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ECA: Control in ecosystems Hans V. Westerhoff1, 2*, Wayne M. Getz1, 3, Frank Bruggeman2, Jan-Hendrik S. Hofmeyr4, Johann M. Rohwer4 and Jacky L. Snoep2, 4 1
Stellenbosch Institute for Advanced Study, 2BioCentrum Amsterdam, De Boelelaan 1087, NL1081 HV Amsterdam, EU, 3Department of Environmental, Science, Policy, and Management, University of California at Berkeley, CA 94720-3112, USA, 4Department of Biochemistry, University of Stellenbosch, South Africa *corresponding author at
[email protected] Abstract Although metabolic control analysis (MCA) cannot be applied directly to microbial ecological systems because of mass conservation and stoichiometric constraints, we demonstrate here that Hierarchical Control Analysis (HCA) can be applied to such systems. We illustrate the approach for a particular ecosystem example of the biological synthesis of acetic acid from glucose, and uncover some surprising aspects to the control of this miniature ecosystem. Introduction Metabolic control analysis (MCA) has been and is useful for understanding how fluxes through and the concentrations in metabolic networks are controlled and regulated. Hierarchical control analysis (HCA) has extended these possibilities to other areas of cell biology, such as gene expression and signal transduction [1]. Due to the difficulty of obtaining quantitative experimental results, applications in the latter fields are only now forthcoming [2]. Yet the theoretical components of Metabolic and Hierarchical Control Analysis are useful in their own right, as they provide properties [3] or even laws to which the control of systems should adhere at steady state. An example is the summation theorem for concentration control. It states that the total control on a concentration by all processes in the system should amount to zero. Hierarchical control analysis adds that processes at one level of cell biology (e.g. transcription) may exert control on a concentration at another level (e.g. metabolism), without necessarily detracting from the control distribution at this other level (metabolism). Deductive laws such as those of MCA have the potential of being equally applicable to systems that differ in details but are similar in their general principles. A trophic chain or a consortium of organisms at steady state would seem to have some properties in common with biochemical networks. Mapping the ‘assimilation of the prey into the biomass of the predator’ onto ‘enzyme catalyzed process’ and ‘organism’ onto ‘metabolic intermediate’ for instance, would seem to make the two systems isomorphic. Giersch concluded however that MCA was not too helpful for ecosystems [4]. In ecological models there is no mass conservation or constant stoichiometries. Noting that in hierarchical control analysis there is no conservation of mass either, we examine here whether ecological networks may be isomorphic with the hierarchical networks of cell biology.
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Methods As discussed in the supplementary material, we first demonstrate an isomorphism between ecological and hierarchical control networks. Because of the isomorphism with the HCA network, one can apply the matrix inversion method of HCA [5]: C = E −1 where C is the matrix of control coefficients and E is the matrix that reflects both the network structure and the degree to which perturbations to concentrations of network variables perturb local rate functions.
Results For an example of a mini-ecosystem of a consortium of two organisms with one metabolite in between, Fig. 1 gives the HCA scheme. This scheme takes into account that there is neither mass conservation, nor a fixed stoichiometry between the rate at which one species in the chain is degraded and the rate at which its predator or product increases. That the two rates are related is reflected by them having elasticities for the same concentration variables. Fig. 1 also demonstrates that species can be biological (yeast and acetobacter) or chemical (ethanol).
va
v -a
Aceto bacter
ve
vy
yeast
ethanol
v -e
v -y
Figure 1. The 3-level ECA representation of acetic acid production that makes it amenable to hierarchical control analysis. Processes are indicated by solid arrows. They produce the species to which they point, or consume the species from which they point. As to the corresponding rates, v i refers to the total rate of synthesis of X i, and v -i to the total degradation rate of X i. The dashed arrows refer to the catalytic influences of species on their own synthesis rates when they are organisms (these arrows are absent when the species is not self replicating. The dotted arrows refer to the cases where the species serves as ‘food’ in process i+1 and is treated as an allosteric effector of that process and a substrate for the corresponding process -i. The concentrations of yeast, ethanol and Acetobacter as well their rates of synthesis (i.e. v y, v e, and v a respectively) and degradation (i.e. v -y, v -e, and v -a respectively) are the only independent variables considered. Acetic acid and glucose concentrations and pH are assumed to be constant (because of excess or buffering).
