Is maximization of molar yield in metabolic networks favoured by evolution?
Introduction
In modern systems biology, there is a range of tools for the theoretical analysis and computer simulation of metabolism (Klipp et al., 2005; Schuster and Fell, 2007). One such tool, stoichiometric network analysis or constraint-based modelling (Clarke, 1981; Schuster et al., 1999, Schuster et al., 2002; Schilling et al., 2000; Edwards et al., 2002; Klamt et al., 2002), has the advantage of not requiring the knowledge of kinetic parameters. Although the conclusions it gives are less general than those obtainable from dynamic simulation, they can still contribute to understanding metabolism for medical, biotechnological and other applications (Bonarius et al., 1997; Papin et al., 2003; Schuster, 2004).
Stoichiometric analysis allows the prediction of all possible metabolic routes through the metabolic network of an organism. It does not, however, predict which of these routes are likely to be used. Thus, the relative fluxes in the reactions of the network cannot be calculated. One method for predicting these fluxes is referred to as flux balance analysis (FBA), which is based on the flux balance at steady state and, in addition, on an optimality principle. This is usually written as a linear programming problem of maximizing output rates under the (sometimes tacitly assumed) side constraint of normalized input rates. The idea is that cells will have flux distributions that maximize the molar yield (molar conversion ratio) of some important product, on the basis that cells that use resources as efficiently as possible are assumed to have a selective advantage. This approach traces back to Papoutsakis (1984), Watson (1986) and Fell and Small (1986). Later, Palsson and co-workers refined and extended the method, applied it to various systems of increasing complexity and coined the term “flux balance analysis” (Varma and Palsson, 1993a, Varma and Palsson, 1993b, Varma and Palsson, 1994; Edwards and Palsson, 1999; Edwards et al., 2001, Edwards et al., 2002; Ibarra et al., 2002; Fong and Palsson, 2004). In most cases, biomass or, less frequently, ATP or other substances have been taken as the products of interest.
An alternative view invokes flux maximization: cells producing essential substances (e.g. ATP and/or biomass) as fast as possible should have a selective advantage (Kacser and Beeby, 1984; Heinrich et al., 1987, Heinrich et al., 1997; Angulo-Brown et al., 1995; Heinrich and Schuster, 1996; Meléndez-Hevia et al., 1994, Meléndez et al., 1999; Stephani et al., 1999; Ebenhöh and Heinrich, 2001). This may be particularly relevant for micro-organisms, which can outcompete other species or strains of micro-organisms when growing fast. Moreover, a principle of flux minimization has been proposed (Holzhütter, 2004, Holzhütter, 2006). This may appear to be in contradiction to flux maximization at first sight. We will comment on this in Section 4.
The two optimization principles of maximizing yield and maximizing flux are not always consistent with each other. If, for example, the entire free energy difference between glucose and pyruvate were transformed into free energy of ATP, the production rate of ATP would drop to zero because the pathway would be at thermodynamic equilibrium (Angulo-Brown et al., 1995; Waddell et al., 1997, Waddell et al., 1999; Aledo and del Valle, 2002; Pfeiffer and Bonhoeffer, 2002). Thus, the number of ATP-producing steps that emerged in the evolution of glycolysis does not optimize yield.
It is worth noting that two optimization problems relevant on different time-scales can be distinguished. Pathways with certain properties can be chosen on evolutionary time-scales or, by regulation, during the life-span of individual organisms. Let us call these two cases the “pathway evolution problem” and the “flux distribution problem”. The question of how many ATP-producing steps can be inserted into a pathway (cf. above) belongs to the former category. Although the present paper alludes to this “pathway evolution problem”, we will here focus on the “flux distribution problem”. We compare the two principles of maximizing yield and maximizing rate and then consider the implications for the applicability of FBA. We will discuss diverse experimental evidence and various theoretical arguments, based, among others, on game-theoretical approaches.
This paper is dedicated to the memory of Reinhart Heinrich. Two of the authors (S.S. and T.P.) had been among his students, and S.S. was his co-worker for many years in the 1980s and 1990s. D.F. had come across Reinhart's work in the early 1980s, but did not meet him until 1989; after that, their common research interests led to frequent meetings at conferences in many different countries. Reinhart taught us systemic thinking in biology and many of the theoretical methods needed to better understand the behaviour of metabolic networks. As cited earlier, Reinhart worked intensely on the optimization principle of flux maximization and was one of the pioneers in evolutionary optimization of metabolic systems in general (Heinrich and Holzhütter, 1985; Heinrich et al., 1987). He also made significant contributions to stoichiometric analysis (e.g. Ebenhöh and Heinrich, 2003; Handorf et al., 2005).
Section snippets
Metabolic flux analysis and flux balance analysis
It is difficult to measure directly intracellular fluxes except those immediately connected to consumption of nutrients or output of products. In a metabolic steady state, the fluxes are related to each other by the equationwhere N and V denote the stoichiometry matrix and vector of net reaction rates, respectively (cf. Clarke, 1981; Heinrich and Schuster, 1996). In metabolic flux analysis, if a sufficient number of fluxes can be measured, some or all of those remaining can be calculated
The role of normalization
In papers on FBA, often maximization of growth rate is mentioned, but this is certainly meant to imply maximization of growth yield (cf. Teusink et al., 2006). The question whether, in FBA, molar yield or growth rate is maximized is not easy to answer. That question is closely connected to the question of how the fluxes in the system are normalized. Some normalization is necessary because otherwise, the linear programming problem would lead to solutions at infinity. However, the normalization
Conclusions
Here, we have critically examined the biological basis of the optimality assumption made in flux balance analysis (FBA). We argue that this assumption is well justified in many, yet not all situations. Nevertheless, FBA can be helpful for many purposes. For example, for checking whether some substance (or biomass) can in principle be synthesized at steady state, it is useful to compute the situation with maximum yield. If that yield is zero or negative, there is no pathway producing that
Acknowledgements
We thank Jan-Ulrich Kreft, Vitor Martins dos Santos and Gunter Neumann for stimulating discussions and Ina Weiß for valuable support in procuring relevant literature. D.A.F. acknowledges support from the Biotechnology and Biological Sciences Research Council (Grant BB/E00203X/1). T.P. thanks the Society in Science/The Branco Weiss Fellowship for support.
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