A stochastic automaton shows how enzyme assemblies may contribute to metabolic efficiency
© Amar et al; licensee BioMed Central Ltd. 2008
Received: 14 November 2007
Accepted: 25 March 2008
Published: 25 March 2008
The advantages of grouping enzymes into metabolons and into higher order structures have long been debated. To quantify these advantages, we have developed a stochastic automaton that allows experiments to be performed in a virtual bacterium with both a membrane and a cytoplasm. We have investigated the general case of transport and metabolism as inspired by the phosphoenolpyruvate:sugar phosphotransferase system (PTS) for glucose importation and by glycolysis.
We show that PTS and glycolytic metabolons can increase production of pyruvate eightfold at low concentrations of phosphoenolpyruvate. A fourfold increase in the numbers of enzyme EI led to a 40% increase in pyruvate production, similar to that observed in vivo in the presence of glucose. Although little improvement resulted from the assembly of metabolons into a hyperstructure, such assembly can generate gradients of metabolites and signaling molecules.
in silico experiments may be performed successfully using stochastic automata such as HSIM (Hyperstructure Simulator) to help answer fundamental questions in metabolism about the properties of molecular assemblies and to devise strategies to modify such assemblies for biotechnological ends.
There is evidence from across the phyla that enzymes are often associated with one another into assemblies or metabolons in which metabolites may or may not be channeled directly from one enzyme to the next in the pathway. Investigation of glycolytic metabolons responsible for the metabolism of glucose has a long and controversial history [1, 2], but it seems clear that in eukaryotes several of the glycolytic enzymes can form oligomers  and associate with other glycolytic enzymes [4, 5]. In the model prokaryote, Escherichia coli, the glycolytic pathway has been isolated as a large equimolar multi-enzyme complex in which metabolites were sequestered [6, 7]. There is also evidence that enzymes may be assembled into much larger hyperstructures , as in the case of cellulolytic enzymes in bacteria such as Acetivibrio cellulolyticus  or compartmentalized into organelles such as glycosomes as in the case of glycolytic enzymes in trypanosomes , or associated with the tubulin cytoskeleton, as in the case of many glycolytic enzymes in mammalian cells . The actual advantages – or lack of them – conferred by metabolons and hyperstructures on metabolic efficiency are unclear. What happens to efficiency when enzymes responsible for import are restricted to a patch of membrane, or when cytoplasmic enzymes are bound to membrane transporters, or when metabolons are brought together in a hyperstructure? If quantitative answers to these questions could be obtained rapidly and cheaply in silico, it would be of value not only to the understanding of existing systems but also to the construction of engineered systems. The development of programs that allow genetic engineers to know in advance which constructions are worth making is therefore biotechnology's equivalent of the quest for the Holy Grail.
Simulation with stochastic automata and multi-agent systems is an attractive alternative to differential equations for studying the diffusion and interaction of the many different enzymes and metabolites in cells which may be present in either small or large numbers [12–18]. A stochastic automaton, HSIM (for Hyperstructure Simulator), has been developed and an early version used to model the assembly of cytoskeletal filaments in a virtual cell . In extending HSIM to the analysis of the assembly, movement and disassembly of large numbers of molecules and macromolecules, we have chosen to base our studies very loosely on two pathways that continue to be intensely studied in E. coli: the phosphoenolpyruvate:sugar phosphotransferase system (PTS), which is responsible for sensing and importing sugars such as glucose and which supplies sugar phosphates to a second intensely studied pathway, the glycolytic pathway [20, 21]. Here, we use HSIM to determine quantitatively the effects of metabolon and hyperstructure formation on a modified version of these systems in which glycolysis is truncated.
Equal numbers of enzymes in varying conditions of association
Pyruvate production at high concentration. The numbers of pyruvate molecules produced are given after 40 s in the simulation when the enzymes are either associated in hyperstructures or free as in Additional Table 1 but at an 8-fold higher concentration (the volume was reduced by this factor).
Gly + PTS
Altering numbers of enzymes in varying conditions of association
Pyruvate production with increasing numbers of EI. Comparison between the production of pyruvate in different metabolon conditions when the numbers of EI are increased fourfold. (other details essentially as in Figure 2).
EI × 4 PFK × 4
EI × 4 PFK × 4
EI × 4 PFK × 4
Gly + PTS
Effects of restricting enzymes to a region of the membrane/cytoplasm
Bacteria are now known to be highly structured and it is interesting to investigate the efficacy conferred by the assembly of metabolons into a hyperstructure. We therefore adapted the program so that only glycolytic enzymes function with the exception of Enzyme IIBC which imports glucose and converts it into P-Glu spontaneously (i.e., without the need for the rest of the PTS). In all cases, the Enzyme IIBC species were restricted to a small patch of membrane while the glycolytic metabolons were positioned 4 nm away. As explained above, direct attachment to EIIBC was avoided because this would have hindered the import of glucose in the present version of the program. It should be noted that PEP accumulates in this experiment because it is consumed by neither EI nor pyruvate kinase.