Using the principles of HCA, we elaborated the matrix method for this particular example, putting in some reasonable values for the elasticities. The control coefficients with respect to the concentrations and fluxes in this ecological network were then calculated, as shown in the supplementary material. The result was: C yJ y y C y C J e ye C y Ja C y C ya
J
J
J
J
C − yy
Ce y
C −ey
Ca y
y −y Je −y e −y Ja −y a −y
y e Je e e e Ja e a e
y −e Je −e e −e Ja −e a −e
y a Je a e a Ja a a a
C C C C C
C C C C C
C C C C C
C C C C C
J C − ay 6 C −ya 5 C −Jae 5 = C −ea 0 C −Jaa 5 C −aa 5
−5
0
0
−5
0
0
−5
1
0
0
0
0
−5
1
−1
−5
1
−1
− 0.83 − 0.83 0.83 − 0.83 0.83 − 1.67 1.67 − 0.67 1.67 − 1.67 1.67 0.83
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Discussion We elaborated how HCA can be applied to at least some ecological systems and demonstrated such applications for a specific example. In forthcoming work [cf. 7] we shall demonstrate that important theorems of MCA can be generalized to ecosystems at steady state for cases where the limited degree of coupling between feeding and growth rates is treated explicitly. This may perhaps further define this new approach of Ecological Control Analysis (ECA). In the results obtained for our example of wine going sour, we note a few points that are quite comparable to what would have been found for a metabolic pathway. One is that both flux and concentration control was distributed. The control on flux totals to 1, whereas that on concentrations totals to zero. As in HCA other levels in the network control a flux or concentration at any one level, but the total control exerted by those other levels amounts to zero. Surprisingly our results indicate that yeast does not control the level of ethanol and that acetobacter strongly controls the concentration of yeast, even when pH and acetic acid concentrations are buffered. The magnitudes chosen for the elasticities may not have been precisely correct, but they may have been realistic enough for these preliminary observations to arouse interest. References 1. Westerhoff H.V. Koster J.G., Van Workum M. & Rudd K.E. (1990) On the control of gene expression. In Control of Metabolic Processes (Cornish-Bowden A. & Cardenas M.L., eds.), pp. 399 – 412, Plenum Press, New York. 2. Snoep J.L., van der Weijden C.C., Andersen H.W., Westerhoff H.V. & Jensen P.R. (2002) DNA supercoiling in Escherichia coli is under tight and subtle homeostatic control, involving geneexpression and metabolic regulation of both topoisomerase I and DNA gyrase. Eur J Biochem.269, 1662-1669 3. Kacser H., Burns J. A. & Fell D. A. (1995) The control of flux. Biochem. Soc. Trans. 23, 341366. 4. Giersch C. (1991) Sensitivity analysis of ecosystems: an analytical treatment. Ecological Modelling 53, 131—146 5. Hofmeyr J.H. & Westerhoff H.V. (2001) Building the cellular puzzle: control in multi-level reaction networks. J Theor Biol. 208, 261-285. 6. Schuster S. & Westerhoff H.V. (1999) Modular control analysis of slipping enzymes. Biosystems 49, 1-15. 7. Westerhoff H.V., Getz W.M., van Verseveld H.W., Hofmeyr J.H. & Snoep J.L. (2002) Bioinformatics, cellular flows, and calculation. Ernst Schering Res Found Workshop. 38, 221-243.
