................................................................
(4) Adaptation ...................................................................................................................................
(5) Maintenance ...............................................................................................................................
( a ) Carriers and regulation .......................................................................................................
( b ) Turnover of structure ..........................................................................................................
( c ) Defence systems ...................................................................................................................
(6) Increase of reserve capacity ........................................................................................................
(7) Morphological control on metabolism .......................................................................................
(8) k-rule and the emergence of cell cycles .....................................................................................
(9) Syntrophy and compartmentalisation ........................................................................................
( a ) Mitochondria .......................................................................................................................
( b ) Membrane plasticity and plastids .......................................................................................
( c ) Genome organisation ..........................................................................................................
(10) Reduction of number of reserves ...............................................................................................
(11) Emergence of life stages: adult and embryo ..............................................................................
(12) Further increase in maintenance costs .......................................................................................
(13) Differentiation .............................................................................................................................
( a ) Ageing and sleeping ............................................................................................................
(14) From supply to demand systems ................................................................................................
( a ) Behaviour and time budgets ...............................................................................................
III. Effects of temperature .......................................................................................................................
IV. Conclusions .......................................................................................................................................
V. Acknowledgements ............................................................................................................................
VI. References .........................................................................................................................................
VII. Appendix a: interacting transformations ..........................................................................................
(1) Reserve dynamics, rejection and homeostasis ...........................................................................
(2) Co-metabolism ............................................................................................................................
(3) Inhibition and preference ...........................................................................................................
( a ) Supply kinetics .....................................................................................................................
( b ) Demand kinetics ..................................................................................................................
(4) Inhibition and social interaction ................................................................................................
(5) Reversion and phototrophy ........................................................................................................
VIII. Appendix b: excretion of digestive enzymes ....................................................................................
(1) Intracellular digestion .................................................................................................................
(2) Social digestion ...........................................................................................................................
(3) Solitary digestion .........................................................................................................................
I. INTRODUCTION
The metabolism of organisms had and still has a strong
influence on the chemical and physical conditions on our
planet (Kooijman, 2004). The organisation of the metabolism
of eukaryotes evolved from that of prokaryotes after a process
of symbiogenesis (Kooijman & Hengeveld, 2005), where freeliving eubacteria were internalised by archaea and eventually
turned into intracellular mitochondria that transferred most
of their genes to the host (Rivera & Lake, 2004). The various
steps in the processes of internalisation and integration are
discussed in Kooijman et al. (2003) on the assumption that the
metabolic organisation of both types of bacteria follows rules
specified by the Dynamic Energy Budget (DEB) theory
(Kooijman, 2000, 2001). It has been shown that this
internalisation could have occurred spontaneously as a result
of incremental changes in the values of particular parameters, in such a way that the merged structure again follows
DEB rules. A full merging comes with a set of constraints on
the metabolic activity for both endosymbionts (mitochondria)
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and the host. Using a mixture of open and closed
handshaking protocols for the enzymes in the tricarboxylic
acid (TCA)-cycle and linking the abundance of this TCA
enzyme-complex to the reserve and the structure of the cell,
the mix of intermediary metabolites and end products that is
released by the mitochondria can exactly match the needs at
the cellular level, which vary with the growth conditions
(Kooijman & Segel, 2005).
The present paper extends the discussion of these
evolutionary events back in time and places them in a wider
context, in an attempt to introduce the various DEB modules
one by one in a realistic way, and to sketch a possible scenario
for the evolution of metabolic organisation. Contrary to most
work in this field, we here focus on the evolution of the
dynamic system called individual, and not on its explicit
biochemistry or phylogeny. We keep the mathematics low key
in the main text (see Table 1 for a description of the symbols
used). The Appendices provide a more detailed quantitative
underpinning, and review recent progress in the development
of synthesising units (SUs) for quantifying the metabolic
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
115
Table 1. Symbols that are used in the text. A link exists between the leading symbol and the dimension
group. ‘Per structural
volume’ is indicated by [], so ½p_ C ¼ p_ C =V and [E] ¼ E/V; ‘per structural surface area’ by {}, so p_ A ¼ p_ A =V 2=3 . Dots above
symbols mean ‘per time’ and do not indicate differentiation. Transposing (i.e. interchanging rows and columns in a matrix or vector)
is indicated with superscript T. Subscript asterisk (*) means a wild card, which can be any compound. Superscript asterisk means
that the value is taken at steady state. In the dimension column: l is length of environment, L is length of structure, # is C-mol, e is
energy
Symbol
_ b_ b,
d*, d3K
_
D
e
E, Em
[EG]
f, f*
g
h_
j*, j*m, j*¢, j*¢¢
jEA, jEM
J_ , J_
J_ EC , J_ EM
_ k_ i , k_ ij
k,
_ k_ k,
k_ A , k_ C
k_ E , k_ M
K, K*
L, LR, L3
mE, mH
M*
n, n*
N
p_ A , p_ Am
p_ C
p_ M , p_ J
r_
S
t
v_
V
w*, w¢*
x
X, Xr
y
y1 2 , ym1 2
Y
Y1 2 , Yg
d
q*, q1 2 , u
k, k*
r, r*
’, j
dim
[1 3 [1
# lt
#l[1
l2t[1
e
eL[3
t[1
##[1t[1
##[1t[1
#t[1
#t[1
t[1
t[1
t[1
t[1
#l[3
L
##[1
#
##[1
#
et[1
et[1
et[1
t[1
l2
t
Lt[1
L3
#l[3
##[1
#l[3
##[1
-
description
affinity or searching rate, – for *
density of *, half-saturation density for the number of SUs
diffusivity of compound *
scaled reserve density: [E]/[Em]
reserve expressed in energy, max –
energy cost per unit of structure
scaled functional response, – for *
energy investment ratio
background expression rate, throughput rate
specific flux of compound *, max –, scaled –, scaled –
specific flux of reserve associated with assimilation, maintenance
arrival flux of compound *, vector of fluxes
flux of reserve associated to catabolism, maintenance
matrix of rates, typical element of this matrix
dissociation rate, – for compound *
normalised assimilation, catabolic rate
reserve turnover rate, maintenance rate coefficient
half-saturation coefficient, – for *
volumetric length of structure, cell radius, diffusion length
reserve, maturity density on the basis of moles
mass of compound *
number per mass of structure, – for SUs of type *
number of synthesising units
assimilation energy flux, max –
catabolic energy flux
somatic, maturity maintenance energy flux
specific growth rate
surface area of environment
time
energy conductance
structure expressed in volume
preference, inhibition parameter for *
scaled substrate or food density
substrate concentration, – in the feed
scaled population density
fixed yield coefficient of compound *1 on compound *2, max –
population density
variable yield coefficient of compound *1 on compound *2, ‘‘true’’ –
aspect ratio
fraction of SU in binding state *, *1*2, vector of fractions
allocation fraction of utilised reserve, – for synthesis of carriers *
binding probability, – for compound *
efficiency, rejection strength
Types of compounds that only can appear as indices of symbols, where VV (the volume of structure) is abbreviated as V and EE (the
chemical potential of reserve) is abbreviated as E. The dot () is used for ‘‘no substrate’’ in enzyme-substrate complexes.
3
F
H
L
R
V
Y,Z
enzyme
fumarate
monomer
photon (light)
reproduction product
structure
population
E
G
J
P,P*
S,S*
X
reserve
glucose
maturity product
pyruvate, product(s)
substrate(s)
food
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
116
behaviour of generalised enzymes; SU dynamics plays an
important role in DEB theory. We start with a chemically and
biologically explicit introduction on the role of metabolic
modules and their fate that helps to explain why it is possible
to have a theory on metabolic organisation that applies to all
organisms. We also need such a theory to motivate the initial
metabolic organisation.
Modern organisms live in a carbohydrate world, but
travelling back in time we encounter a lipid world (Segré
et al., 2001), a protein/DNA world, an RNA world (Stryer,
1988; Duve, 1984) or an RNA/peptide world (Lahav, 1993)
and then it becomes more speculative (Orgel, 1998). We
start the discussion somewhere in the RNA world and
follow the quantitative aspects of the various steps that
metabolism might have taken.
that all cycles in the central metabolism of typical modern
heterotrophs ran in the opposite direction in the evolutionary past.
This reconstruction suggests that lateral gene exchange
between eubacteria and archaea occurred during the
evolution of central metabolism. Initially cells could
exchange RNA and early strands of DNA relatively easily
(Woese, 2002); restrictions on exchange became more
stringent with increasing metabolic complexity. Many
authors suggest that considerable lateral gene exchange
occurs in extant prokaryotes (Maynard Smith et al., 1993;
Gupta, 1998; Koonin, Makarova, & Aravind, 2001; Martin
et al., 2003) by conjugation, plasmids and viruses (Sullivan,
Waterbury & Chisholm, 2003).
Phototrophy probably was invented more than 3.2 Ga
ago; the green sulphur bacterium Chlorobium runs the TCA
cycle (and glycolysis) in the opposite direction compared to
typical (modern aerobic) organisms (Madigan, Martinko &
Parker, 2000), indicating an early type of organisation
(Hartman, 1975; Wächtershäuser, 1990; Morowitz et al.,
2000). Recent evidence suggests that phototrophy is also
possible near hydrothermal vents at the ocean floor (Dover,
2000), where the problem encountered by surface dwellers,
namely that of damage by ultraviolet (UV) radiation, is
absent.
In view of its availability, methane is likely to have been
an important substrate (and/or product) during life’s origin
(Hayes, 1994). Methanogenesis and (anaerobic) methylotrophy are perhaps reversible in some archaea (Hallam
et al., 2004); their metabolic pathways share 16 genes, and
are present in some archaeal and eubacterial taxa. The
most probable scenario for its evolutionary origin is that it
first evolved in the planctomycetes, which transferred it to
the proteobacteria and the archaea (Chistoserdova et al.,
2004). This remarkable eubacterial taxon is unique in
sporting anaerobic ammonium oxidation (anammox). The
II. STEPS IN METABOLIC EVOLUTION
The evolution of central metabolism is discussed in
Kooijman & Hengeveld (2005) and its possible prokaryotic
start is summarised in Fig. 1. The examples of contemporary models (Lindahl & Chang, 2001; Romano & Conway,
1996) illustrate that the metabolic systems themselves are
not hypothetical, but the evolutionary links between these
systems obviously are. This is not meant to imply, however,
that the taxa also would have these evolutionary links. Some
species of Methanococcus have most genes of the glycolysis;
Thermoproteus possesses a variant of the reversible EmbdenMeyerhof-Parnas and the Entner-Douderoff pathways;
Sulfolobus has oxidative phosphorylation. These contemporary models are not just evolutionary relicts; the picture is
rather complex.
Some important features are that heterotrophy evolved
from phototrophy, which itself evolved from lithotrophy, and
chemolithotrophy
fatty acid
metabolism
phototrophy
heterotrophy
iRC
iRC
iPP
iPP
iPP
Eubacterial roots
PP
Gly
Archaeal roots
ACS
iTCA
isoprenoid−ether
metabolism
iGly
iTCA
carbohydrate
metabolism
TCA
RC
Gly
TCA
RC
oxidative
phosphorylation
Fig. 1. Evolution of central metabolism among prokaryotes that formed the basis of eukaryotic organisation of central
metabolism. ACS ¼ acetyl-coenzyme A synthase pathway, iPP ¼ inverse pentose phosphate cycle ( ¼ Calvin cycle), PP ¼ pentose
phosphate cycle, iTCA ¼ inverse tricarboxylic acid cycle, TCA ¼ tricarboxylic acid cycle ( ¼ Krebs cycle), iGly ¼ inverse glycolysis,
Gly ¼ glycolysis, iRC ¼ inverse respiratory chain, RC ¼ respiratory chain. The arrows indicate the directions of synthesis to show
where they reversed; all four main components of eukaryote’s heterotrophic central metabolism originally ran in the reverse
direction to store energy and to synthesise metabolites. The approximate time scale is indicated above the scheme (i.e. the origin of
life, and that of cyanobacteria and eukaryotes). Contemporary models: A1 Methanococcus; A2 Thermoproteus; A3 Sulfolobus; E2
Nitrosomonas; E3 Chloroflexus; E4 Prochlorococcus; E5 Escherichia. Modified from Kooijman & Hengeveld (2005).
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
anammox clade has ether lipids in their membranes and
a proteinaceous cell wall like the archaea (Strous & Jetten,
2004). They have advanced compartmentation and
a nuclear membrane like the eukaryotes (Lindsay et al.,
2001), and are abundant in (living) stromatolites (Papineau
et al., 2005). Fossil stromatolites resemble the living ones
closely (Dill et al., 1986) and date back some 3.5 Ga ago
(Walker, 1994). Although, this points to a key role in early
evolution, planctomycetes seem too complex as a contemporary model for an early cell. Moreover, anaerobic
methane oxidation (amo) involves sulphate reduction.
Isotope data indicate that sulphate reduction originated
3.47 Ga ago (Shen & Buick, 2004). Sulphate was rare by
then (Canfield, Habicht & Thamdrup, 2000) and might
have been formed photochemically by oxidation of volcanic
SO2 in the upper atmosphere, or phototrophically by green
and purple sulphur bacteria (Chlorobiaceae, Chromatiacea),
(Pierson, Mitchell & Ruff-Roberts, 1993).
In an attempt to imagine the metabolic start of life, Fig. 2
illustrates some of the evolutionary steps that are discussed
in the next subsections, where the origin of reserves is linked
to the evolution of homeostasis, and enhanced by
maintenance.
(1) Variable biomass composition
We start with a living (prokaryotic) cell, surrounded by
a membrane. Although it remains hard to define what life is
exactly, it represents an activity and, therefore, requires
energy. ATP generation via a proton pump across the outer
membrane is probably one of the first steps in the evolution
of metabolism. The energy for this ATP generation
probably came from some extracellular chemoautotrophic
process (Wächtershäuser, 1988; Russell & Hall, 1997;
Russell & Hall, 2002). Since genome size might quantify
117
metabolic complexity, it helps to note that some chemoautotrophs have the smallest genome size of all organisms
(Kooijman & Hengeveld, 2005); e.g. the togobacterium
Aquifex aeolicus has a genome size of 1.55 Mbp and utilises
H2, S0 or S2O3[ as electron donors, and O2 or NO3[ as
electron acceptors. Nanoarchaeum equitans has an even smaller
genome size, 0.5 Mbp (Huber et al., 2002), but lives
symbiotically which complicates the comparison. The freeliving a-proteobacterium Pelagibacter ubique, with a genome
size of 1.3 Mbp is probably phototrophic (using proteorhopsin) and uses organic compounds as carbon and
electron source (Giovannoni et al., 2005; Rappé et al.,
2002). Its metabolic needs are uncertain, since it is difficult
to culture.
