Opinion
TRENDS in Biochemical Sciences
Vol.30 No.1 January 2005
Game-theoretical approaches to
studying the evolution of biochemical
systems
Thomas Pfeiffer1,2 and Stefan Schuster3
1
Computational Laboratory, ETH Zurich, CH-8092 Zurich, Switzerland
Ecology & Evolution, ETH Zurich, CH-8092 Zurich, Switzerland
3
Department of Bioinformatics, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany
2
Evolutionary optimization has been successfully used to
increase our understanding of key properties of biochemical systems. Traditional optimization is, however,
often insufficient for gaining deeper insights into the
evolution of such systems because usually there is a
mutual relationship between the properties optimized
by evolution and the properties of the environment.
Thus, by evolving towards optimal properties, organisms change their environment, which in turn alters the
optimum. Evolutionary game theory provides an appropriate framework for analyzing evolution in such
‘dynamic fitness landscapes’. We therefore argue that
it is a promising approach to studying the evolution of
biochemical systems. Indeed, recent studies have
applied evolutionary game theory to key issues in the
evolution of energy metabolism.
towards an optimal strategy (i.e. optimal properties in a
biochemical system), the properties of the environment
can change, and this in turn can change the optimal
strategy [7,8]. This factor is particularly important if the
environment includes coevolving competitors that optimize their own strategies. For example, the optimal
strategy in the use of energy resources might depend on
how other competitors use the energy resources that are
present in the environment.
By reviewing recent publications, we illustrate how
traditional optimization can be extended by methods
adopted from evolutionary game theory to study central
issues in the evolution of energy metabolism. To illustrate
the wide scope of evolutionary game theory, we further
discuss its recent application to several highly relevant
phenomena in biochemistry.
The physiological properties of living cells are the result of
interactions between compounds in highly complex biochemical systems, and it is a major challenge in biology to
understand the central features of such complex systems.
Because cellular biochemical systems are a result of
Darwinian evolution, a promising approach is to study
how evolutionary forces and processes shape these
systems.
Evolutionary optimization has been applied to metabolic pathways in several studies [1–6]. Optimization of
properties such as rates, enzyme concentrations and
intermediate concentrations has been used to determine
essential features of metabolic pathways, such as the
order of reactions in a pathway, the kinetic properties
of the enzymes and their expression patterns [1–6].
Traditional optimization (see Glossary) is, however,
often insufficient for a deeper understanding of
complex phenomena in the evolution of metabolic pathways. These phenomena include, for example, the
emergence of crossfeeding in microbial communities
and the balance between cooperation and competition
in spatially structured microbial communities such
as biofilms.
Approaches based on simple optimization usually
neglect the fact that, during the course of evolution
Optimization principles in the analysis of energy
metabolism
One of the key compounds in cellular energy metabolism is adenosine triphosphate (ATP). The hydrolyzation of ATP is used to balance energetically unfavorable
reactions such as transport processes and biomass
synthesis. Because ATP is present in the cell at
catalytic levels and is not used to store energy, it
needs to be regenerated continuously. Thus, the properties of ATP production can have a large impact on the
fitness of an organism [9,10].
A simple but revealing way in which to derive
the optimal properties of ATP-producing pathways
(see Box 1, Figure I) has been proposed by Waddell
et al. [11]. Using linear flux–force relationships
between the flux of a pathway and the difference
between the free energies of substrates and products,
Waddell et al. have shown that the energy yield that
maximizes the rate of ATP production is 0.5; in other
words, half of the energy provided by a resource is
conserved as ATP and half of it is used to drive the
pathway. Note that this result implies that the ATP
yield (in units of ATP per unit of resource) and
production rate (in units of ATP per unit of time)
cannot be optimized simultaneously, because for a
yield larger than 0.5 the ATP production rate
decreases with increasing ATP yield.
Corresponding author: Schuster, S. ([email protected]).