Glycolytic metabolons in hyperstructures. PEP production after 10 s when the EIIBC enzymes are in a patch and i/when the glycolytic enzymes diffuse freely or ii/when these enzymes are assembled into 60 glycolytic metabolons in a hyperstructure with or without channeling (a single enzyme space was left between the patch of EIIBC enzymes that import and phosphorylate glucose and the glycolytic hyperstructure).
Glucose per sec.
Hyperstructure no channeling
Hyperstructure + channeling
There is increasing evidence that in both eukaryotic and prokaryotic cells, enzymes are organized into higher order assemblies. In metabolism, such assemblies may take the form of metabolons  which may themselves be associated into still higher order hyperstructures [24, 25]. In signaling, chemotaxis proteins may be organized into arrays in the poles of certain bacteria , while in eukaryotes, signal transduction proteins may be assembled into transducons or signalosomes . Exploring and exploiting the consequences of such organization is a major challenge. What are the advantages, if any, of the colocalization of successive enzymes in a pathway? Can these advantages be quantified? What are the effects of increasing the number of copies of a particular enzyme or of the efficiency of the reaction that it catalyzes? What happens when enzymes responsible for import are restricted to a patch of membrane or when cytoplasmic enzymes are bound to membrane enzymes or transporters? To help provide quantitative answers to such questions, we have developed a stochastic automaton, HSIM , to simulate realistically the diffusion of metabolites and enzymes in both the membrane and the cytoplasm, the reactions between enzymes and their substrates, and interactions between enzymes leading to their assembly into metabolons and hyperstructures. For a typical workstation, HSIM takes 3 minutes to simulate 1 minute of real time for a cell, irrespective of its size, containing a total of 1000 molecules (or enzymes), irrespective of the number of classes of these molecules.
The system that we have investigated with HSIM is a simplified one loosely based on the PTS and glycolysis in E. coli which, although intensely studied for many years, is still imperfectly understood. For example, how many rate-limiting steps are there, and how might these be related to the formation of metabolons? Our version of glycolysis has 7 enzymes that act successively to produce only one PEP from each G6P. As in the real PTS, our version has 4 enzymes that pass a phosphate group from one another to the final membrane-bound enzyme, EIICBGlc, that imports and phosphorylates glucose to make G6P and completes the loop in the synthesis of one PEP from one G6P. This system has the advantage of avoiding the complication of the rate of PEP production being non-linear due to both synthesis of two PEP molecules per G6P and consumption of PEP also depending on pyruvate kinase (which metabolizes it into pyruvate). Enolase, the final glycolytic enzyme in our version, catalyzes the production of pyruvate from PEP and transfers a phosphate to EI, the first enzyme in the PTS. Pyruvate production is used as a measure of the efficacy of the system. Enzymes can either diffuse freely and release their products immediately or be in metabolons. Enzymes in metabolons can either release their products immediately to diffuse away or channel them obligatorily to the next enzyme in the pathway, provided this recipient enzyme is itself in a state to receive the metabolite (i.e., has passed its own product on).
The initial concentrations of PEP at which increased efficiency is conferred by assembly of either the PTS or the glycolytic enzymes into separate metabolons are readily revealed by HSIM. Moreover, descriptions that would otherwise be qualitative can be turned into quantitative ones. For example, at intermediate initial levels of PEP (around 500 molecules), the PTS is limiting while glycolysis is not, and the formation of PTS metabolons increases pyruvate production fourfold. At low levels of PEP (< 300), both glycolytic and PTS metabolons are advantageous, and the formation of both sets of metabolons can increase pyruvate production up to eightfold. We then tested the prediction that attaching the PTS and glycolytic metabolons to form pairs would increase efficiency relative to these metabolons that are not physically associated. A relatively modest increase of 150% was observed for some PEP concentrations below 1000 per cell, and no increase was observed over that. HSIM can be used to investigate the possibility that a particular enzyme is rate-limiting in a pathway, and that increasing the number of copies of the enzyme will improve efficiency (but see ). To illustrate this, we increased the number of EI fourfold, which corresponds to the increase observed when glucose is added. This resulted in a 40% increase in pyruvate formation under conditions where the PTS enzymes were already limiting. The association of enzymes into hyperstructures much larger than metabolons either exists already or can be created by genetic engineering by modifying enzymes so that they associate either with themselves into homopolymers or with other enzymes into heteropolymers or with cytoskeletal structures or with membrane domains. Surprisingly, there was little advantage to be gained in terms of efficiency of pyruvate production from a hyperstructure unless channeling was allowed. This doubled efficiency. It is, of course, conceivable that the advantage of grouping enzymes into hyperstructures lies elsewhere in, for example, signaling . We therefore positioned a hyperstructure at a pole and explored the consequences on the distribution of PEP under conditions in which PEP is consumed in a reaction catalyzed by a freely diffusible enzyme. This showed that a gradient of PEP can be generated by a broad range of values of different parameters; in other words, if enzymes are altered so as to assemble into a hyperstructure, it is easy to form a gradient.