A possible scenario for the earliest metabolism is
presented in Fig. 3, which may be found in the archaeum
Pyrodictium occulatum (Wächtershäuser, 1988). Irrespective of
the biochemical ‘‘details’’, which are still controversial
(Österberg, 1997), it rightly places membrane activity
central to metabolism, which means that cell size matters.
Since the number of active enzyme molecules is proportional to the amount of membrane and thus to cell
surface area, and the change in concentrations of dissolved
substrate and product at the cytosol side depends on cellular
volume, the size of the cell is ‘‘known’’ at the local
molecular level; the ratio of cytosol volume to membrane
surface area gives a length measure that affects metabolic
rates. Prokaryotes span a huge cellular size range; the
largest is the colourless sulphur bacterium Thiomargarita
namibiensis with a cell volume of 2 10[10 m3 (Schulz &
Jørgensen, 2001), the smallest is Pelagibacter ubique at 10[20
m3. This small size has the remarkable implication that it
has less than a single free proton in its cell if its internal pH
is 7 as is typical for bacteria. This has peculiar consequences
FeS2
FeS
H2
2H+
out
S0
H2S
ADP
2e−
S0
Pi
ATP
in
H2S
2H2O
2H+
2OH−
Fig. 2. Steps in the evolution of the organisation of
metabolism of organisms, as described in the text; the numbers
in parentheses refer to sections. Symbols: S substrate, E reserve,
V structure, J maturity, R reproduction, PV and PJ somatic and
maturity maintenance products. We show only two types, of
the several possible ones. Font size reflects relative importance.
Stacked dots mean sloppy coupling.
Fig. 3. A possible early ATP-generating transformation,
based on pyrite formation FeS ] S / FeS2 (Taylor, Rummery
& Owen, 1979; Wächtershäuser, 1988), that requires a membrane and only three types of enzyme: proto-hydrogenase,
proto-ATP-ase and S0-reductase; modified from Madigan et al.
(2000). Sulphur is imported in exchange for H2S.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
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for the molecular dynamics of metabolism (Kooijman,
2001).
Another implication of this scenario is that the acquisition
of energy and (probably several types of) building blocks to
synthesise new structure were separated and the first cells
suffered from multiple limitations; they could only flourish if
all necessary compounds were present at the same time.
Initially there were no reserves and hardly any maintenance costs. A cell’s chemical composition varied with the
availability of the various substrates. As soon as the
membranes were rich in lipids (eubacteria) or isoprenoid
ethers (archaea), the accumulation of lipophilic compounds
could have been rather passive. The occurrence of lipids and
isoprenoid ethers among prokaryotes is only easy to
understand if the archaea and eubacteria were already
separated before these compounds had a role in metabolism.
The excretion of waste products was not well organised.
(2) Strong homeostasis
In a stepwise process, the cells gained control over their
chemical composition, which became less dependent on
chemical variations in the environment. One mechanism is
coupling of the uptake and use of different substrates. How
uncoupled uptake of supplementary compounds can
gradually change into coupled uptake of complementary
compounds is discussed in Kooijman et al. (2003). With
increasing homeostasis, stoichiometric restrictions on
growth become more stringent; the cells could only grow
if all essential compounds were present at the same time in
the direct environment of the cell. The activity of the cells
varied with the environment at a micro-scale, which will
typically fluctuate wildly. The reduction in variability of the
chemical composition of the cell came with an increased
ability to remove waste products, i.e. with a process of
production of compounds that are released into the
environment.
Although the mechanisms of acquiring homeostasis are
understood only partially (Kooijman et al., 2003), it gradually
became more perfect and biomass can be considered as
being composed of a single generalised compound called
structure. A generalised compound is a mixture of a set of
chemical compounds of fixed composition. This (idealised)
condition is called strong homeostasis.
(3) Reserves
The increased stoichiometric constraints on growth result in
a reduction of possible habitats in which the cell can exist. By
internalising and storing the essential compounds before use,
the cells became less dependent on the requirement for all
essential compounds to be present at the same time. In this
way, they could smooth out fluctuations in availability at the
micro-scale. Most substrates are first transported from the
environment into the cell across the membrane by carriers
before further processing. By reducing the rate of this further
processing, storage develops automatically. We will return to
this in more detail below. Initially the storage capacity must
have been small to avoid osmotic problems, which means
S. A. L. M. Kooijman and T. A. Troost
that the capacity to process internalised resources is large
relative to the capacity to acquire them from the environment. By transforming stored compounds to polymers, these
problems could be avoided, and storage capacity could be
increased further to smooth out fluctuations more effectively.
This can be achieved by increasing the acquisition rate or
decreasing the processing rate.
(a) Partitionability and weak homeostasis
Since the use of reserves is the motor behind metabolism,
we give it special attention here. According to the DEB
theory, the change in the amount of reserve E is the
difference between the assimilation flux and the mobilised
reserve flux, called the catabolic flux, i.e. dtd E ¼ p_ A [ p_ C as
expressed in energy fluxes. The DEB rules link assimilation,
i.e. the transformation of substrate into reserve, to the
surface area of the organism; it obviously also depends on
substrate availability. The DEB rules state that the catabolic
flux can only be a function of the amounts of reserve and
structure and it should not depend on the details of
transformations that further process the products. This
function is obtained from the requirements that it must be
weakly homeostatic, i.e. the reserve density does not change
during growth in constant environments. The function
must also be partitionable, meaning that the volumespecific use of reserve must be first-degree homogeneous in
the reserve density, and in the specific maintenance and
growth costs, and zero-degree homogeneous in the
structure. So, for an arbitrary factor k between 0 and 1
we must have
k p_ C ð½E; V p_ M ; EG Þ ¼ p_ C ðk½E; V k p_ M ; k EG Þ;
ð1Þ
where ½p_ C ¼ p_ C =V is the use of reserve per unit of
structural volume V, [E] ¼ E/V the reserve density, ½p_ M the volume-specific maintenance costs, and [EG] the costs
for synthesising a unit volume of structure from reserve.
The DEB rules treat the latter two quantities as constant
parameters; since they are details of the allocation, the
function can only depend on these quantities via the
(specific) growth rate. The derivation of the result that weak
homeostasis together with partitionability fully specify reserve
dynamics (Kooijman, 2000) is not the easiest part of DEB
theory. As shown in Kooijman et al. (2003), the partitionability requirement is essential to understand how two
individuals (or populations of individuals) can evolutionarily
merge into a single one that still follows the same rules. Since
this happened quite a few times (Hirose et al., 1996; Palmer,
2003; Schmid, 2003; Pimentel-Elardo et al., 2003; Dubilier
et al., 2001; Dohlen et al., 2001), the requirement is in fact
a consistency requirement that must apply to all models that
are not species-specific. The partitionability requirement is
also essential to reduce the number of different reserves in
a smooth way. Although the result is mathematically very
simple, and the empirical support substantial (see e.g. Fig. 4), it
proved to be quite a challenge to uncover a plausible and
simple mechanism. Only in retrospect do we realise that it
has been there but unrecognised for decades. The next
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
subsections describe how the use of reserve could have been
evolved mechanistically.
(b) First-order dynamics
The simplest catabolic flux that partially obeys the weak
homeostasis and partitionability requirements is firstorder kinetics, ½p_ C ¼ k_ E ½Efor reserve turnover rate k_ E ,
which is implied if all reserve molecules have a constant
probability rate of being used by metabolism for maintenance and growth. This results
specific growth
in the
ðk_ E ½E [ p_ M Þ= EG and reserve density
rate r_ ¼ dtd ln V ¼
kinetics dtd ½E ¼ p_ A [ ðk_ E ] r_ Þ½E. As long as surface area
is proportional to volume (these morphs are called V1morphs, see Section II.7), this kinetics is weakly homeostatic because reserve density [E] settles for constant
½p_ A ¼ p_ A =Vat a value that does not depend on the size of
the organism. It is not weakly homeostatic for other
morphs, such as isomorphs, i.e. organisms that do not
change in shape during growth; surface area is then
proportional to volume2/3. This specific catabolic flux is
first-degree homogeneous in the reserve density and zerodegree homogeneous in structure, but also in the specific
maintenance and growth costs (because that latter three
quantities do not occur in the specific catabolic flux). Firstorder dynamics is, therefore, not partitionable. We now
discuss a scenario that leads to the DEB dynamics given in
equation (2).
(c) Rejection of mobilised reserve
Since reserve primarily consists of polymers (RNA, proteins,
carbohydrates, lipids), an interface exists between reserve
and structure. Section II.3d explains structural homeostasis,
which is the phenomenon that sub-organismal (or subcellular for unicellulates) structures grow in harmony with
the whole structure in constant environments. For isomorphs this means that the surface area of the reservestructure interface is proportional to the ratio of the amount
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of reserve and a length measure for the structure; for V1morphs this means that it is proportional to the amount of
reserve. The mobilisation rate of reserve is now taken to be
proportional to the surface area of the reserve-structure
interface and allocated to the SUs for maintenance and
growth, called the catabolic SUs. The mobilised reserve flux
that cannot be bound to these units is returned to the
reserve, while the rest is further processed for maintenance
and growth. Maintenance is demand-driven and the flux is
proportional to the amount of structure, while growth is
supply-driven. The amount of SUs is such that weak
homeostasis results, which turns out to be proportional to
the amount of structure. Originally the density of SUs
would not have been the value that results in weak
homeostasis, so the setting of their abundance is an
evolutionary achievement. Appendix A.1 gives a more
detailed explanation and introduces the dimensionless
rejection strength, which not only involves the density of
SUs, but also the costs for growth, the mobilisation rate of
reserve and the dissociation rate of the SU-reserve complex.
Fig. 5 illustrates that the standard deviation of the specific
use of reserve for growth and somatic maintenance during
a stochastic feeding process is very sensitive for the value of
the rejection strength.
Polymers as such do not take part in metabolism as
substrates, their use as substrate involves monomerisation.
The decomposition of many types of source polymers and
other compounds into a limited number of types of central
metabolites before polymerisation into biomass (growth) is
known as the ‘funnel’ concept (Kluyver & Donker, 1926).
The next step in the evolutionary development of reserve
dynamics is to avoid the rejection of mobilised reserve by
the creation of local pools of monomers from which the
SUs take their substrate, and linking the pool size of the
monomers of reserves to that of the polymers (this is implied by the strong homeostasis assumption). The reserve
0.15
0.12
8
0.09
6
0.06
4
0.03
2
0
0
0
10
20
30
Fig. 4. Poly-b-hydroxybutyrate (PHB) density (C-mol/C-mol)
in starving activated sludge from an aerobic sewage treatment
plant at 20°C. The fitted curve is exponential with reserve
turnover rate k_ E ¼ 0:15 h. Data from Beun (2001).
0
0.5
1
1.5
2
Fig. 5. The standard deviation of the specific use of reserve
density as a function of the rejection strength, if the assimilation
rate k_ A jumps randomly between 0 and 1 h[1. The hazard rate
at level 0 is 2, 10 and 50 h[1 and at level 1 is 10 h[1. The
standard deviations are estimated from Monte Carlo simulations over 200 h, using reserve turnover rate k_ E ¼ 1:5 h[1
and maintenance rate coefficient k_ M ¼ 0:01 h[1.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
120
dynamics probably matured early in the evolution of
prokaryotes, but the avoidance of rejection of mobilised
reserve is especially important for large body sizes (in
multicellular eukaryotes) and for reproduction via eggs or
seeds (so much later in evolution). Appendix A.1 gives
details of this derivation. The result for V1-morphs is that
½p_ C ¼ ðk_ E [ r_ Þ½E, with the consequence that the specific
k_ E ½E [ ½p_ growth rate amounts to r_ ¼ ½E ] ½EGM and the reserve
density kinetics to dtd ½E ¼ p_ A [ k_ E ½E.
The single-reserve kinetics for V1-morphs results in
Droop’s empirical relationship (Droop, 1973) for cell quota
(i.e. nutrients in reserve plus structure of algae) as a function
of the specific growth rate in steady state situations.
Although nutrient storage does not always involve polymers, carbohydrate storage does and the kinetics of different
reserves is coupled (see Section II.3e). Direct empirical
support exists for this reserve kinetics for organic compounds (see Fig. 4). The fit is remarkable, not only because
it concerns a non-steady state situation, but also an
exponential decay of a density. This is much less easy to
understand than such a decay for an amount. With firstorder kinetics for reserve, reserve density would initially
decrease faster, and then slower than an exponential decline
for reasonable costs of growth and maintenance.
The net effect of this reserve dynamics is that reserve
density increases with growth rate, which produces
a particular relationship between 1/yield coefficient of
biomass on substrate and 1/specific growth rate (see Fig. 6).
This can be seen after some rescaling. For a large substrate
concentration X, relative to the saturation constant K, the
scaled functional response f ¼ X/(X ] K) assumes the value
1, and the specific assimilation energy flux ½p_ A ¼ f ½p_ Am reaches its maximum value, so does the reserve density
½E/½Em ¼ ½p_ Am =k_ E . The scaled reserve density e ¼ [E]/
[Em] is equal to f at steady state. Substitution of the
maintenance rate coefficient k_ M ¼ ½p_ M =½EG and the
energy investment ratio g ¼ [EG]/[Em] gives the specific
0.04
0.03
0.02
0.01
0
3
6
9
12
Fig. 6. The yield of biomass on substrate as function of the
specific growth rate in the bacterium Streptococcus bovis. Data
from Russel & Baldwin (1979) and Russel & Cook (1995). The
deviations from a constant yield are caused by maintenance
and reserve.
kM g
at steady state. The yield
growth rate r_ ¼ kE ff [
]g
coefficient Y}_r=f for biomass on substrate relates to the
_
k_ E
scaled functional response as Y ¼ Yg fg f [f k]M g=
g , where Yg is
a reference value that applies without maintenance and
reserve, called the ‘true yield’ in the microbiological literature.
This means that the yield relates to the specific growth rate
[ r_ =k_ E
, which fits the data well. This graph of the
as Y ¼ Yg 11 ]
k_ M =_r
yield in Fig. 6 is U-shaped; the right arm is due to
maintenance and is well understood. The explanation for
the left arm is more problematic. The traditional explanation
is a supposed down-regulation of specific maintenance costs at
high growth rates (Russel & Cook, 1995). The explanation
offered by the DEB theory is that at high growth rates reserves
are more abundant, and they do not require maintenance
costs. The consequence is that more substrate is fixed in
biomass. This has a deep relationship with why respiration
scales approximately as body weight3/4, see Section II.7; these
very different phenomena share a common cause.