Available online 7 December 2004
www.sciencedirect.com 0968-0004/$ - see front matter Q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tibs.2004.11.006
Opinion
TRENDS in Biochemical Sciences
Glossary
Dynamic fitness landscape: A dynamic fitness landscape depends on the
properties of the population; therefore, the fitness landscape changes as the
population evolves. A dynamic fitness landscape can result from interactions
between population and environment and from the presence of coevolving
competitors in the environment [8]. (The term ‘fitness landscape’ refers to a
function describing the relationship between the fitness of an organism and its
properties (i.e. strategy) in the typically multidimensional space of possible
properties.)
Evolutionarily stable strategy: A strategy that, if adopted by a population,
cannot be invaded by any other strategy. An evolutionarily stable strategy is
always an optimal strategy in its environment.
Evolutionary game theory: A mathematical framework for modeling evolution
in dynamic fitness landscapes [8]. Game theory was originally developed to
study the optimal behavior of agents in games such as the prisoner’s dilemma.
Fixed fitness landscape: A fixed fitness landscape does not depend on the
properties of the evolving populations, which implies that the environment
(including the strategies of competitors) does not change with the evolving
population. In a fixed fitness landscape, the population evolves towards
properties that maximize its fitness [8]. (The term ‘fitness landscape’ refers to a
function describing the relationship between the fitness of an organism and its
properties (i.e. strategy) in the typically multidimensional space of possible
properties.)
Flux balance analysis: An approach for analyzing pathway fluxes on the basis
of stoichiometric restrictions and maximization of product yields [39].
Optimal strategy: A strategy that maximizes fitness in a fixed fitness landscape.
Pathway flux: Reaction rates in a metabolic pathway at steady state.
Population dynamical model: A mathematical model for describing the
dynamics of population growth. Population dynamical models enable the
interplay between environment and population to be studied [7].
Prisoner’s dilemma: A situation in which two individuals simultaneously have
the choice of cooperative or selfish behavior (defection). If both players
cooperate they gain a high payoff (‘reward’), but if both players choose
defection they gain a low payoff (‘penalty’). If one player cooperates and the
other player defects, however, the cooperator gains the lowest payoff (‘suckers
payoff’) and the defector gains the highest possible payoff (‘temptation’). Thus,
it always pays for a player to defect, irrespective of what the other player is
choosing. Rational players therefore end up defecting, although in principle
they could gain a higher reward. The prisoner’s dilemma is frequently used to
study the evolution of altruistic behavior [15].
Rock–scissors–paper game: A game in which two players simultaneously have
the choice between three strategies, namely ‘rock’, ‘scissors’ and ‘paper’. Each
strategy beats one strategy but in turn is beaten by the third strategy: paper
beats rock, rock beats scissors, and scissors beats paper. Thus, there is no
optimal strategy in this game.
Signaling theory: A framework in evolutionary theory for studying the
evolution of communication [49,50].
Traditional optimization: Mathematical procedures to find an optimum of an
objective function.
Tragedy of the commons: A framework in evolutionary game theory for
explaining the evolution of inefficient use of common resources [52]. It is a
generalized form of a multiplayer prisoner’s dilemma.
Other approaches that incorporate mechanistic
details of ATP production lead to similar results and
enable a theoretical explanation of the order of reactions
in ATP-producing pathways. In line with patterns
observed in glycolysis, it has been predicted that
ATP-consuming reactions in an ATP-producing pathway
might, counter to intuition, increase the overall rate of
ATP production [12]. The ATP-consuming reactions and
ATP-producing reactions are correctly predicted to be
located at the upper and lower ends of the pathway,
respectively. Thus, it seems to be advantageous to ‘invest
energetically’ in the upper part of the pathway. An
analogous finding has been obtained in studies that apply
optimization principles to enzyme expression in simple
linear pathways (see Box 1, Figure I). For example, it has
been shown that if the rate of a pathway is maximized under
the assumption that the expression of enzymes involves
costs or is restricted, then it is advantageous to invest more
enzymes in the first steps of a pathway [1,2].