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Supplementary material
A
v1
X1
v2
X2
v3
v4
X3
B
v1
X1
v2
v2’
X2
v3
v3’
X3
v4
Figu re A 1. A three level ecological system in a n inco mplete (A ) a nd a more co mplete (B) represe ntatio n. T hree species coexist in the sense of eating one a nother. X i refe rs to a species, v i to the rate of synthesis of X i, and v 4 to the degradatio n rate of X 3 . A : the simple diagra m that is similar to that of a bioche mical pathway. B: a diagra m that is more realistic indicating that species 1 and 2 are connected by two flu xes rather tha n one. v 2 refers to species X 2 gro wing because it feeds on species X 1. v 2 ’ refe rs to species X 1 being re moved because it is eaten by species X 2 . T he ovals indicate that the feeding a nd gro wth processes are incomplete ly coupled. T he dashed, curved arro w fro m any species back onto the process that produces that species, represents the phe no me no n that the gro wth of an orga nis m is autocatalytic, the orga nis m stimulating its ow n gro wth rate. A ll arrows correspo nd to elasticities, solid arrows correspond to flu xes.
Fig. A1A illustrates a trophic chain of ecology where organisms Xi serves as prey for organism Xi+1. The diagram is similar to that of a metabolic pathway, except for the positive autocatalytic stimulation of the synthesis rate of each organism by itself. This is already a shortcut of what is in fact an application of hierarchical control analysis. Fig. A1B emphasizes the problem noted by Giersch [4], i.e. that not all biomass of Xi that is lost ends up as biomass in Xi+1. Fig. A1B notes that this is similar to the cases in MCA of two incompletely coupled (‘slipping’) reactions. Applying Hierarchical Control Analysis has made such slipping reactions in metabolic networks tractable [6]. In Fig. 1 we have given credit to these two ventures into HCA by immediately writing the ecological network as a diagram of HCA. Because of the isomorphism with the HCA network, one can apply the matrix inversion method of HCA [5]: C = E −1 where the matrix of control coefficients C is defined as: C yJ y C −J yy C eJ y C −Jey C aJ y C −Jay y C −yy C ey C −ye C ay C −ya C y C J e C J e C J e C J e C J e C J e −y e −e a −a C = ye e e e e e C C C C C C y e a −y −e −a Ja C −J ya C eJ a C −Jea C aJ a C −Jaa C y C ya C −a y C ea C −ae C aa C −aa and the matrix E should reflect elasticities and network structure as follows:
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1 1 0 E= 0 0 0
− ε yy 0 − ε y− y 0 − ε ye 1 0 1 0 0 0 0
0 −ε −ε −ε −ε 0
−y e e e −e e a e
0 0 0 0 1 1
− ε a−e − ε aa − ε a− a
0 0 0
We shall here merely consider an example: −1 0 0 0 0 1 1 − 1.2 0 − 0.1 0 0 0 −1 1 0 0 0 E= − 0.5 0 −1 0 1 0 0 − 0.6 1 −1 0 0 − 1 0 0 0 1 0 The three non-zero elasticities in the second column reflect the respective assumptions that the synthesis of new yeast cells is autocatalytic, that the death of yeast cells is supposed to be a little bit more than proportional to the yeast concentration (this may be realistic close to the stationary phase), and that the synthesis rate of ethanol is proportional to the yeast cell concentration. The three nonzero elasticities in the fourth column signify the assumed slight toxicity of ethanol for yeast, the assumption that the ethanol concentration is around the Michaelis constant at which Acetobacter consumes ethanol, and the assumption that that concentration is just below the Monod constant of Acetobacter, respectively. The three 1’s in the sixth column reflect the respective assumptions that ethanol degradation, Acetobacter growth rate and Acetobacter death rates are all proportional to the Acetobacter concentration. Matrix inversion leads to the following control coefficients: −5 − 0.83 0 0 0.83 6 5 −5 − 0.83 0 0 0.83 5 −5 − 0.83 1 0 0.83 C= − 1.67 1.67 0 0 0 0 5 −5 −1 − 0.67 1 1.67 −5 −1 − 1.67 1 1.67 5