_
_
(d) Structural homeostasis
Most prokaryotes are close to V1-morphs (see Fig. 7). For
a constant size of the granules of the reserve, the interface
with the cytoplasm is proportional to the amount of reserve
and the reserve mobilisation rate is linked to this interface.
Isomorphs cannot support weak homeostasis in this way.
Isomorphs cannot use a constant reserve turnover rate k_ E ,
but must decrease it for increasing length as v_ V [ 1=3 ; the
proportionality constant v_ is called the energy conductance.
A plausible mechanism is the development of structural
homeostasis, where the size of granules of reserve is linked
to cell size. There is some direct empirical support for this
strategy (see Anderson & Dawes, 1990), while there is strong
indirect empirical support in the form of the resulting
reserve dynamics (Kooijman, 2000) (Fig. 8). It amounts to
an interface of a surface area proportional to EV[1/3. If the
monomerisation activity is directly proportional to the
surface area of this interface, the specific reserve mobilisation rate
(also called the specific catabolic energy flux)
becomes p_ C ¼ ð_vV [ 1=3 [ r_ Þ½E and the specific growth
v_ V [ 1=3 ½E [ ½p_ M rate r_ ¼
. The rate at which the monomers
½E ] ½EG are processed is inversely proportional to a length if the
polymerisation SUs are bound to a membrane when active.
The structural homeostasis argument means that the (mean)
travelling time for monomers in the cytosol to a membrane
increases with length, so the transformation rate is inversely
proportional to length; this is consistent with the reserve
mobilisation rate. The growth process might also include
transportation of the resulting polymer to other parts of the
body; the required transportation time also increases with
a length measure. We need this argument to ensure that the
monomer density remains proportional to the reserve
density, and is independent of size (i.e. weakly homeostatic).
The reserve density becomes
d
½E ¼ ðf p_ A g [ ½E_vÞV [ 1=3 ;
dt
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
ð2Þ
Quantitative steps in the evolution of metabolic organisation
121
5.5
8
4.5
6
3.5
4
2.5
2
0
1.5
11
13
15
17
19
21
23
0
10
20
30
40
50
0
20
40
60
80
100
1.6
0.8
1.4
0.7
1.2
0.6
1
0.5
0.8
0.6
0.4
0
20
0
1
40
60
80
100
4
3
2
1
0
2
3
4
Fig. 7. Dynamic-energy-budget-based growth curves for cells in constant environments starting from V1-morphs (A), through
mixtures of V1- and V0-morphs (B,C), to almost pure V0-morphs (D), in comparison with isomorphs (E). Species A–D grow
in length only, the diameter remaining constant. The larger the aspect ratio, d, the more the growth curve changes from
an exponential to a (special) satiation type, reflecting the different surface area/volume relationships. Modified from Kooijman
(2000).
for p_ A ¼ p_ A V [ 2=3 the surface-area-specific assimilation
energy flux. Since the steady state value of the reserve
density [E] does not depend on size, it is weakly homeostatic
for isomorphs.
Shape correction functions can be used to evaluate the
kinetics for other morphs (Kooijman, 2000), cf. Fig. 7, but
this does not affect the principles discussed here. This
reserve dynamics is the only dynamics that satisfies the
requirements imposed by the DEB theory; all other types of
dynamics are either not weakly homeostatic and/or not
partitionable, which will cause deep theoretical problems in
applications for quantitative metabolism that are not
species-specific.
The reason for evolutionary selection towards partitionability might well be in the incremental change in the
number of different types of reserves, so in the organisational
aspects of metabolism. These changes not only occur within
individuals, but also during the internalisation of symbionts.
The resulting dynamics differs substantially from first-order
dynamics, e.g. eggs and seeds, which initially consist of almost
pure reserve (so the reserve density is very high initially) and
start their development slowly rather than explosively.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
122
80
20
60
16
12
40
8
20
4
0
0
0
20
40
60
80
0
20
40
60
80
Fig. 8. The embryonic development of the New Guinea soft-shelled turtle Carettochelys insculpta clearly illustrates a nonpathological transition in an isomorph that initially consists of almost pure reserve to mainly structure. The respiration data show
that maintenance is obviously linked to structure, not to reserve. The yolk data show that reserve dynamics differs from first order
kinetics. Data from Webb, Choquenot & Whitehead (1986), fitted to the dynamic energy budget (DEB) model by Zonneveld &
Kooijman (1993).
(e) Excretion
The DEB reserve dynamics implies that the amount of the
most limiting reserve co-varies with growth, and the
amounts of non-limiting reserves can or cannot accumulate
under conditions of retarded growth, depending on the
excretion of mobilised reserve that is not used; excretion is
an essential feature of multiple reserve systems to avoid
accumulation without boundary. This is because assimilation does not depend on the amount of reserve, so also not
on the use of reserve; it only depends on the amount of
structure and substrate availability. The excretion process
can be seen as an enhanced production process of chemical
compounds, but its organisation (in terms of the amounts
that are excreted under the various conditions) differs from
waste production. Waste production is proportional to the
source process (assimilation, maintenance, growth). Excretion, on the contrary, reflects an unbalanced availability of
resources. The flux is proportional to a fixed fraction of
what is rejected by the SU for growth. The theory of SUs
quantifies the rejection flux (Appendix A quantifies
rejection for different special cases).
Empirical evidence has so far revealed that the various
reserves have the same turnover rate (Kooijman, 2000). The
reason might be that mobilisation of different reserves
involves the same biochemical machinery. This possibly
explains why the use of e.g. stored nitrate follows the same
dynamics as that of polymers such as carbohydrates and
lipids, although the use of nitrate obviously does not involve
monomerisation.
Together with waste, excretion products serve an
important ecological role as substrate for other organisms.
Most notably polysaccharides that are excreted by phototrophs in response to nutrient limitation provide energy
and/or carbon substrate for heterotrophs, so they fuel
a production process that is known as the microbial loop
(see Section III.) Adaptive dynamics analysis has indicated
the importance of syntrophy in evolutionary speciation
(Doebeli, 2002).
Other excretion products are toxic for potential competitors, such as domoic acid produced by the diatom
Pseudonitzschia spp. in response to nitrogen surplus, which
can be highly toxic to a broad spectrum of organisms,
including fish. Nitrogen enrichment of the environment by
human activity enhances the formation of nitrogen reserves,
and so the production of toxicants that contain nitrogen by
some algae.
(4) Adaptation
Carriers in the outer membrane typically only transport
particular substrates from the environment into the cell.
This comes with the requirement to regulate gene
expression for carriers of substitutable substrates to match
the substrate availability in the environment. Data strongly
suggest that allocation to the assimilation machinery is
a fixed fraction of the utilised reserve flux, and that the
expression of one gene for a carrier inhibits in some cases
the expression of another gene (see Appendix A.3a for
quantitative details). Inhibition strength is linked to the
workload of the carriers. This regulation mechanism has
similarities to that of differentiation (see Section II.13).
Fig. 9 illustrates this for two data sets on the uptake by
E. coli K21 of fumarate and pyruvate (Fig. 9A) and of
fumarate and glucose (Fig. 9B). Unlike pyruvate, glucose
suppresses the uptake of fumarate. The background expression of carrier synthesis and the maintenance requirements
were set to zero, because the data provide little information
on this. The yield of structure on reserve was fixed (because
the data give no information on biomass composition). The
data were fitted simultaneously to ensure that the uptake
parameters for fumarate and the reserve turnover rates
are identical in the two data sets (so removing degrees of
freedom). Apart from the initial conditions, 12 parameters
were estimated for six trajectories. The fit is quite good,
despite the constraint for the parameter values for fumarate
to be identical. The data in Fig. 9A clearly show continued
growth after depletion of substrates, which requires
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
123
1.2
2.5
1
2
0.8
1.5
0.6
1
0.4
0.5
0
0.2
0
2
4
6
8
0
0
2
4
6
Fig. 9. The uptake of fumarate (F) and pyruvate (P) (see Fig. A), and of fumarate (F) and glucose (G) (see Fig. B) by E. coli K12 in
a batch culture. Data from Narang, Konopka & Ramkrishna (1997). Parameters: saturation coefficients (g l[1) KF ¼ 0.089, KP ¼
0.012, KG ¼ 0.013; yield coefficients (g g[1 dry weight) yEF ¼ 0.577, yEP ¼ 0.015, yEG ¼ 0.446, yEV ¼ 1.2 (fixed); max. specific
uptake rates (g(h g dry weight)[1), jFm ¼ 1.138, jPm ¼ 40.15, jGm ¼ 2.59; reserve turnover rate (h[1) k_ E ¼ 4:256; maintenance rate
coefficient (h[1) k_ M ¼ 0 (fixed); preference parameter (-) wP ¼ 0.941wF for pyruvate versus fumarate; wG ¼ 12.15wF for glucose versus
fumarate; background expression (h[1) h_ ¼ 0 (fixed). Initial conditions: (A) F(0) ¼ 2.0 g l[1, P(0) ¼ 2.1 g l[1, E. coli(0) ¼ 0.037 g
l[1, kF(0) ¼ 0.96, mE(0) ¼ 0.288 g g[1 dry weight; (B) F(0) ¼ 0.81 g l[1, G(0) ¼ 1.11 g l[1, E. coli(0) ¼ 0.013 g l[1, kF(0) ¼ 0.99,
mE(0) ¼ 1.3 g g[1 dry weight.
reserves to capture; this cannot be done with e.g. a Monod
model.
(5) Maintenance
The storing of ions, such as nitrate, creates concentration
gradients of compounds across the membrane that have to
be maintained. These maintenance costs might originally
have been covered by extra-cellular chemoautotrophic transformations, but this requires the presence of particular
compounds (e.g. to deliver energy). Maintenance can only be
met in this way if the organism can survive periods without
having to meet such costs, i.e. facultative rather than obligatory
maintenance. Most maintenance costs are obligatory,
however. The next step is to pay the maintenance costs
from reserves that are used for energy generation to fuel
anabolic work and thus to become less dependent on the
local presence of chemo-autotrophic substrates. Although
extreme starvation, causing exhaustion of reserves, can still
affect the ability to meet maintenance costs (see Appendix
A.3), such problems will occur much less frequently.
Maintenance requirements were increased further and
became less facultative in a number of steps, which we will
discuss briefly.
(a) Carriers and regulation
Originally carriers (which transport substrate from the
environment into the cell across the membrane) were less
substrate-specific and less efficient, meaning that the cell
required relatively high concentrations of substrate. The cell
increased the range of habitats in which it could exist by
using carriers that are not fully structurally stable, meaning
that a high-efficiency machinery changes to low efficiency
autonomously. The maintenance of a high efficiency
involves a turnover of carriers.
High-performance carriers are also more substratespecific, which introduces a requirement for regulation of
their synthesis and for adaptation to substrate availability in
the local environment. The expression of genes coding for
the carriers of various substitutable substrates becomes
linked to the workload of the carriers (see Appendix A.3).
The principle that allocation occurs according to relative
workload seems to be general and conserved; we will discuss
it again in allocation to organs in relation to multicellular
eukaryotes (see Section II.13).
(b) Turnover of structure
Not only carriers, but many chemical compounds (especially proteins with enzymatic functions) suffer from
spontaneous changes that hamper cellular functions. The
turnover of these compounds, i.e. breakdown and resynthesis from simple metabolites, restores their functionality (Levine & Klionsky, 2004), but increases maintenance
requirements. This mixture of conversion machineries with
high and low efficiencies is present in structure and so, due
to turnover, to maintenance, it is converted into structure
with high-efficiency machinery. The biochemical aspects of
the process are reviewed in Klionsky (2004).
These increased requirements made it even more
important to use reserves, rather than unpredictable external
resources to cover them. When such reserves do not suffice,
maintenance costs are met from structure, and cells shrink.
Paying maintenance from structure is less efficient than from
reserve directly, because it involves an extra transformation
(namely from reserve to structure). The preference for reserve
as the substrate rather than structure, would have been weak
originally, later becoming stronger (see Appendix A.3b).
Since the turnover rate of compounds in structure depends
on the type of compound (some rates are possibly very low),
the metabolites derived from these compounds do not
necessarily cover all metabolic needs.
The waste (linked to maintenance and the overhead of
growth) and the excreted reserves (linked to stoichiometric
restrictions on growth due to homeostasis) serve as
substrates for other organisms, so life becomes increasingly
dependent on other forms of life even at an early stage.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
124
Some of these products were transformed into toxins that
suppress competition for nutrients by other species.
(c) Defence systems
The invasion of (micro)habitats where toxic compounds are
present, and the production of toxic waste and excretion
products by other organisms, required the installation of
defence systems, which increase maintenance costs. Phototrophy requires protection against UV radiation and these
two systems must have been evolved simultaneously (Dillon
& Castenholz, 1999). Phototrophy possibly evolved from
UV protection systems Pierson et al. (1993), although it is
unlikely that it appeared at the start of evolution, as some
authors suggest (Woese, 1979; Cavalier-Smith, 1987;
Hartman, 1998; Blankenship & Hartman, 1992, 1998). The
pathways for anaerobic methane oxidation and methanogenesis possibly evolved from a detoxicifation system for
formaldehyde; this is another illustration of a change in
function of a protection system. A general-purpose protection system against toxic compounds consists of proteins
that encapsulate toxic molecules. Another general system is
to transform lipophilic compounds into more hydrophilic
(and so more toxic) ones to enhance excretion. The
development of a complex double cell membrane in the
didermata (Gram-negative eubacteria) was possibly
a response to the excretion of toxic products by other
bacteria (Gupta, 1998), although the outer membrane is not
a typical diffusion barrier (Lengeler, Drews & Schlegel,
1999). When dioxygen first occurred in the environment as
a waste product of oxygenic photosynthesis, it must have
been toxic to most organisms (Dismukes et al., 2001; Lane,
2002); the present core position of carbohydrates in the
central metabolism of eukaryotes and its use in energy
storage is directly linked to this waste product. The reactive
oxygen species (ROS) play an important role in ageing
(Leeuwen, Kelpin & Kooijman, 2002), and induced the
development of defence systems using peroxidase dismutases to fight their effects. Viruses probably arose early in
the evolution of life, and necessitated specialised defence
systems that dealt with them. These defence systems further
increased maintenance costs.
(6) Increase of reserve capacity
Substrate concentration in the environment is not constant,
which poses a problem if there is a continuous need to cover
maintenance costs. An increase in maintenance costs
therefore requires increased storage capacity in order to
avoid situations in which maintenance costs cannot be met.