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Vol.30 No.1 January 2005
21
Optimization versus game theory in evolutionary
biology
The above optimization approaches enable us to understand important patterns observed in pathways of energy
metabolism. They are, however, based on the assumption
that populations evolve in a fixed fitness landscape.
During the course of evolution, populations are assumed
to increase in fitness until they reach a fitness maximum.
As indicated above, this assumption neglects the fact that
the evolving population often has an impact on its
environment, resulting in a dynamic fitness landscape
rather than a fixed fitness landscape.
Evolutionary game theory (including the framework of
adaptive dynamics [13]) provides a powerful tool with
which to analyze the dynamics of evolution in a dynamic
fitness landscape [7,8]. Developed about 40 years ago,
evolutionary game theory is now a major theoretical
framework in evolutionary biology and has been successfully applied to studying important evolutionary phenomena such as the evolution of altruistic behavior [14,15],
behavior in conflicts and fights [16], the evolution of sex
ratios [17], host–parasite interactions [18] and optimal
dispersal strategies [19,20]. Originally, game-theoretical
studies focused mainly on the behavior of higher animals,
but more recently it has been realized that evolutionary
game theory can be also applied to properties of genes [21],
viruses [22] and microorganisms [23–28].
The design of biochemical systems has been traditionally analyzed on the basis of optimization rather than
game theory, but game theory has proved to be a very
valuable framework for analyzing a wide range of
interesting biochemical phenomena [23–28]. Below we
discuss recent studies that have used such methods to
study the evolution of energy metabolism and compare
them to approaches based on simple optimization.
Crossfeeding in microbial populations
In a crossfeeding interaction, two or more strains (or
different microbial species) coexist on a single limiting
energy resource. One of the strains is superior in growth
on this primary energy resource, but it degrades it only
partially and excretes one or more intermediates that
can be used as an energy resource by the other strains
(see Box 1, Figure Ic). Crossfeeding interactions have been
observed in several important ecosystems such as
methanogenic environments [29] and in microbial communities that are involved in degrading xenobiotic
compounds [30,31]. Furthermore, crossfeeding reproducibly emerges in long-term evolution experiments on
Escherichia coli in continuous culture (chemostat) on a
single limiting resource [32,33].
On the basis of the competitive exclusion principle [34]
– a fundamental ecological principle that predicts that
a single limiting resource can maintain only a single
competitor – crossfeeding is not expected to evolve.
This raises the question: what advantages do crossfeeding strains have over a single competitor that
completely degrades the primary resource? Optimization
is not sufficient to answer this question, but it provides
valuable information. For a given environment with
constant fitness-relevant properties (such as resource
Opinion
22
TRENDS in Biochemical Sciences
Vol.30 No.1 January 2005
Box 1. From optimization to game theory: crossfeeding in microbial populations
Maximizing the rate of an unbranched pathway with m intermediates
and mC1 enzymes with linear, irreversible kinetics (Figure Ia) under
restrictions for the total concentration of enzymes (SEi%E*, where *
denotes upper limit values) and intermediates (SXi%X*) results in an
optimal expression of the enzymes given by E1ZE*X*/(X*CSm2) for
the first enzyme of the pathway, and EiZSE*m/(X*CSm2) for the other
enzymes [2], where S denotes the concentration of the substrate in the
environment and the rate constants k of the reactions are assumed to
be equal to each other. The corresponding flux of the pathway is JZ
kE*X*S/(X*CSm2).
If the pathway involves nATP ATP-producing steps (Figure Ib), the
rate of ATP production is given by JATPZnATPkE*X*S/(X*CSm2). If we
assume that the more a substrate is degraded (i.e. the longer the
pathway is), the more ATP can be produced, then this relationship
implies that an optimal length of the ATP-producing pathway exists.
For a linear relationship between the pathway length and the number
of ATP-producing steps, the optimal pathway length is moptZ(X*/S)1/2.
The existence of such an optimum implies that it might be
advantageous to degrade an energy resource partially (Figure Ic).
(a)
S
(i) Complete resource degradation
(c)
E1
X1
E2
...