In sum, our results show how HSIM may help answer fundamental problems and evaluate biotechnological objectives. Here we have applied HSIM to the analysis of a simplified model system in steady state conditions. However, it may also prove useful where parameters such as numbers of enzymes and substrates fluctuate. Continued development of HSIM and other stochastic automata and multi-agent systems should show that the dream of meaningful experiments in silico is not a pipedream.
The stochastic automaton
At the start of the simulation, the cytoplasmic enzymes are grouped together in blocks and the integral membrane enzymes in a patch. To form spatially distant metabolons, the enzymes are allowed to diffuse randomly before they are given an affinity for one another, whereas to form a single hyperstructure, they are given affinities before being allowed to diffuse. A hyperstructure itself can move with a speed that is inversely related to its size.
dE1/dt = k'1r·E1S1 + k'2r·E1S2 - (k'1f·S1 + k'2f·S2).E1 (1)
dE2/dt = k'3r·E2S2 + k'4r·E2S3 - (k'3f·S2 + k'4f·S3).E2 (2)
dS1/dt = k'1r·E1S1 - k'1f·E1.S1 (3)
dS2/dt = k'2r·E1S2 + k'3r·E2S2 - (k'2f·E1 + k'3f·E2).S2 (4)
dS3/dt = k'4r·E2S3 - k'4f·E2.S3 (5)
dE2S3/dt = k'4f·E2.S3 + k'10f·E2S2 - k'4r·E2S3 - k'10r·E2S3 (6)
which together with the 3 mass balance algebraic equations:
E1 + E1S1 = E1t (7)
E2 + E2S2 = E2t (8)
S1 +S2 + S3 + E1S1+ E1S2+ E2S2+ E2S3= S1 0 + S2 0 + S3 0 (9)
allow calculation of the concentrations of the 9 species involved in the system: E1, E2, S1, S2, S3, E1S1, E1S2, E2S2 and E2S3 using the initial conditions S1 (0) = S1 0, S2 (0) = S2 0, S3(0) = S3 0, E1 (0) = E1t, E2 (0) = E2t, the initial concentrations of the other species being zero. In these equations, k'if, k'ir, are the forward and reverse rate constants, E1t and E2t are the constant total concentrations of the enzymes, S1 0, S2 0 and S3 0 are the initial concentrations of S1, S2 and S3 and t is the time. Moreover, the rate constants have to satisfy the principle of detailed balance, i.e.
(k'1f·k'2r·k'3f·k'4r·k'9f·k'10f)/(k'1r·k'2f·k'3r·k'4f·k'9r·k'10r) = K (10)
in which K is the equilibrium constant of the overall reaction of S1 into S3.
Numerical resolutions were performed using the MAPLE 9.5 software to solve the above non-linear ODE system with the given initial conditions.
Ki = k'if/k'ir with i = 1, 2, 3, 4 and 10 (11)
We then arbitrarily chose
k'ir = 1.τ (12)
k'if = Ki.k'ir (13)
Finally, to compare HSIM with a published cellular automaton model, we used the same parameters and values as in a 2-D cellular automaton . HSIM gave exactly the same results as in their Figure 2 (data not shown).
The simplified metabolic model
Our version of glycolysis acts on glucose-6-phosphate to convert it into phosphenolpyruvate via a pathway containing 7 enzymes (Figure 1). The real glycolytic pathway yields two PEPs per G6P which are converted into pyruvate by pyruvate kinase. To limit our study, pyruvate kinase and triose phosphate isomerase are omitted and, unlike real glycolysis, aldolase splits fructose-6-bisphosphate into one glyceraldehyde-3-phosphate without an accompanying dihydroxyacetone phosphate so that our pathway yields one and not two PEP per G6P. Enolase, in our version of glycolysis, is the final enzyme and catalyzes the production of PEP. Interactions between successive enzymes in the PTS result in a reaction catalyzed by EI that entails the phosphate from PEP being transferred to HPr to EIIA to EIICBGlc. Membrane-bound EIICBGlc then imports and phosphorylates glucose to yield G6P as the substrate for glycolysis. The initial PEP concentration is an important parameter here because PEP is the basis for the phosphate transfer that leads to glucose being imported. The spatial position and number of any enzymes and any metabolites, including all metabolic intermediates, can be displayed. The number of PTS and glycolytic enzymes in a real E. coli range roughly from about 2000 for IICBGlc to about 40000 for HPr (depending on growth conditions) . Because our objective here is to test general hypotheses rather than those restricted to a specific system such as the PTS where important information is still missing, the initial system explored here has 64 copies of each class of enzyme, irrespective of whether they are membrane-bound or cytoplasmic, and the volume of our model bacterium is a quarter of an E. coli growing relatively slowly. The mean volume of this cell is about 1 cubic micron in glucose minimal medium. [Note that simulations were also performed with the same numbers of enzymes but in a volume that was reduced eightfold]. For simplicity, the operation of the system under different conditions is analyzed here only in terms of the numbers of pyruvate molecules formed although details of all other molecules are available.
We thank Russ Doolittle and the reviewers for helpful comments and the Epigenomics Project and the 'Complexité du Vivant' programme for support.
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