The solution is to further delay the conversion of substrate
metabolites to structure, creating a pool of intermediary
metabolites. The optimal capacity depends on the variability of substrate availability in the environment and (somatic)
maintenance needs. Transformation to polymers (proteins,
carbohydrates) and lipids will reduce concentration gradients
and osmotic problems, and thus maintenance costs, but
involves machinery to perform polymerisation and monomerisation. The development of vacuoles allows spatial separa-
S. A. L. M. Kooijman and T. A. Troost
tion of ions and cytoplasm to counter osmotic problems.
One example is the storage of nitrate in vacuoles of the
colourless sulphur bacteria Thioploca spp. (Jørgensen &
Gallardo, 1999), which use it to oxidise sulphides first to
sulphur, for intracellular storage, and then to sulphate for
excretion together with ammonium (Otte et al., 1999).
Cyanobacteria only develop vacuoles at low pH (Zhao et al.,
2001). Organelles like acidocalcisomes also play a role in
the storage of cations (Docampo et al., 2005).
A further step to guarantee that obligatory maintenance
costs can be met is to catabolise structure (see Appendix
A.3b). This is inefficient and involves further waste production (so requiring advanced excretion mechanisms), but at
least it allows the organism to survive lean periods.
Reserves can contribute considerably to the variability of
biomass composition; phytoplankton composition greatly
affects the rate at which phytoplankton bind atmospheric
carbon dioxide and transport carbon to deep waters (Omta
et al., 2006), known as the biological carbon pump. The
activity of the biological carbon pump strongly influences
climate.
(7) Morphological control on metabolism
Morphology will influence metabolism for several reasons:
assimilation rate is proportional to surface area, maintenance
rate to volume and catabolic rate to the ratio of surface area
and volume. This means that surface-area-volume relationships are central to metabolic rates, as beautifully illustrated
by the set of five growth curves (Fig. 7) for two species of
ascomycetes and three species of eubacteria, The samples
represent (static) mixtures of V1- and V0-morphs, i.e.
organisms for which surface area is proportional to volume
to the power of 1 and 0, respectively. The shape of the growth
curve is directly related to the changes in morphology of the
cell, i.e. to what extent it is a V1- or a V0-morph. See
Kooijman (2000) for the derivation and application of DEB
theory in these cases. Organisms like crustose saxicolous
lichens make the transition from a V1- to a V0-morph
dynamically during growth, because the outer annulus acts as
a V1-morph and the inner part as a V0-morph and the latter
increases in importance during growth. This causes their
diameter to increase linearly with time in constant environments (Kooijman, 2000), as was confirmed empirically by
Clark et al. (2000).
At the asymptotic size resource acquisition by assimilation just matches maintenance requirements. Unicellulates
don’t have an asymptotic size, because they reset their size
at division, but multicellulates generally do reach an
asymptote, since most are approximately isomorphic. Since
their cells are similar, the main difference in size between
a whale and a mouse is in the assimilation capacity, not in
the mass-specific maintenance needs (apart from thermal
considerations). Asymptotic body size represents the ratio of
assimilation and maintenance and is a consequence rather
than a cause of how physiological rates, and in particular
the respiration rate, depends on body size.
Respiration (i.e. the use of dioxygen, or the production of
carbon dioxide or heat, three different definitions that all
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
work out slightly different) has contributions from assimilation, maintenance and growth. Recent attempts to explain
why respiration scales approximately with body weight3/4
fail to consider how the comparison of body sizes is actually
made (see Meer (2006) for a critical review). Young (small)
and old (large) individuals of the same species in the same
environment have equal reserve densities, but differ in what
they do, metabolically; small individuals grow at a higher
rate, and therefore respire more per unit of weight. If the
small and large individuals are adults that have ceased
growing (typically of different species), they differ in
parameter values and reserve density will increase with
asymptotic body size for reasons discussed in Kooijman
(2000). Small individuals then respire more per unit of
weight because a larger fraction of their body is structure;
reserve does not require maintenance, and so does not
involve respiration, as clearly demonstrated in Fig. 8. This
difference in body composition can also be seen in Fig. 6,
where yields are plotted for cells growing in different
environments; the ones growing at low growth rates
experience low substrate levels, and have less reserve, hence
a different yield coefficient.
(8) k-rule and the emergence of cell cycles
Control on morphology and cell size will increase stepwise.
Initially the size at division would be highly variable among
cells. This variance will be decreased by the installation of
a maturation process, where division is initiated as soon as
the investment in maturation exceeds a threshold level. This
state of maturity creates maturity maintenance costs.
Allocation to this maturation program is a fixed fraction 1 –
k of the catabolic flux, gradually increasing from zero (see
Appendix A.1 for a mechanism). Such an allocation is only
simple to achieve if the catabolic flux does not depend on
the details of allocation, see Section II.3. If the SUs for
maturation operate similar to those for somatic maintenance and growth, the fraction k is constant and depends
on the relative abundance and affinity of the maturation
SUs.
The metabolic relevance of cell size is in membranecytoplasm interactions; many catalysing enzymes are only
active when bound to membranes (Baltscheffsky, Schultz &
Baltscheffsky, 1999), and cellular compartmentalisation
affects morphology and metabolism. The turnover of
reserve decreases with a length measure for an isomorphic
cell, which comes with the need to reset cell size. Apart from
the increase of residence time of compounds in the reserve
with a length measure, the cell’s surface area to volume
ratio decreases with increasing cell size, as does the growth
potential. The increase in metabolic performance requires
an increase in the amount of DNA and in the time spent on
DNA duplication. The trigger for DNA duplication is given
when investment into maturation exceeds a given threshold,
meaning that a large amount of DNA leads to large cell
sizes at division. Prokaryotes partly solved this problem by
telescoping generations (DNA duplication is initiated before
the previous duplication cycle is completed) and by deleting
unused DNA (Stouthamer & Kooijman, 1993).
125
The existence of a maturity investment threshold can be
deduced phenomenologically. If the specific maturity
maintenance costs ½p_ J relates to the somatic maintenance
k
costs ½p_ M as ½p_ J ¼ ½p_ M 1 [
k , the threshold is exceeded if
the amount of structure exceeds a threshold; k represents
the fraction of the utilised reserve that is allocated to
somatic maintenance plus growth. For all other values of
½p_ J , the amount of structure at the transition depends on
the nutritional history. This argument can be used in reverse
to estimate the specific maturity maintenance costs from
size-at-transition data. If the cells are separated at the twocell stage of the embryo sea urchin Strongylocentrotus
droebachiensis, the embryonic period is hardly affected, but
the size at the transition to the larval stage is halved (Hart,
1995). Thus it is possible to manipulate the threshold value
experimentally, meaning that its biochemical identification
is within reach.
(9) Syntrophy and compartmentalisation
While prokaryotes passed metabolic properties from one
taxon to another by lateral exchange of genes, eukaryotes
specialised in symbiotic relationships and even internalisation of whole organisms to acquire new metabolic
properties.
(a) Mitochondria
It is now widely accepted that all eukaryotes have or once
had mitochondria. The first endosymbionts that evolved
into mitochondria must have been an extremely rare
coincidence (Fig. 10). Epibiotic symbioses will have been
quite abundant, but the first penetration [probably of an agroup purple bacterium (Andersson et al., 1998) in an
archaeum (Martin & Muller, 1998; Martin & Russel, 2003;
Baldauf et al., 2004)] required membrane rupture and
healing, without causing cell death. This possibly occurred
once only, some 2.7 Ga ago, cf. Fig. 10, which explains the
metabolic similarity among all eukaryotes, compared to
the diversity among prokaryotes. Since opisthokonts were
the first to branch, and animals probably first appeared
in the sea, this internalisation event presumably occurred
in a marine environment. The fungi, notably the chytrids,
diverged from the animals (unicellular relatives of the
choanoflagellates) some 0.9 - 1.6 Ga ago Taylor et al. (1979).
In view of the biology of modern nucleariids and chytrids,
this might have occurred in a freshwater environment.
(b) Membrane plasticity and plastids
The subsequent development of membrane plasticity has
been a major evolutionary step, that allowed phagocytosis;
cells no longer needed to excrete enzymes to split large
molecules of substrate into smaller metabolites for uptake
with low efficiency, but digestion could be carried out
intracellularly, avoiding waste and the necessity for
cooperative feeding. (See Fig. 11 and Appendix B for
quantitative details.) Fungi possibly never developed this
ability and animals evolved from fungi (Martin et al., 2003)
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
126
Eubacterial
heterotroph
(symbiont)
Eukaryotic
heterotroph
(host)
Eubacterial
phototroph
iRC (symbiont)
PP
Gly
iPP
iPP
PP
Gly
Gly
TCA
TCA
TCA
RC
RC
TCA
RC
iRC
PP
Gly
PP
RC
iGly
iRC
Gly
iPP
Eukaryotic
phototroph
iTCA
TCA
Gly
PP
RC
TCA
Archaeal
methanogen
(host)
Eukaryotic
heterotroph
(host)
RC
Eukaryotic
phototroph
(symbiont)
Fig. 10. Scheme of symbiogenesis events; the first two primary inclusions of prokaryotes (to become mitochondria and
chloroplasts respectively), were followed by secondary and tertiary inclusions of eukaryotes. Each inclusion comes with a transfer of
metabolic functions to the host. The loss of endosymbionts is not illustrated. See Fig. 1 for definitions of the modules of central
metabolism and for the ancestors of mitochondria and chloroplasts. The outer membrane of the mitochondria is derived from the
endosymbiont, and that of the chloroplasts from the host; mitochondria were internalised via membrane rupture, chloroplasts via
phagocytosis. Modified from Kooijman & Hengeveld (2005).
suggesting that the animal lineage developed phagocytosis
independently. Recent phylogenetic studies (Steenkamp,
Wright & Baldauf, 2006) place the phagocytotic nucleariids
at the base of the fungi, however, suggesting that the fungi
lost phagocytosis, and that it only developed once. Most
animals also excrete enzymes (like their fungal sisters), but
since this is in the gut environment, most metabolites arrive
at the gut epithelium for uptake. Plantae (glaucophytes,
rhodophytes and chlorophytes) gave up phagocytosis, but
chromophytes, which received their plastids in the form of
rhodophytes, still sport active phagocytosis (Andersen, 2004)
despite their acquired phototrophic abilities. Phagocytosis
allowed the more efficient use of living and dead organisms
as a resource. Scavenging, predation and new forms of
endosymbioses became widespread.
The protein clathrin plays a key role in membrane
invagination, and is not known in prokaryotes. We are
beginning to understand the evolutionary roots of cell
motility, including changes in shape in response to
environmental stimuli, and extension of protrusions like
lamellipodia and filopodia to allow particles to be enclosed in
a phagocytotic cup, which is based on the spatially controlled polymerisation of actin. The eubacterial pathogens
Listeria monocytogenes and Shigella flexneri exhibit actin-based
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
Fig. 11. The yield of substrate X that is taken up on excreted enzyme 3 relative to the yield for intracellular digestion. The yield for
social extracellular digestion builds up slowly, while that for solitary digestion is much slower still and reaches a lower asymptote. The
dark arrows indicate enzyme flux, the light arrows metabolite flux. Model details and parameter values are given in Appendix B.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
movement in the host cytoplasm (Pantaloni, Clainche &
Carlier, 2001). Actin and tubulin have also been isolated
from the togobacterium Thermatoga maritimum (Ent, Amos &
Löwe, 2001); apart from their role in motility, these proteins
also play a key role in the cytoskeleton of eukaryotes, which
is used by transporters for the allocation of metabolites to
particular destinations.
The evolution of membrane plasticity must have taken
place in a time window of some 700 Ma, since biomarker
data suggest that the first eukaryotic cells appeared around
2.7 Ga (Brocks et al., 1999) ago (around the time
cyanobacteria evolved), while rhodophytes appeared 2.0
Ga (Saunders & Hommersand, 2004; Tappan, 1976) ago.
Sequence data suggest that glaucophytes received the first
plastids, and that rhodophytes evolved from them, while
chlorophytes diverged from rhodophytes 1.5 Ga ago; the
secondary endosymbiosis event that seeded the chromophytes was some 1.3 Ga ago (Yoon et al., 2004) (see Fig. 10).
The glaucophytes have a poor fossil record, and now consist
of a few freshwater species. The internalisation of
a cyanobacterium as a plastid was possibly also an event
that occurred once only (Yoon et al., 2002; Delwiche et al.,
2004; Rodriguez-Ezpeleta et al., 2005), but this is controversial (Stiller, Reel & Johnson, 2003). The present
occurrence of glaucophytes weakly suggests that this
occurred in a freshwater environment. The rhodophytes
have their greatest diversity in the sea, and most of their
hosts are most diverse in the sea, while chlorophytes and
their hosts are most diverse in fresh waters.
Before the arrival of plastids, eukaryotes were heterotrophic. Cyanobacteria are mixotrophic, which makes it likely
that their plastids before internalisation were mixotrophs as
well. Very few, if any, eukaryotes with plastids became fully
specialised on phototrophy, remaining mixotrophic to some
extent. Theoretical studies show that the spontaneous
evolutionary specialisation of mixotrophs into organo- and
phototrophs is difficult in spatially homogeneous environments (Troost, Kooi & Kooijman, 2005b). In spatially
heterogeneous environments, however, such as in the water
column where light extinction favours phototrophy at the
surface and heterotrophy at the bottom, such specialisation
is relatively easy (Troost, Kooi & Kooijman, 2005a).
Membrane plasticity had a huge impact on cellular
organisation. The presence of vacuoles increased the
capacity to store nutrients (Leigh & Sanders, 1997), and
vesicle-mediated intracellular transport reorganised metabolism (Duve, 1984). By further improving intracellular
transport using the endoplasmatic reticulum and further
increasing storage capacity, cells could grow bigger and be
more motile. Bigger size favours increased metabolic
memory, and increased motility allows the organism to
search for favourable sites.
(c) Genome organisation
The organisation of the genome in chromosomes enhanced
the efficiency of cell propagation by reducing the time
needed to duplicate DNA (Chela-Flores, 1998), and
harnessed plastids, whose duplication is only loosely
coupled to the cell cycle in prokaryotes. Since animals such
127
as the ant Myrmecia croslandi and the nematode Parascaris
univalens have only a single chromosome (Kondrashov,
1997), acceleration of DNA duplication is not always vital.
It allows more efficient methods of silencing viruses, by
changing their genome and incorporating it into that of the
host (half of eukaryotic ‘‘junk DNA’’ consists of these
silenced viral genomes). Eukaryotes had to solve the
problem of how to couple the duplication cycles of their
nuclear genome and that of their mitochondria and
chloroplasts. Dynamin-related guanosine triphosphatases
(GTPases) seem to play a role in this synchronisation
(Osteryoung & Nunnari, 2003). The nuclear membrane of
eukaryotes and planctomycetes possibly allows a better
separation of the regulation tasks of gene activity and
cellular metabolism by compartmentalisation, which might
have been essential to the development of advanced gene
regulation mechanisms.