Em
Xm
Em+1
P
S
E1
X1
E2
Xk, int
...
n1x ADP ATP
(b)
S
This is an essential precondition for crossfeeding, in which one strain
degrades an energy resource partially and excretes an intermediate
that serves as energy resource for a second strain. Specific conditions
for the emergence of crossfeeding can be studied by using gametheoretical approaches [35,36].
The interplay between relevant properties of the population and
properties of the environment is specified by using mathematical
models for the population dynamics in the chemostat. The steadystate population size (N), and resource (S) and intermediate (X)
concentrations in the chemostat can be calculated for a given
population [35,36] (Figure IIa). Thereafter, the optimal biochemical
properties in the chemostat can be calculated. If a mutant with
such optimal properties (N2) invades the resident population (N1),
it either out-competes the resident strain or coexists with it. After
invasion, the chemostat runs into a new steady state with different
metabolite concentrations (Figure IIb), and new mutants might
invade. By repeated invasion, the population evolves towards a
state in which no further invading strain (N3) can establish itself
(Figure IIc).
E1
X1
E2
...
Em
Xm
Em+1
P
nATP x ADP ATP
Em
Xm
Em+1
P
n2x ADP ATP
Em+2
(ii) Partial resource
degradation
...
(iii) Degradation of
the intermediate
Xk, ext
Ti BS
Figure I. Pathway schemes. (Ei, enzymes.) (a) Linear (unbranched) metabolic pathway. (b) Linear ATP-producing pathway. (c) Branched ATP-producing pathway. The
intermediate Xk can be excreted or taken up [Xk, ext/int (for Xk external and internal)]. If there are ATP-producing steps upstream and downstream of this branching point,
then there are three modes of ATP production: (i) substrate S is degraded completely into product P; (ii) substrate S is partially degraded into the intermediate Xk;
(iii) intermediate Xk is degraded into product P. In a crossfeeding interaction, two different strains that use mode (ii) and mode (iii) coexist in the population. Reproduced,
with permission, from Ref. [36].
(c)
(b)
(a)
X
N2
S
N2
N1
N
S
X
N1
Time
Time
S
X
N3
Time
Ti BS
Figure II. The evolution of crossfeeding in microbial populations. (a) Invasion by the first strain leads to a steady state in the chemostat. (b) Invasion by a second strain
might lead to coexistence. (c) Evolutionarily stable situation in which invasion by another strain is not possible. Abbreviations: S, substrate concentration; X, intermediate
concentration; N, population density.
concentrations), traditional optimization enables us to
calculate optimal strategies in the use of energy resources.
If organisms with such optimal properties emerge in the
population, however, they change their environment. For
example, if a particular substrate starts being used
because of some selective advantage, its concentration
decreases and an alternative substrate might become
more valuable. Thus, it is essential to include the impact of
the evolving population on its environment by using, for
example, a population dynamical model.
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Invasion and replacement by novel mutants might
occur until the population generates an environment
that no other mutants can invade; in other words, all
strains present in the population have optimal properties in the environment that they generate. Such a
population is evolutionarily stable, that is, uses an
evolutionarily stable strategy [7,8]. Note that an
evolutionarily stable population might consist of subpopulations with different properties such as a population with crossfeeding interactions [35,36]. An
Opinion
TRENDS in Biochemical Sciences
example of applying game theory to the evolution of
crossfeeding is described in Box 1.
Interactions similar to crossfeeding emerge when
microbial populations evolve on two different resources
(e.g. glucose and acetate) in batch culture, where the
population is periodically transferred into fresh medium
[37]. Unlike in chemostat culture, which has a continuous
influx of resources and steady-state metabolite concentrations, the concentrations permanently change in batch
culture. After being transferred into fresh medium, the
population first consumes glucose. As the glucose concentration decreases, acetate becomes more valuable and it
thus might pay to switch from glucose to acetate. If all
competitors switch, however, it might be better to continue
growth on glucose. As in crossfeeding interactions, the
optimal strategy depends on the strategy of the competitors. In contrast to the findings of optimization studies on
optimal switching [6], game-theoretical analysis indicates
that subpopulations with different switching strategies
might coexist in the population [37].