(10) Reduction of number of reserves
Many eukaryotes started feeding on dead or living biomass
with a chemical composition similar to themselves. This covariation in time of all required metabolites for growth
removed the necessity to deal with each of those reserves
independently. By linking the uptake of various metabolites,
the various reserves co-vary fully in time because their
turnover times are equal, as was discussed above. This
improved homeostasis, and allowed further optimisation of
enzyme performance (see Kooijman et al. 2003).
From an organisation point of view, reserves play a key
role in product formation. If biomass would have a constant
composition (so no reserve), one of the three basic (energy)
fluxes of assimilation, maintenance and growth would
follow from two of them plus the mass balance. Reserve
provides the degree of freedom that is essential to uncouple
the three energy fluxes, meaning that all products in single
reserve - single structure systems can be written as
a weighted sum of these three energy fluxes. These products
include water, carbon dioxide, nitrogen waste, faeces, but
also products that remain useful to the individual like chitin
(in fungi), cellulose and wood (in plants) and carbonates (in
corals). These products differ from biomass by not requiring
maintenance, which is why for example fungi, like trees,
have low maintenance costs when expressed on the basis of
total dry weight. Non-limiting resources, such as dioxygen
in aerobic environments, and heat also follow these kinetics.
This explains why indirect calorimetry is successful, where
dissipating heat is taken to be a weighted sum of the
dioxygen, carbon dioxide and nitrogen waste fluxes. The krule means that new allocation destinies (maturity maintenance and maturation or reproduction) do not affect the
simple rule that all mass and heat fluxes are weighted sums
of the three basic fluxes if we extend the maintenance flux
to include the collection of transformations that do not
relate to synthesis of biomass.
The three basic energy fluxes turn out to be cubic
polynomials of length for isomorphs, where the polynomial
coefficients depend on reserve density. Weak homeostasis
ensures that the coefficients become constants in constant
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
128
environments. Intra-species comparisons of fluxes necessarily concern a limited size range only, in which case cubic
polynomials very closely resemble the popular, but poorly
understood, allometric functions.
(11) Emergence of life stages: adult and embryo
Several groups of bacteria evolved to a multicellular state in
the form of reproductive bodies (myxobacteria, actinomycetes), chains with cell differentiation (cyanobacteria),
mats, films or flocs (Brandt & Kooijman, 2000). When
eukaryotisation had occurred, multicellularity became
complex and arose independently in almost all major taxa.
This came with the invention of reproduction by eggs in the
form of packages of reserve with an very small amount of
structure (cf. Fig. 8): the juvenile state thus gave rise to both
the adult and the embryo state. Embryos differ from
juveniles by not taking up substrates from the environment.
That is to say they do not (yet) use the assimilation process
for energy and building-block acquisition, although most do
take up dioxygen. The spores of endobacteria can be seen
as an embryonic stage for prokaryotes. Adults differ from
juveniles by allocation to reproduction, rather than further
increasing the state of maturity. Unlike dividing juveniles,
adults do not reset their state (i.e. the amount of structure,
reserve and the state of maturity). Animals, notably
vertebrates, and embryophytes, notably the flowering
plants, provide the embryo fully with reserve material.
Egg size, relative to adult size, has proved highly adaptable
in evolutionary history.
If the cumulative investment into maturity exceeds
a threshold, further allocation to maturity is ceased and
mobilised reserve is redirected to reproduction. Logically,
and perhaps also biochemically, this threshold corresponds
with that of cell division by unicellulates.
The fact that allocation to reproduction is incremental,
and eggs are not incrementally small, implies the installation of a buffer with destiny reproduction, and a set of
buffer-handling rules. Some organisms produce an egg as
soon as this buffer allows, as in some rotifers, while others
accumulate over a year, as in corals or mussels, or over
several years, as in some trees.
Foetal development in some animals (notably mammals)
is a further variation on this theme. Vegetative propagation
was invented independently in many taxa; even animals as
advanced as the sea cucumber Holothuria parvula sport
propagation by division (Emson & Mladenov, 1987).
Quite a few references suggest the existence of determinate and indeterminate growth patterns, especially
in animals, where no growth occurs during reproduction in
determinate growers. These patterns could be captured, in
principle, by a change in the value of k (Lika & Kooijman,
2003). The combination of weak homeostasis and partitionability still allows that k is a function of the amount of
structure. However, the von Bertalanffy growth curve fits
most growth data for isomorphs at constant food availability
very well, which means that growth is not at the expense of
reproduction and that k is constant. It simply depends on
the value of the maturity threshold for puberty whether or
S. A. L. M. Kooijman and T. A. Troost
not growth still proceeds during reproduction, so there need
not be a fundamental difference in metabolic organisation
between these patterns.
If growth is of the von Bertalanffy type at a constant low
food density, and food availability increases after growth
ceases, will an organism resume growth? Many species fail
to do so, depending how long growth has already ceased.
This loss of metabolic flexibility is possibly linked to the
ageing process, and follows similar patterns as, for example,
the occurrence of post-reproductive periods in many
species. These patterns can be included in the DEB theory
by linking parameter values to ageing-induced damage,
similar to the strategy that has been shown to be effective
for the effects of toxicants (Kooijman & Bedaux, 1996).
The holometabolic insects are a clear (and possibly the
only) example of determinate growers; they insert an extra
embryo stage (called the pupal stage) in their life cycle and
do not grow as adults. Some species of Octopus and Oikopleura
and some flowering plant species sport suicide reproduction, where growth is suddenly interrupted and some of the
structure is rapidly converted to eggs or seeds, typically
followed by death. Like torpor and migration, these
strategies probably evolved to survive bleak periods.
(12) Further increase in maintenance costs
Multicellular organisation and an active life-style, especially
in eukaryotes, results in a series of extra maintenance costs.
Concentration gradients across the more abundant and
dynamic membranes become more important, as well as
intracellular transport and movements of the individual.
The invasion of the fresh-water habitat required a solution
to the osmotic condition. Many eukaryotes use pulsating
vacuoles for this purpose. Invasion of the terrestrial habitat
required an answer to the problem of desiccation. Many
animals and some plants elevate the temperature of parts of
their body metabolically to enhance particular physiological
functions. Birds and mammals have taken this to extremes.
Most maintenance costs are proportional to the amount of
structure, but some (osmotic and thermoregulatory work)
are proportional to organism’s surface area. All these
processes increased maintenance requirements further, but
also improved the metabolic performance. Such organisms
became less dependent on the local chemical and physical
conditions.
Some animals developed ovovivipary, i.e. they carry their
eggs inside the body during the embryonic stage. This offers
much better protection, and the mother is not confined to
a particular site during breeding or parental care. Some
animals (e.g. Peripatus, some sharks, placentalia) developed
a placenta to transfer reserve from the mother to the foetal
system. Foetal development is similar to that of embryos
inside eggs, but their developmental rate is no longer
restricted by the availability of reserve (Kooijman, 2000).
Many parental animals feed their offspring in the early
juvenile stages, which is important for nutrition and for the
inoculation of symbiotic digestive microorganisms.
Some animals and plants increase their body temperature temporarily (some flowers during gamete and fruit
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
development, insects during flight, some fishes in certain
regions of their body) or more permanently (mainly birds
and mammals). This was a next step in the evolution of
homeostasis, and resulted in a considerable further increase
in maintenance costs that had to be balanced by an
equivalent increase in their ability to acquire resources.
Since cooling is linked to surface area, the impact on the
energy budget depends on the body size; this has been
quantified in the DEB theory (Kooijman, 2000) in
a straightforward way.
(13) Differentiation
The transition to a multicellular state has been made in
almost all large taxa, including prokaryotes. It comes with
cell differentiation into tissues and organs; the organs then
take the role of organelles. The dynamics of organ sizes can
be quantified effectively by further partitioning the flux of
mobilised reserve (the k-rule). If the fraction that is allocated
to a particular body part is fixed, and the specific
maintenance costs equal those of other body parts,
isomorphic growth results. Although this frequently covers
the main patterns, deviations can be observed that can be
understood by linking the allocation fraction to the relative
workload. This even holds for tumours, where the workload
is quantified as their maintenance requirement, relative to
that of the host (Leeuwen, Zonneveld & Kooijman, 2003).
This allocation produces realistic predictions for how
tumour growth depends on the physiological state of the
host; tumour growth is more aggressive in young (small)
individuals, compared to old (large) ones, and in well-fed
individuals, compared to those that experience caloric
restriction. It also gives realistic predictions for how velum
versus gut size in bivalve larvae depends on food availability
(Kooijman, 2006) (see Fig. 12). The relative workload of the
velum, which functions in filtering, equals one minus the
relative workload of the gut, which functions in food
processing. The relative size of the velum and gut adapts
rather quickly to the feeding conditions, growth is isomorphic after the adaptation period. The feeding rate of
adapted individuals depends on food density according to
the Hill’s equation, rather than the Holling type II
relationship that would result if the relative organ size was
Fig. 12. Macoma baltica larvae develop a large velum and small
gut at low food levels (left), and the reverse at high food levels
(right).
129
constant. By partitioning food handling into a mechanical
phase that is sequential to food searching and a digestion
phase that is parallel to food searching, the observed
differences from the standard Holling type II functional response in fish larvae can be understood (Lika &
Papandroulakis, 2004).
Likewise, the allocation to adipose tissue can be linked to
feeding, allocation to the liver to particular dietary components (e.g. alcohol in humans), and allocation to muscles in
sportsmen, etc.
Plants differentiated their structure into a root for
nutrient uptake linked to water uptake and a shoot for
gas exchange, photon acquisition and evaporation of water;
the latter dominates water uptake by the root, which means
that the ratio of the surface area of the root and the shoot
appears in the saturation coefficient for nutrient uptake by
the roots. In addition plants developed translocation of
reserves between root and shoot, which are fixed fractions
of the mobilised flux (consistent with the k-rule for
allocation). These links between root and shoot imply
compensating development of both types of structure
(Kooijman, 2000); a reduction in light affects the root more
than the shoot, and a nutrient reduction affects the shoot
more than the root. Plants typically alter their morphology
in predictable ways; they start as V1-morphs immediately
after germination, then undergo an isomorphic phase,
finally ending as a V0-morph when the neighbouring plants
in their habitat prohibit further extension of functional
surface area of the roots and shoots. Leaves typically last
one year, and fall after recovering (some of) the reserve. This
means that plants live syntrophically with the soil biota
(especially bacteria and fungi), that feed on this organic rain
and release the locked nutrients as waste for renewed
uptake by the plants. Moreover, almost all plant species
have an endomycorrhiza, i.e. specialised fungi of the phylum
Glomeromycetes that are probably involved in drought
resistance and nutrient uptake. The Brassicacaea, which
are specialists on nutrient-rich soils, do not have endomycorrhizae. Some 30 % of plants also have an ectomycorrhiza and many use animals for pollination and dispersal.
(a) Ageing and sleeping
When the cyanobacteria eventually enriched the atmosphere
with dioxygen, many species adapted to this new situation
and energy acquisition from carbohydrates was greatly
improved by using dioxygen for oxidation in the respiratory
chain. Although means to cope with free radicals, such as
reactive nitrogen species (RNS), were already present, the
handling of reactive oxygen species (ROS) became important
to reduce damage to the metabolic machinery and especially
to DNA. This especially holds true for tissues of cells with
non-reversible differentiation; this excludes e.g. plants.
Specialised proteins (peroxidase dismutases) were developed
and their effectiveness was tuned to compromise between
survival of the juvenile period and the use of ROS to
generate genetic variability among gametes. The latter is
important to allow adaptation to long-term environmental
changes that are too large for adaptation within a given
genome. Big-bodied species are vulnerable; the body size
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
130
scaling relationships implied by the DEB theory show that
the length of the juvenile period scales with body length
among species whereas the reproductive rate decreases with
length. Therefore large-bodied species must have efficient
peroxidase dismutases and, therefore, reduce the genetic
variability among their gametes, while having few offspring.
High feeding levels for an individual mean high respiration
rates and a short lifespan. Survival probability changes with
age in predictable ways Leeuwen et al. (2002) and involves
acceleration of ageing. This acceleration is linked to
mitochondrial damage (in aerobic eukaryotes), which produce ROS, the amount of mitochondria per cell is upregulated to achieve an adequate production of intermediary
metabolites from the TCA cycle (Kooijman & Segel, 2005);
the TCA cycle and the respiratory chain are both inside the
mitochondria.
The various evolutionary lines to multicellularity arose
with a variety of communication strategies among cells.
Many fungi merged their cells in hyphae; heterokonts and
plantae (rhodophytes and chlorophytes) linked their cells via
protoplasm connections and the latter (especially the
embryophytes) developed transport systems via apoptosis
to reallocate metabolites. Animals continued the use of
(prokaryotic) gap junctions, which allow for limited transport of particular metabolites only, and developed both
a transport system (blood and lymph) and a (relatively) fast
signalling system (the neuronal system). The latter allowed
for the development of signal processing from advanced
sensors (light, sound, smell, electrical field, pain) in
combination with advanced locomotory machinery for
food acquisition (mostly other organisms or their products).
Advanced methods for food acquisition also came with
a requirement for learning and the development of parental
care. The neuronal system is, however, sensitive to ROS,
and requires sleep for repair (Siegel, 2001, 2003). Since the
required sleeping time tends to be proportional to the
specific respiration rate, large-bodied species have more
time to search for food. Their speed and the diameter of
their home range increases with length, which enhances
their ability to cope with spatial heterogeneity. Because the
maximum reserve density also increases with length, the
time to death by starvation will increase which enhances
their ability to copy with temporal heterogeneity. At the
extreme, the largest whales leave their Antarctic feeding
grounds, swim to oligotrophic tropical waters to calve, feed
the calf some 600 l of milk per day for several months, and
then swim back with their calf to their feeding grounds
where they resume feeding. Such factors partly compensate
for the disadvantages of a large body size and the associated
high minimum food densities.
In summary, the link between ageing, sleeping and
energetics is via the respiration rate (which is fully specified
by DEB theory) and the time-budget.
(14) From supply to demand systems
Plants evolved extreme forms of morphological and biochemical adaptations to the chemical and physical conditions
in their direct environment and remained supply systems.
S. A. L. M. Kooijman and T. A. Troost
Animals, by contrast, especially birds and mammals, excel in
behavioural traits designed to meet their metabolic needs.
They evolved into demand systems, ‘‘eating what they need’’,
with those needs having reduced variability. This co-evolved
with an increase in the difference between standard and
peak metabolic rates, closed circulation systems, advanced
forms of endothermy, immune systems and hormonal
regulation systems. The physical design of these organisms,
such as the capacity of transport networks, is designed to
meet the peak metabolic performance, but gives little
information about standard metabolic performance. The
minimization of transport costs in space-filling fractallybranching (closed) circulation systems has been suggested as
the reason why animals tend to respire at rate proportional
to their weight to the power 3/4 (West, Wooddruff &
Brown, 2002). Since this pattern for standard metabolic
performance applies to all organisms, while most species do
not have a closed circulation system, it is possibly the other
way around: if transport costs scale as weight3/4, such
a branching circulation system is efficient, especially at peak
performance, when these transport costs might matter. The
evolutionary history of demand systems makes clear that we
can only understand their metabolic performance in the
light of that of supply systems. The demand of demand
systems represents an evolutionary fixation of the performance of supply systems under ‘‘typical’’ environmental
conditions.
(a) Behaviour and time budgets
Animals, especially those functioning at the demand-end
of the supply-demand spectrum, can acquire their food
so efficiently that time is available for behaviour other
than food acquisition and food processing, such as social
interaction. The Holling type II functional response for how
feeding rate depends on food density is identical to
the Michealis-Menten product formation by enzymes
because individuals and enzymes use their time in either
searching for substrate or processing of substrate. If
other behaviour traits compete for time, predictable
deviations from this relationship result. Since specific food
uptake is no longer a function of food density only but
also of population density, stable coexistence is possible
of two species that compete for a single substrate, even
in spatially homogeneous and constant environments (see
Appendix A.4).
III. EFFECTS OF TEMPERATURE
Temperature affects physiological rates as described by the
Arrhenius relationship for a species-specific temperature
range, but usually with lower rates at the borders of this
range; many organisms have the ability to switch to a state of
torpor at low temperatures. Together with irradiation (and
water in terrestrial systems), temperature controls abundance
and geographic distribution of many species. Nutrient
availability controls primary production and has indirect
relationships with temperature and water availability.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
Section II described how life evolved from a multiple
reserve - single structure system to plants with two structures
and animals with a single reserve. The evolutionary path
that led to animals shows a reduction in metabolic flexibility
coupled to increased homeostasis and improved foodacquisition that release spare time for behaviour. This has
a deep relationship with how temperature effects physiological rates. Because multiple reserve systems have to deal with
excretion, assimilation is much more loosely coupled to
maintenance and growth compared to single reserve
systems. The way temperature affects photosynthesis (i.e.
the formation of carbohydrates from photons, carbon
dioxide and water) differs from how it affects growth
(synthesis of structure), with the consequence that the
excretion of carbohydrates (mobilised from its reserve, but
rejected by the SUs for growth) depends on temperature.
This means that the importance of the microbial loop is
temperature dependent. Single reserve systems, by contrast,
do not excrete in this way and so do not have this degree of
freedom, with the consequence that all their rates (assimilation, maintenance, growth, reproduction, respiration)
depend on temperature in (more or less) the same way.
The logic is in the biochemistry behind the transformation
from food to biomass (growth, reproduction). This machinery does not have the flexibility to operate with a temperature-dependent efficiency. Studies on the temperature
dependence of rates typically do not consider the mass
balance of the system. If temperature affects animal
assimilation differently than growth, body composition or
product formation would depend on temperature as well;
this has never been observed in ‘‘lower’’ animals to our
knowledge. Temporal heterogeneity, acclimatisation, the
role of reserve in body composition and the separation of
effects of food intake and temperature hamper this line of
research.
IV. CONCLUSIONS
(1) Early life forms showed little homeostasis or maintenance. With an increase in homeostasis, stoichiometric
constraints are imposed that require a reserve per substrate
for smoothing out fluctuations in substrate availability. The
early reserves were created by delaying the processing of
substrate that has been taken up.
(2) Homeostatic control and increased uptake enhanced
maintenance, which was internalised by meeting requirements from reserve. Reserve capacity was simultaneously
increased to enhance the smoothing out of fluctuations,
which came with the need to transform reserves to polymers
or store it in vacuoles to avoid osmotic problems. Some of
the reserves (such as RNA) adopted metabolic functions.
The overhead costs and stoichiometric constraints
enhanced product formation, and induced syntrophic
interactions.
(3) A mechanism for reserve dynamics in DEB theory is
proposed, where the mobilisation rate is proportional to the
interface between reserve and structure, and the constant
relative amount of SUs for growth plus maintenance is such
131
that the ratio of the rejected and accepted fluxes matches
the existing reserve density. This ensures weak homeostasis;
several mechanisms for homeostatic control are discussed.
Adaptation processes evolved via tunable gene expression,
which matches the machinery for the uptake of substrates to
the availability of substrates.
(4) A maturation program was installed parallel to the
allocation to growth and maintenance to organise cell cycle
events, such as the initiation of cell division. This requires
a sequence of actions, such as DNA replication, membrane
synthesis etc. The maturation program came with a control
of size and shape; since uptake is linked to surface area, and
maintenance to volume, this control has profound consequences for changes in metabolic rates.
(5) Reciprocal syntrophic interactions were intensified by
internalisation events, which gave rise to the eukaryotes.
Such recombinations are relatively easy to make because of
the modular organisation of metabolism that we discussed
at several places. Phagocytosis was invented, which
enhanced digestion efficiency and facilitated further internalisation events.
(6) Multicellularity was invented, which came with
differentiation of cell types. The juvenile stage gave rise to
the embryonic and adult stages. This reproduction evolved
from the maturation program, so parallel to growth and
somatic maintenance, and requires the formation of
a reproduction buffer and buffer handling rules.
(7) Organisms (especially animals) that specialised on living
from other organisms, experienced a coupling of availability of
the various substrates and responded by reducing the number
of reserves. The isolation of their cells came with the evolution
of a neuronal system for rapid long-distance communication
within the body, which allowed them to develop sensors and
complex behaviour. This required sleep to repair damage by
ROS.
(8) Endothermic animals (birds and mammals) made extra
evolutionary steps towards demand systems, and increased
the difference between standard and peak metabolic rates.
Many aspects of their metabolic organisation still testify to
their supply-driven ancestors.
V. ACKNOWLEDGEMENTS
We thank Peter van Bodegom, Marc Strous, Maarten van
Wieren, Anne-Willem Omta, João Rodrigues and Bob Kooi
for helpful discussions.
Information about the DEB research program and its
results can be found at http://www.bio.vu.nl/thb/deb/.
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YOON, S. H., HACKET, J. D., CINIGLIA, C., PINTO, G., &
BHATTACHARYA, D. (2002). The single, ancient origin of chromist
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15507–15512.
YOON, S. H., HACKET, J. D., CINIGLIA, C., PINTO, G., &
BHATTACHARYA, D. (2004). A molecular timeline for the origin
of photosynthetic eukaryotes. Molecular Biology and Evolution 21,
809–818.
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(2001). Vacuolation induced by unfavorable pH in cyanobacteria. Progress in Natural Ncience 11, 931–936.
ZONNEVELD, C. & KOOIJMAN, S. A. L. M. (1993). Comparative
kinetics of embryo development. Bulletin of Mathematical Biology 3,
609–635.
VII. APPENDIX A: INTERACTING
TRANSFORMATIONS
We here present some recent developments in the
quantification of interactions in biochemical transformations, based on synthesising units (SUs) kinetics (Kooijman,
1998) for various applications in cellular physiology and
ecology (Muller et al., 2001). SUs are generalised enzymes
that follow the rules of classic enzyme kinetics with two
modifications: transformation is based on fluxes, rather
than on concentrations of substrates, and the backward
fluxes are assumed to be negligibly small in the transformation
S ] 34S34P34P ] 3;
ð3Þ
where S stands for substrate, P for product and 3 for
enzyme. The backward fluxes might be small, not because
of enzyme performance as such, but because of the spatial
organisation of the supply of substrate and the removal of
product by transporters. The differences from classic
enzyme kinetics do not affect the simple one-substrate
one-product conversion in spatially homogeneous environments, but do affect more complex transformations. The
arrival flux can be taken to be proportional to the density in
spatially homogeneous environments. So for compound S
present in concentration S, with binding probability r and
affinity b,_ the arrival flux J_ S relates to the concentration as
_
rJ_ S ¼ bS
The substrates are classified as substitutable or complementary and binding schemes as sequential or parallel.
These four classes comprise the standard kinetics. Let us
characterise the states of the SUs in bounded fractions with
vector u, while 1T u ¼ 1and 0 O qi < 1 for all states i. The
change in bounded fractions of SUs can be written as
d
_ for a matrix of rates k_ with diagonal elements
dtu ¼ ku, P
k_ ii ¼ [ j6¼i k_ ij , while k_ ij P0, so 1T k_ ¼ 0. Using a time
scale separation argument, a flux of metabolite X can be
135
T
written as J_ X ¼ J_ u , with weight coefficients J_ and
_ . Mixtures of the four classes
fractions u such that 0 ¼ ku
P
of standard kinetics have the property that k_ ¼ i k_ i ,
where k_ is the matrix of rates of the mixture, and k_ i that of
a standard type. Such mixtures are discussed in Kooijman
et al. (2003) in connection with the gradual transition from
substitutable to complementary compounds.
SUs can be organised in a metabolic chain or network,
sometimes they are spatially organised in a metabolon and
pass intermediate metabolites to each other by channelling.
They might use the open-handshaking protocol for
dissociation, meaning that the dissociation process is
independent of the binding state of the neighbouring SUs,
the closed-handshaking protocol, meaning that dissociation
only occurs if the neighbouring SUs are in the free
unbounded state, or a mixture of both protocols. Closed
handshaking involves communication, and typically physical contact (so spatial structure). If handshaking is fully
closed, the whole metabolon acts as if it is a single SU. For
an application of this to the TCA cycle see Kooijman &
Segel (2005).
We here discuss first a single substrate transformation
illustrating the rejection principle that is inherent to fluxbased transformations and then describe some forms of
interactions in transformations.
(1) Reserve dynamics, rejection and
homeostasis
The derivation of the reserve dynamics has the following
steps: (i) reserve and structure are spatially segregated; (ii)the
mobilisation of reserve is at a rate proportional to the
surface area of the reserve-structure interface and allocated
to catabolic SUs; (iii) rejection of mobilised reserve occurs
because if the catabolic SUs are busy, the rejected flux
returns to the reserve; (iv) the bounded (somatic) catabolic
SUs dissociate to the demand-driven maintenance SUs and
via growth; and (v) the number of catabolic SUs is such that
weak homeostasis is achieved, which depends on the rate of
reserve mobilisation relative to dissociation rate.
The surface area of the interface of reserve and structure
is proportional to the amount of reserve ME for V1-morphs
and to ME/L for isomorphs (for which length L } MV1/3,
where MV is the amount of structure) if structural
homeostasis applies. The reserve is mobilised at rate
J_ EC ¼ ME k_ E for V1-morphs and at rate J_ EC ¼ ME v_ =L
for isomorphs. So the reserve turnover rate k_ E is constant
for V1-morphs, but its equivalent for isomorphs, v_ =L,
changes in time because the energy conductance v_ remains
constant, while length L changes in time.
The dynamics of the fraction of unbounded SUs, q _ for
V1-morphs is
d
q ¼ ð1 [ q_Þk_ ] jEM =n [ q_k_ E mE =n;
dt _
ð4Þ
where n ¼ N/MV denotes the specific number of SUs, k_ the
dissociation rate of the SUs, mE ¼ ME/MV the reserve
density, jEM ¼ J_ EM =MV the specific somatic maintenance
costs and J_ EM the somatic maintenance costs. Because
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
136
maintenance is a demand process that has a fixed specific
rate, and growth a supply process with a varying rate,
maintenance has absolute priority above growth and takes
mobilised reserve instantaneously at the moment it arrives
at the SUs. It, therefore, appears with the term jEM/n in the
change of the unbounded fraction.
The steady state fraction of unbounded SUs then
amounts to
q_ ¼
k_ ] jEM =n
;
k_ ] k_ E mE =n
ð5Þ
while
the
specific_
growth
rate
equals
[ jEM
for rejection strength
r_ ¼ ð1 [
q_ Þn yVE k_ ¼ mmEEkjE ]
y
EV
_
j ¼ yEVnk_kE and yield of structure on reserve yVE ¼ yEV[1.
The mobilised reserve flux of size ME k_ E is partitioned into
the flux ME ðk_ E [ r_ Þthat is accepted and used for somatic
maintenance at rate jEMMV and growth (i.e. structure is
synthesised at rate r_ MV ), and the flux ME j_r that is rejected
and returned to the reserve. The latter flux can be seen
(formally) as a synthesis of reserve, which helps to see that
_ homeostasis is most effective
for j ¼ 1 (so n ¼ yEV k_ E =k),
because reserve is then synthesised at the same specific rate
as structure, so the reserve density is not affected.
The dynamics of the reserve density becomes
d
mE ¼ jEA [ mE ðk_ E ] r_ ð1 [ jÞÞ;
dt
Table 2. Three steps in the evolution of reserve dynamics, and
the implications for the specific catabolic flux, the specific
growth rate and the dynamics of the reserve density. Symbols:
[E] ¼ E/V reserve density, V structural volume, L ¼ V1/3
structural length, ½p_ C ¼ p_ C =V specific catabolic flux,
½p_ M ¼ p_ M =V specific somatic maintenance
flux, ½p_ A ¼ p_ A =V
(volume-)specific assimilation flux, p_ A ¼ p_ A =L2 , surfacearea-specific maintenance flux, [EG] specific costs for structure,
k_ E reserve turnover rate, v_ energy conductance
ð6Þ
where jEA is the specific assimilation rate, which depends on
substrate density and so typically fluctuates in time. The
catabolysing SUs at the reserve-structure interface experience a local
chemical environment that changes with
d
, so with k_ E ] r_ ð1 [ jÞ. Let us call this
[ dtln mE jEA ¼0
quantity k_ C, the normalised catabolic rate. Fig. 5 gives the
standard deviation of k_ C as function of rejection strength j,
when the assimilation rate k_ A jumps randomly between
0 and some fixed value; so the assimilation process follows
an alternating Poisson process with the consequence that
the reserve density changes in time as does the specific
growth rate r_. The standard deviation of k_ C equals zero for
j ¼ 1 (because k_ C ¼ k_ E in that case), but increases almost
proportional to the deviation from this value. The tuning of
the number of SUs n can then be seen as one of the
mechanisms organisms use to improve homeostasis.