Yield versus rate in the ATP production of heterotrophs
Evolutionary game theory has been further applied to
energy metabolism in analyses of the consequences of
tradeoffs between the rate and yield of ATP-producing
pathways. As mentioned above, such tradeoffs arise from
thermodynamic principles [3,11] and from the presence of
alternative pathways of ATP production [38], such as
respiration and fermentation, that have opposing
Vol.30 No.1 January 2005
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properties in terms of ATP yield and rate. The existence
of such tradeoffs raises the issue of whether it is favorable
to produce ATP fast but inefficiently (i.e. with a low ATP
yield) or slow but efficiently (i.e. with a high ATP yield).
Notably, this issue is usually not discussed in traditional
optimization studies, where the property that is optimized
is defined a priori. For example, many studies on optimal
pathway design explicitly assume that the rate of a
pathway is maximized [2,5,12]. By contrast, flux balance
analysis [39] is based on the assumption that yields are
maximized. In both cases, however, it should be considered
that in the presence of a tradeoff between rate and yield,
maximization of one property is achieved at the cost of the
other property.
Studying the tradeoff between the rate and yield of ATP
production on the basis of evolutionary game theory
indicates that competition for shared energy resources
should lead to the evolution of fast but inefficient ATP
production (Box 2), even though slow but efficient ATP
production would be more beneficial to all users of the
resource. This paradoxically implies that the tendency of
the users to maximize their fitness actually results in a
decrease in their fitness – a result that cannot be obtained
from traditional optimization. The resulting evolutionary
dilemma is analogous to the tragedy of the commons [24]
and the prisoner’s dilemma [26].
In the framework of evolutionary game theory, slow and
efficient ATP production can be seen as altruistic cooperative behavior, whereas fast and inefficient ATP production
Box 2. Yield versus rate in ATP production
Imagine a group of cells that share a limited amount of resource.
Fitness is assumed to depend on the amount of ATP that is
synthesized until the resource is exhausted. If all users of a shared
resource use efficient rather than fast ATP production (blue
symbols), then a large amount of ATP is produced from the
resource (Figure Ia). Thus, at the level of the group it is
advantageous to use efficient rather than fast ATP production. If,
(a)
however, there is a user that consumes the resource fast but
inefficiently (red symbols), then this user will produce the most
ATP until the resource is exhausted (Figure Ib). Whereas all users
of the group share the disadvantage of inefficient resource use,
the advantage of faster ATP production applies only to the
individual. As a consequence, fast and inefficient resource users
can out-compete efficient resource users (Figure Ic).
(b)
(c)
Sugar
Sugar
ATP
ATP
ATP
ATP
Sugar
ATP
ATP
ATP
ATP
ATP
ATP
ATP
ATP
Ti BS
Figure I. Yield versus rate in ATP production. Microbes convert sugar into different amounts of ATP. (a) All strains use the resource efficiently. (b) One strain uses the
resource fast but inefficiently. (c) All strains use the resource fast but inefficiently.
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Opinion
TRENDS in Biochemical Sciences
can be seen as selfish behavior (see Box 2, Figure I). In
studies based on the maximization of pathway flux, it has
been tacitly taken into account that selfish behavior is the
evolutionarily stable situation. If there is no competition
for shared resources, however, maximization of yield
rather than rate is more advantageous.
The prediction that fast and inefficient ATP production
evolves if organisms are in competition for shared
resources is in line with the observation that many
microbes that usually grow on shared resources use fast
but inefficient pathways for producing ATP [40]. For
example, Saccharomyces cerevisiae and Lactobacilli use
respiro-fermentation and fermentation, respectively,
rather than pure respiration, even under aerobic conditions. By contrast, multicellular organisms that internalize resources before degrading them produce ATP
more efficiently. Thus, cells in multicellular organisms
can be considered to behave cooperatively. An interesting
exception are cancer cells, which rely on glycolysis much
more than do most healthy cells in higher animals [41].