The specific flux that is mobilised from the reserve, the
specific catabolic flux ½p_ C , relates to the energy costs per
unit of structure [EG] and the specific maintenance costs
½p_ M as ½p_ C ¼ ½EG _r ] ½p_ M ¼ ½Eðk_ E [ j_rÞ, where [E] is
the reserve density. It is partitionable for all positive values
¢
h i
½EG k_ E ] j½p_ ½EG k_ E ] ½p_ M of j because p_ C ¼ ½E j½E ] ½EG M ¼ ½E
¢ . So,
½E ] ½EG rejection strength j only affects the apparent growth costs,
[EG]¢ ¼ [EG]/j. The abundance of SUs n, therefore, affects
parameter values, not model structure.
Table 2 presents the specific catabolic flux and the
reserve density dynamics for the first-order process; this is
compared with that for V1 and isomorphs according to
DEB rules. We here suggest that this sequence of three types
Module
Specific
catabolic
flux ½p_ C First-order
½Ek_ E
V1-morphs
½Eðk_ E [ r_ Þ
Isomorphs
½Eð_v=L [ r_ Þ
Specific
growth
rate r_ ¼ dtd ln V
½Ek_ E [ ½p_ M ½EG ½Ek_ E [ ½p_ M ½E ] ½EG ½E_v=L [ ½p_ M ½E ] ½EG Reserve
density dtd ½E
½p_ A [ ½Eðk_ E ] r_ Þ
½p_ A [ ½Ek_ E
ð p_ A [ ½E_vÞ=L
of dynamics represents an evolutionary sequence. The
specific growth rate appears in either the specific catabolic
flux or the reserve density dynamics because of the chain
rule for differentiation (i.e. dilution by growth).
The k-rule can be understood in terms of competition for
mobilised reserve by of two types of SUs, the somatic SUs
(which deal with somatic maintenance and growth) and the
maturity SUs (which deal with maturity maintenance and
maturation). If their relative abundance is constant (on the basis
of a strong homeostasis argument), the partitioning fraction k is
constant. We here suggest that maturation is only initiated after
the somatic metabolic organisation was established during
evolution, so k gradually decreased from a value of 1.
Monomers being part of the reserve, the strong homeostasis assumption implies that the amount of monomers MH
is a fixed fraction of the reserve, MH } ME. It might be by
rapid inter-conversion of the first-order type. The problem is
then how the cost is met, because the energy drain that is
involved should be evident in the respiration rate. This drain
should be large in eggs that start development, because they
consist of almost pure reserve, but such eggs hardly respire (cf.
Fig. 8). A more likely possibility is that monomerisation is
product inhibited and ceases if the monomers per polymer
reach a threshold level. The monomerisation cost is then
covered by maintenance and growth. For an individual with
an amount of structure MV and reserve ME, the kinetics of
the amount of monomers MH could be
d
mH _
ME ¼ [ ME ðk_ EH [
kHE Þ;
dt
mE
d
mH _
MH ¼ yHE ME ðk_ EH [
kHE Þ;
dt
mE
ð7Þ
with reserve density mE ¼ ME/MV and maturity density
mH ¼ MH/MV , while k_ EH and k_ HE are the specific fluxes
from E to H and vice versa. This kinetics implies the steady
_
m
state mH ¼ kk_ EH . The monomerisation occurs at the E-V
HE
E
interface, which has a surface area proportional to E/L in
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
isomorphs, where structural length L } MV1/3. This makes
that k_ EH and k_ HE are proportional to L[1 as well.
This concludes the transformation from reserve to
structure. The next sections discuss important other
variations on SU-mediated transformations.
(2) Co-metabolism
Suppose that substrates S1 and S2 are substitutable and are
bound in parallel and that the binding probability of each
substrate depends on binding with the other substrate as
described and applied by Brandt, Leeuwen & Kooijman
(2003). We study the process 1 S1 /yPS1 P and 1 S2 /yPS2 P
(see Fig. 13). So we have three binding probabilities of each
substrate; for substrate S1 we have the binding probabilities:
(1) 0 if S1 is already bound, (2) rS1 if S1 and S2 are not
bound, (3) rS1 S2 if S2 is bound, but S1 is not.
No interaction occurs if rS1 ¼ rS1 S2 ; full co-metabolism
occurs if rS1 ¼ 0. Sequential processing occurs if
rS1 S2 ¼ rS2 S1 ¼ 0. The dissociation rates k_ S1 and k_ S2 of
product P, and the stoichiometric coefficients yS1 P and yS2 P ,
might differ for both substrates. The binding period is
measured as the period between arrival of substrate and
dissociation of product, so it includes the production period.
For scaled fluxes jS¢ 1 ¼ jS1 rS1 , jS² 1 ¼ jS1 rS1 S2 , jS¢ 2 ¼ jS2 rS2 ,
²
jS2 ¼ jS2 rS2 S1 , the fractions of bounded SUs follow the
dynamics
1 ¼ q:: ] q: S1 ] qS2 _ ] qS1 S2 ;
d
q ¼ [ ð jS¢ 1 ] jS¢ 2 Þq:: ] k_ S1 q: S1 ] k_ S2 qS2 : ;
dt ::
d
qS : ¼ jS¢ 1 q:: [ ðk_ S1 ] jS² 2 Þq: S1 ] k_ S2 qS1 S2 ;
dt 1
ð8Þ
ð9Þ
ð10Þ
137
d
q S ¼ jS¢ 2 q:: [ ðk_ S2 ] jS² 1 ÞqS2 : ] k_ S1 qS1 S2 ;
dt : 2
ð11Þ
d
qS S ¼ jS² 2 qS1 : ] jS² 1 qS2 : [ ðk_ S1 ] k_ S2 ÞqS1 S2 :
dt 1 2
ð12Þ
*
Assuming pseudo steady state (i.e. dtd q ¼ 0 for q** ¼ q**
),
the production flux amounts to
jP ¼ jP;S1 ] jP;S2 ¼ yPS1 k_ S1 ðqS1 : ] qS1 S2 Þ
] yPS2 k_ S2 ðq ] q Þ
S2 _
yPS1 k_ S1 ðjS¢ k_ S2 ] jS² jS²
¼
1
1
1 S2
ð13Þ
S1 S2
Þ ] yPS2 k_ S2 ðjS² 2 k_ S1 ] jS¢ 2 jS² 2 S1 Þ
j² j¢ k_ S ] j¢ j¢ k_ S ] j² j² ðj¢ ] j¢ Þ
S1 S2 2
S2 S1 1
S1 S2 S1
S2
j² ] j² ] k_ S ] k_ S
1
2
S1
S2
jS¢ ðk_ S1 ] k_ S2 Þ ] jS² ðjS¢ ] jS¢ Þ
2
2
1
2
jS² ] jS² ] k_ S1 ] k_ S2
1
2
jS¢ ðk_ S1 ] k_ S2 Þ ] jS² ðjS¢ ] jS¢ Þ
1
1
1
2
jS² ] jS² ] k_ S1 ] k_ S2
1
2
jS¢ k_ S2 ] jS¢ k_ S1 ] k_ S1 k_ S2 ]
1
2
with jS² 1 S2 ¼
and jS² 2 S1 ¼
ð14Þ
:
If S2 represents a xenobiotic substrate, and S1 a natural one,
the case rS1 ¼ rS1 S2 and rS2 ¼ 0 is of special interest. The
use of S1 is not affected by S2, but S2 can only be processed if
S1 is present. The expression for the product flux simplifies
for jS¢ 1 ¼ jS² 1 and jS¢ 2 ¼ 0 to
yPS1 k_ S1
jP ¼
1 ] k_ S1 jS¢ 1[ 1
jS² 2 jS¢ 1 ] k_ S1 ] k_ S2
yPS2 k_ S2
:
]
1 ] k_ S1 jS¢ 1[ 1 jS² 2 ðjS¢ 1 ] k_ S2 Þ ] k_ S2 ðjS1 ] k_ S1 ] k_ S2 Þ
ð15Þ
The accepted flux of substrate S2, so the specific bio[1
,
degradation rate of S2, is jS]2 ¼ yS2 P jP;S2 with yS2 P ¼ yPS
2
and jP;S2 is given by the second term in the expression for jP .
We need this scheme for co-metabolism to describe for
example that the conversion of grass to cow and of sheep
brain to cow is much less efficient than of the combination
of grass and sheep brain to cow.
(3) Inhibition and preference
Fig. 13. Changes in the fraction of bounded synthesising
units (SUs) q* in three important transformations. Left: the
interaction between the conversions from reserve E to
maintenance products PV and structure V. See Section A.1.
Middle: the scheme for general co-metabolism of the transformations S1 /P1 with S2 /P2 . See Section A.2. Right:
coupled photosynthesis – photorespiration in the transformations from carbon dioxide C plus photons (light) L (plus water)
to hydrocarbon H (plus dioxygen), C ] L/H, and from
dioxygen O plus photons plus hydrocarbon to carbon dioxide
(plus water), O ] L ] H/C. See Section A.5.
We here deal with interacting substitutable substrates that
are bound in a parallel fashion. Standard inhibition makes
part of the SUs unavailable for catalysing transformations
(Fig. 14). Stronger forms of interaction can occur if one
substrate is able to replace another that is already bound to
an SU (Fig. 14).
Let jS1 and jS2 be the fluxes of substrate S1 and S2 that
arrive at an SU, and rS1 and rS2 be the binding
probabilities. The binding kinetics, i.e. the changes in the
bounded fractions of SUs, for scaled fluxes jS¢ 1 ¼ rS1 jS1 ,
jS¢ 2 ¼ rS2 jS2 and 1 ¼ q_ ] qS1 ] qS2 are
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
138
d
S1
S1 ¼ [ jS1 X; jS1 ¼ kS1 jS1 m fS1 ; fS1 ¼
;
dt
S1 ] KS1
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Fig. 14. Left: Interaction between the conversions S1 /P and
S2 /P, with preference for the first transformation. q*
indicates the fraction of synthesising units that are bound to
substrates. The graph gives the accepted fluxes jS]1 and jS]2 of
substrate S1 and S2, respectively, as functions of the supply flux
jS1 for a demand-driven transformation; parameter values:
preference parameter w ¼ 0 and 0.1, dissociation flux
k_ P ¼ 0:04 h[1, yield coefficients yPS2 ¼ 0:12, yPS1 ¼ 0:1,
binding probabilities rS1 ¼ rS2 ¼ 1. Right: The standard
inhibition scheme, where S2 inhibits the transformation S1 /P.
d
qS ¼ jS¢ 2 q_ [ ðjS¢ 1 ] k_ S2 ÞqS2 ;
dt 2
ð16Þ
d
qS ¼ jS¢ 1 ðq_ ] qS2 Þ [ k_ S1 qS1 ;
dt 1
ð17Þ
where k_ S1 and k_ S2 are the dissociation constants of the
SU-substrate complexes.
ð20Þ
d
S2 ¼ [ jS2 X; jS2 ¼ kS2 jS2 m fS2 ;
dt
S2
fS2 ¼
; kS2 ¼ 1 [ kS1 ;
S2 ] KS2
ð21Þ
k_ E mE [ k_ M
d
X ¼ r_ X; r_ ¼
;
dt
mE ] yEV
ð22Þ
d
mE ¼ yES1 jS1 ] yES2 jS2 [ mE k_ E ;
dt
ð23Þ
!
w¢S1 kS1 fS1
d
_
kS ¼ ð_r ] hÞ
[ kS1 ;
dt 1
w¢S1 kS1 fS1 ] w¢S2 kS2 fS2
ð24Þ
where j*m is the maximum specific uptake flux of substrate *,
f* is the scaled functional response and K* the half-saturation
coefficient for substrate *. The coefficient yE* is the yield of
reserve E on substrate *, k_ E the reserve turnover rate, k_ M the
maintenance rate coefficient and r_ the specific growth rate.
The fraction kS1 between 0 and 1 quantifies the relative gene
expression for the carrier of substrate S1 and w¢S1 the
inhibition of the expression of the gene for the carrier of
substrate S1 by the expression of the gene for the carrier of
substrate S2; without loss of generality we can assume that
1 ¼ w¢S1 ] w¢S2 . Notice that a single substrate induces full
gene expression ( kS1 /1 if fS2 ¼ 0). The typically very low
background expression rate h_ serves an antenna function for
substrates that have been absent for a long time. This
readily extends to an arbitrary number of substrates. See
Fig. 9 and Brandt et al. (2004) for an illustration of the
application of this theory.
(a) Supply kinetics
For the binding fraction at steady state, the production flux
of P equals jP ¼ yPS1 jS]1 ] yPS2 jS]2 , while the fluxes of S1 and
S2 that are used are
jS]1
¼
k_ S1 qS1
jS]2 ¼ k_ S2 qS2 ¼
¼
k_ S1 jS¢ 1
k_ S1 ] jS¢ 1
(b) Demand kinetics
If the flux of P is given (and constant), we require that
jP ¼ yPS1 k_ S1 qS1 ] yPS2 k_ S2 qS2
ð18Þ
;
k_ S1 k_ S2 jS¢ 2
k_ S2 ] jS¢ 1 ] jS¢ 2
ð25Þ
is constant at value k_ P , say, by allowing k_ S1 and k_ S2 to
depend on q*. The following rates fulfil the constraint:
k_ S1 ¼ k_ P =q and k_ S2 ¼ wk_ P =q with q ¼ yPS1 qS1 ] yPS2 wqS2 ;
:
ð26Þ
ð19Þ
Although their derivation has been set up slightly
differently, this formulation is used in Brandt et al. (2004)
to model substrate preference and diauxic growth in
microorganisms. The use of genes coding for substratespecific carriers is here linked to the use of carriers; the
expression of one gene inhibits the expression of the other.
When embedded in a batch culture, the uptake rate of
substrates S1 and S2 by biomass X (of V1-morphs) with
reserve density mE in a batch culture is given by
where the preference parameter w ¼ k_ S2 =k_ S1 has the
interpretation of the ratio of dissociation rates. For the
fractions in steady state, the fluxes of S1 and S2 that are used
to produce P are
jS]1 ¼ ðk_ P [ yPS2 jS]2 ÞyS1 P and
ð27Þ
qS
2ak_ P =yPS2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jS]2 ¼ wk_ P 2 ¼
;
2
q
2a ] yPS1 ð b [ 4ac [ bÞ
with a ¼ wjS¢ 2 k_ P yPS2 , b ¼ yPS1 c ] ðð1 [ wÞjS¢ 1 ] jS¢ 2 Þk_ P ,
c ¼ [ jS¢ 1 ðjS¢ 1 ] jS¢ 2 Þ. Tolla (2006) proposed this model to
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
Quantitative steps in the evolution of metabolic organisation
quantify the preference to pay maintenance (flux jP) from
reserve (flux jS1 ) rather than from structure (flux jS2 ). Only
under extreme starvation conditions, when reserve is
insufficient, is maintenance met from structure. This is less
efficient, because structure is synthesised from reserve. Since
the structure-specific maintenance cost is constant, this
formulation is demand driven, rather than the more typical
supply driven. The DEB model specifies the reserve flux jS1 .
Since the turnover of structure is constant, jS2 is constant.
We want to minimise payment of maintenance costs from
structure; the worst case is that jS1 ¼ 0 and all must be paid
from structure, which gives jS¢ 2 ¼ k_ P yS2 P . The preference
parameter w allows for an absolute preference for reserve
for w/0, and a preference for structure for w/N. Fig. 14
presents a numerical study that shows that this model can
mimic a switch model, without having a switch.
Another variation on the demand version of (partly)
substitutable compounds was studied by Kuijper, Anderson
& Kooijman (2003), where carbohydrate reserve is preferred above protein reserve for paying the energymaintenance in zooplankton, but protein reserve is required
to pay the building-block maintenance. This increase in
metabolic flexibility has the consequence that a nutrientlight-phytoplankton-zooplankton system evolves to a situation in which it becomes both energy and nutrient limited,
rather than a single limitation only.
(4) Inhibition and social interaction
Social interaction (especially among animals at the demand
end of the supply-demand spectrum) can be seen as an
association between two individuals that dissociates without
transformation; the effect on the feeding rate is via loss of
time that depends in a particular way on the population
density; the process is formally equivalent to an inhibition
process of a special type. Fig. 15 shows schemes for the cases
that socialisation can be initiated during food processing
(parallel case) or can not (sequential case), while searching
Fig. 15. The various associations of an individual of species Y
with substrate X, which leads to a conversion X/E, or with
other individuals of species Y or Z. The left scheme treats food
processing as a process sequential to socialisation, the right one
as parallel.
139
for food cannot be initiated during socialisation. Socialisation can be intra- and/or interspecific.
For species Y that interacts intraspecifically only and feeds
on food X, the possible ‘‘binding’’ fractions are 1 ¼ q ]
qX ] qY ] qXY. The changes in the ‘‘binding’’ fractions for
the parallel case are
d
q ¼ k_ X qX ] k_ Y q_Y [ ðb_ X X ] b_ Y YÞq:: ;
dt ::
ð28Þ
d
qX ¼ b_ X Xq:: ] k_ Y qXY [ b_ Y YqX ;
dt
ð29Þ
d
q ¼ b_ Y Yq:: ] k_ X qXY [ b_ X Xq_Y ;
dt _Y
ð30Þ
where b_ are the affinities and k_ dissociation rates. For the
sequential case, we exclude all double binding.
The scaled functional response equals f ¼ q*x with
q_ ¼ ð1 ] x ] yÞ [ 1 sequential case
¼
xy
1]x]y]
1 ] w¢ ] w¢ y
ð31Þ
[1
parallel case
ð32Þ
where scaled food density x ¼ X/KX and scaled population
density y ¼ Y/KY are scaled with saturation constants
KX ¼ k_ X =b_ X and KY ¼ k_ Y =b_ Y , i.e. ratios of the dissociation
rates and the affinities. The socialisation parameter
w¢ ¼ k_ X =k_ Y is the ratio of the dissociation rates for food and
social interaction and plays the role of an inhibition parameter.
If food X is supplied to a population of socially interacting
_ the
consumers Y in a chemostat run at throughput rate h,
changes in food and population densities are given by
d
_ r [ XÞ [ fjXm Y;
X ¼ hðX
dt
ð33Þ
d
_
Y ¼ ð_r [ hÞY;
dt
ð34Þ
_
k_ M g
_
with specific growth rate r_ ¼ kE ff [
] g , where kM is the
maintenance rate coefficient, k_ E the reserve turnover rate and
g the energy investment ratio. At steady state we have h_ ¼ r_.
Fig. 16 illustrates the effects of socialisation in a singlespecies situation. After finishing a food-processing session,
a sequentially interacting individual starts food searching,
but one interacting in parallel first has to complete any
social interaction that started during food processing. If
social interaction is parallel, it can always be initiated; if
sequential, it can only be initiated during searching. This
explains the substantial difference between both models;
sequential socialisation has relatively little impact because
low growth rates accompany low densities (because of
maintenance), and so rare social encounters, whereas high
growth rates accompany high food levels, so most time is
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
S. A. L. M. Kooijman and T. A. Troost
140
10
1.8
8
1.5
1.2
6
0.9
4
0.6
2
0
0.3
0
0.02
0.04
0.06
0.08
0.1
0
0
0.02
0.04
0.06
0.08
0.1
Fig. 16. No socialisation (0), and sequential (s) and parallel (p) socialisation in a single-species population in a chemostat.
Parameter values: substrate concentration in the feed Xr ¼ 10 mM, maximum specific substrate uptake rate jxm ¼ 1 h[1, energy
investment ratio g ¼ 1, maintenance rate coefficient k_ M ¼ 0:002 h[1, reserve turnover rate k_ E ¼ 0:2 h[1, half-saturation
coefficients Kx ¼ 0.1 mM and Ky ¼ 0.1 mM, socialisation w¢ ¼ 0.01. The latter parameter only occurs in the parallel case.
spend on food processing and not on social interaction. The
models are more similar for higher values of K and/or w¢.
While the sequential model is well known (Beddington,
1975; DeAngelis, Goldstein & O’Neill, 1975), the parallel
model is here formulated for the first time.
(5) Reversion and phototrophy
Where inhibition reduces the rate at which a transformation
proceeds, reversion interchanges the roles of substrate and
product. Reversible transformations are an obvious example, but phototrophy provides an example of a different
type. Possibly due to the fact that rubisco originated in an
anaerobic environment when binding to dioxygen was not
an issue, under aerobic conditions it can operate in two
modes on the substrate ribulose-1,5-biphosphate (RuP2):
Carboxylase activity : RuP2 ] CO2 ] H2 O/
2½3P [ glycerate
Oxygenase activity : RuP2 ]O2 /
1½3P [ glycerate
ð35Þ
substrate into product (metabolites) X; the resulting
metabolites can be taken up and used for metabolism. We
here compare this digestion mode with endocellular
digestion, assuming that the concentration of (solid)
substrate is very large relative to biomass (so the decrease
of solid substrate is negligibly small) and the enzyme
molecules have a limited active lifespan.
At lower substrate concentrations, extracellular feeding
rapidly becomes even less efficient, because enzymes lose
time in their unbound state.
(1) Intracellular digestion
Suppose the digestive enzyme becomes inactive at constant
the mean production time per product
specific rate k_ 3,[and
1
molecule is k_ X . The maximum yield of product per
enzyme molecule thus amounts to ymX3 ¼ k_ X =k_ 3 and serves as
a reference for extracellular digestion. Although no
metabolites are lost, this mode of digestion comes with
costs of phagocytosis, and processing of inactive enzymes.
The latter might represent a cost or a further benefit.
ð36Þ
]1½2P [ glycolate:
The net result is that photorespiration counteracts photosynthesis, which implies a compensation point, i.e. a photonflux at which there is no net synthesis, nor breakdown of
hydrocarbon. Fig. 13 gives the scheme for the SUs that are
involved and Kooijman (2000) presents the dynamics. The
further processing of hydrocarbons involves nutrients, and
hydrocarbons can be excreted if nutrients are insufficiently
available, see Section II.3e.
VIII. APPENDIX B: EXCRETION OF DIGESTIVE
ENZYMES
Prokaryotes have no phagocytosis and therefore have to
excrete enzymes to digest substrate molecules that cannot
pass through their membrane. These enzymes 3 transform
(2) Social digestion
Suppose now that bacteria are tightly packed in a one-cellthick layer on a solid substrate,
and they excrete enzyme
molecules at specific rate J_ 3 (moles per surface area of
cell per time). If the cells are
with radius LR, they
spherical
excrete enzymes at rate J_ 3 4pL2R . One cell occupies
surface pL2R in the layer, so a unit surface area has
[1
cells. Let d*(L) denote the density of * per length
ðpL2R Þ
of medium at length L from the cell center. In surfacearea S
of medium, enzymes are excreted at rate J_ 3 ¼ 4 J_ 3 S
(moles time[1). Assuming that the cells are half embedded
in
the medium and the maximum specific uptake rate
J_ Xm is large enough to ensure that the concentration
d3 ðLR Þ=Sat the
is small, the uptake rate of
cell membrane
RÞ
,
where
d3K is the half-saturation
a cell is J_ Xm pL2R d3dðL
3K
density. In surface area S of medium the uptake rate is
RÞ
J_ X ¼ J_ Xm Sd3dðL
(moles time[1). The yield of metabolite
3K
fJ_ g 3 ðLR Þ fJ_ Xm gd3 ðLR Þ
on enzyme equals yX3 ¼ fJ_Xmg d4d
¼ J_
d3K .
3K
3
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
3
Quantitative steps in the evolution of metabolic organisation
The change in densities of enzyme and product
_X
_ 3 and D
concentrations is for diffusivities D
_ 3 v d3 ðLR ; tÞ;
0 ¼ J_ 3 ] D
vL
0¼
ð37Þ
2
v
_ 3 v d3 ðL; tÞ;
d3 ðL; tÞ ] k_ 3 d3 ðL; tÞ [ D
vt
vL2
per cell’s
enzyme molecules at specific rate J_ 3 (moles
surface area per time) or at rate J_ 3 ¼ J_ 3 4pL2R in total.
RÞ
, so
The cell’s uptake rate of metabolites is J_ Xm 2pL2R d3dðL
3K
fJ_ Xm gd3 ðLR Þ
the yield of metabolites on enzyme is y3X ¼ fJ_ g 2d3K .
3
The change in densities of enzyme and product
_X
_ 3 and D
concentrations is for diffusivities D
_3
0 ¼ J_ 3 ] D
_ X v dX ðLR ; tÞ;
0 ¼ J_ X [ D
vL
0¼
ð38Þ
v
d3 ðLR ; tÞ;
vL
ð43Þ
ð39Þ
2
v
_ X v dX ðL; tÞ:
dX ðL; tÞ [ k_ X d3 ðL; tÞ [ D
vt
vL2
ð40Þ
The steadyRstate profiles follow from the balance for enzyme
N
molecules LR d3 ðLÞ dL ¼ J_ 3 =k_ 3 , which have solution
qffiffiffiffiffiffiffiffiffiffiffiffi
J_ 3 L3
LR [ L
_ 3 =k_ 3 ; ð41Þ
d3 ðLÞ ¼
exp
for L3 ¼ D
_3
L3
D
dX ðLÞ ¼
141
_ 3 J_ L3
k_ X D
3
[ d3 ðLÞ :
_X D
_3
k_ 3 D
v
0 ¼ d3 ðL; tÞ ] k_ 3 d3 ðL; tÞ ]
vt
2
_
_ 3 v d3 ðL; tÞ [ 2D3 v d3 ðL; tÞ;
[D
2
vL
L vL
ð44Þ
_ X v dX ðLR ; tÞ;
0 ¼ J_ X [ D
vL
ð45Þ
v
0 ¼ dX ðL; tÞ [ k_ X d3 ðL; tÞ ]
vt
2
_
_ X v dX ðL; tÞ [ 2DX v dX ðL; tÞ:
[D
2
vL
L vL
ð42Þ
ð46Þ
_ _
d
We have dL
dX ðLR Þ ¼ DJ_ 3 kk_X . The uptake equals
X 3
_ X d dX ðLR ; tÞ, while J_ k_ X =k_ 3 metabolites is proJ_ X ðtÞ ¼ D
3
dL
duced when the extracellular enzyme buffer is full. The
difference is lost to the environment. The yield coefficient at infinite time is yX 3 ¼ J_ X =J_ 3 . We define the rela_ _
tive efficiency to be ’ ¼ yyXm 3 ¼ JJ_ Xk_k3. Initially, when
X3
3 X
d3 ðL; 0Þ ¼ dX ðL; 0Þ ¼ 0, we have ’ ¼ 0; it takes a long
time to build up to ’ ¼ 1, when all of the medium (apart
from the direct neighbourhood of the bacteria) has
metabolite density dX(N).
(3) Solitary digestion
Suppose now that a single spherical cell of radius LR lives
half embedded on a homogeneous medium and excretes
The steady state profile of the enzyme and metabolite is
J_ L3 L2R =L
LR [ L
d3 ðLÞ ¼ 3
exp
;
_ 3 L3 ] LR
L3
D
ð47Þ
_ 3 J_ 3 L3 LR
k_ X D
[ d3 ðLÞ :
dX ðLÞ ¼
_X D
_ 3 L3 ] LR
k_ 3 D
ð48Þ
Fig. 17 compares enzyme and metabolite profiles for the
social and solitary digestion modes. Although the results
depend on parameter values, a lot of metabolite is unavailable for the cell, and the problem is much worse for solitary
cells. It also takes a long time to build up yield, compared
with intracellular digestion. The different enzyme and
60
10
8
40
6
4
20
2
0
0
2
4
6
0
0
2
4
6
Fig. 17. The enzyme (3) and metabolite (X) profiles for social (left) and solitary (right) digestion for times 100, 200,., 500 h. The steady
_3 ¼
state profiles for enzyme and metabolite are indicated. Parameters: excreted enzyme flux J_ 3 ¼ 1mmol h[1, diffusivities D
_ X ¼ 0:03 mm2 h[1, specific enzyme decay rate k_ 3 ¼ 0:01 h[1, dissociation rate of metabolite k_ X ¼ 0:01h[1, specific
0.03 mm2 h[1, D
maximum uptake rate k_ ¼ 20h[1, cell radius LR ¼ 0.5 mm. See Fig. 11 for the yield of metabolite on enzyme as function of time.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society
142
metabolite profiles correspond to the same choices for time
points. So, the enzyme profile reaches its steady state much
earlier than the metabolite profiles, especially in solitary
feeding; the metabolites first must flush the whole medium
before a steady state profile can build up. If flocs of
microorganisms feed in spatially homogeneous environ-
S. A. L. M. Kooijman and T. A. Troost
ments, metabolite profiles inside the floc build up that follow
the same principles (Brandt & Kooijman, 2000). The
metabolic implication is that the half-saturation coefficient
of flocculated growth is several orders of magnitude larger
than that of free suspensions. Microbial activity in sewage
treatment plants is almost exclusively by flocculated growth.
Biological Reviews 82 (2007) 113–142 Ó 2007 The Authors Journal compilation Ó 2007 Cambridge Philosophical Society