Thus, we might speculate that cancer cells shift towards
a more selfish behavior. Game-theoretical methods
seem to provide a promising approach for analyzing
cancer cells [42].
Several yeast genera such as Kluyveromyces mainly
rely on efficient ATP production by respiration [43]. It
would be interesting to analyze why, in contrast to other
unicellular organisms, these organisms evolved towards a
more cooperative use of resources. A mechanism that
potentially facilitates the evolution of cooperative resource
use is that of kin selection in spatially structured
environments [44,45]. Evolution in such environments
has interesting implications for the structure and properties of microbial biofilms [46] and the evolution of multicellular organisms [24,47].
Future perspectives
Evolutionary game theory provides an excellent tool with
which to analyze biochemical phenomena such as biofilms
that show strong interactions between the population and
its environment. An interesting phenomenon that is
closely associated with biofilm formation is quorum
sensing [48]. By excreting auto-inducing compounds into
the environment, quorum sensing enables bacteria to
measure population densities. This might promote
density-dependent interactions between bacteria such as
those that occur in biofilm formation. The evolution of
quorum sensing can be studied by using game-theoretical
approaches such as signaling theory [49,50].
A more detailed knowledge of crossfeeding might
enable us to design and to manipulate the properties of
microbial consortia, such as those that are involved in
degrading xenobiotic compounds. These compounds are
particularly interesting because they can be beneficial and
detrimental to microorganisms at the same time: for
example, they can be used as substrates but might be
poisonous in larger concentrations. Thus, there is a subtle
balance between competition and cooperation in degrading xenobiotic compounds [30,31].
One of the most prominent applications of evolutionary
game theory to biochemical phenomena has been in the
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Vol.30 No.1 January 2005
analysis of bacteriocin production [23,25,51]. Many bacteria produce antimicrobial toxins to defend themselves
against competitors. These toxins often act specifically
against bacteria that are closely related to the toxinproducing strain. Notably, there are numerous bacteriocins and there is variety both in toxin resistance and in
toxin production: some strains produce a particular toxin,
some strains do not produce the toxin but are resistant to
it, and some strains are sensitive to the toxin.
The coexistence of these three types of strain resembles
a rock–scissors–paper game [7]. Assuming that both toxin
production and toxin resistance are associated with
metabolic costs, each of the three strains described above
can invade another type but is also susceptible to invasion
by the remaining type. In short, toxin-producing strains
can invade a sensitive population because the sensitive
population is poisoned by the toxin. In turn, toxinproducing strains can be invaded by resistant strains
that do not carry the costs for toxin production. Resistant
strains, however, can be invaded by sensitive strains that
do not carry the costs for resistance. Such mutual
vulnerability might lead to cycling between the three
strategies in a population or, in a spatially structured
environment, to stable coexistence. Thus, the presence of
specific biochemical pathways in a given strain cannot be
explained simply by optimality criteria applied to a single
isolated strain.
A similar example is provided by the production of
siderophores by bacteria [27,28]. Siderophores are proteins that are excreted by bacteria to import iron. In
pathogenic bacteria, the production of siderophores can be
associated with virulence. If the production of siderophores is associated with fitness costs, a situation
similar to the ‘tragedy of the commons’ arises [52]. A
siderophore-producing population can be invaded by
variants that do not produce siderophores but benefit
from the siderophore production of their competitors.
In maintaining siderophore production, spatial interactions and kin selection between the bacteria play a
crucial role [27,28].
In summary, we argue that evolutionary processes need
to be considered for a detailed understanding of complex
biochemical systems. Evolutionary processes do not
simply optimize these systems because there are interactions between the evolving population and its environment. Thus, traditional optimization is often insufficient
for understanding the dynamics of evolutionary processes.
Evolutionary game theory provides a more appropriate
tool with which to study the dynamics and outcome of
evolution of biochemical systems.
Acknowledgements
We thank S. Bonhoeffer (Zurich) and V. Martins dos Santos (Brunswick)
for helpful discussions.
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