Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
Review Article
Deciphering the regulation of metabolism with dynamic optimization: an overview of recent advances Jan Ewald1, Martin Bartl1 and Christoph Kaleta2 1
Research Group Theoretical Systems Biology, Department of Bioinformatics, Friedrich-Schiller-Universität Jena, Jena, Germany and 2Research Group Medical Systems Biology, Christian-Albrechts-Universität zu Kiel, Kiel, Germany Correspondence: Jan Ewald ( [email protected])
Understanding optimality principles shaping the evolution of regulatory networks controlling metabolism is crucial for deriving a holistic picture of how metabolism is integrated into key cellular processes such as growth, adaptation and pathogenicity. While in the past the focus of research in pathway regulation was mainly based on stationary states, more recently dynamic optimization has proved to be an ideal tool to decipher regulatory strategies for metabolic pathways in response to environmental cues. In this short review, we summarize recent advances in the elucidation of optimal regulatory strategies and identification of optimal control points in metabolic pathways. We discuss biological implications of the discovered optimality principles on genome organization and provide examples how the derived knowledge can be used to identify new treatment strategies against pathogens. Furthermore, we briefly discuss the variety of approaches for solving dynamic optimization problems and emphasize whole-cell resource allocation models as an important emerging area of research that will allow us to study the regulation of metabolism on the whole-cell level.
Development of approaches to understand the regulation of metabolic pathways
Received: 21 April 2017 Revised: 21 June 2017 Accepted: 29 June 2017 Version of Record published: 28 July 2017
A precise control of metabolism in response to external and internal cues is of central importance for growth, development and survival of unicellular and multicellular organisms. However, the complexity of the metabolic networks as well as the myriad of potential environmental conditions most organisms are facing make the determination of optimal targets of pathway regulation a challenging task. Since metabolic pathways are often considered as chemical reaction chains, the first ideas about optimal points of control in metabolic pathways focused on the slowest reaction as the rate-limiting or -determining step, an idea originating from chemistry [1], and formulated in the concept of pacemaker enzymes catalyzing the corresponding reactions [2,3]. In a similar manner, other qualitative approaches to pathway regulation like the first committed step, which is the first irreversible reaction in a pathway, consider specific enzyme traits that determine flux controlling steps [3,4]. In contrast with this, metabolic control analysis (MCA) was developed to quantify the control of each enzyme across a pathway on the flux and discovered that, in many biological systems, multiple enzymes control pathway flux [5,6]. While this approach allows a much more fine-grained analysis of the distribution of control throughout a metabolic pathway, it is limited in the sense that it is focusing on steady-state conditions neglecting the dynamics of varying conditions. Additionally, MCA has been developed to deterministically investigate the effect of small changes and therefore fails to estimate the effect of large deviations, for example, after a change in environmental conditions that requires large adjustments in pathway flux [7].
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Especially, the consideration of dynamic conditions is of high relevance since fluxes through metabolic pathways often need to be readjusted following a change in conditions or cyclic events like the cell cycle or circadian processes. The time frame for this environmental changes and the subsequent metabolic adjustments can be within seconds or minutes (e.g. heat-shock) or within hours or days (e.g. cell cycle). Depending on the time frame, the cell has to optimize the different levels of regulation like transcriptional regulation (rather slow) or post-translational modification (rather fast) to minimize its response time, which is of major evolutionary advantage especially in fluctuating environments [8]. It is important to note that this way of reasoning is distinct from approaches such as flux balance analysis [9] that uses linear optimization to identify the most likely flux through a metabolic network based on an a priori formulated evolutionary objective [9] and thus is not primarily concerned with pathway control. Complementary to MCA, approaches based on dynamic optimization problems, which have their roots in solving engineering problems [10], have also been used to identify optimal programs of metabolic pathway regulation. These approaches are used to investigate optimization problems in which a time-dependent behavior needs to be optimized after a perturbation, for instance, by the minimization of response times or the minimization of deviation of a system from a preferred target state. The interest in these approaches has particularly been emphasized by studies of short-term gene expression responses to environmental perturbations in which it has been found that after such perturbations, often very complex regulatory programs are induced like in metabolic pathways [11,12] or flagellum assembly pathways [13]. This shows that time-dependent principles in pathway regulation are of high importance in conditions in which cells are not in steady state. In this short review, we summarize studies that have used dynamic optimization to elucidate the optimal timing patterns in the control of metabolic networks. While we focus mainly on regulatory strategies for pathway activation and control of product dilution, we further discuss their functional implications such as the genomic organization of genes. Beyond this, we briefly elaborate solution approaches for dynamic optimization problems and an outlook on models investigating optimality principles on the whole-cell level.
Determinants of optimal regulatory programs controlling metabolism defined by dynamic optimization Optimal activation patterns of metabolic pathways In principle, the optimal solution during the response to an environmental stimulus is to produce all proteins that are required for this response at once. However, this approach is often not feasible due to physiological constraints such as the immediate availability of building blocks for protein biosynthesis, a limited cellular volume, general constraints in protein biosynthetic capacity and other physiological constraints. The principle that functionally related proteins are induced in a precisely timed manner was shown for the first time for the flagellum assembly pathway [13]. In this experimental study, it was shown that the proteins along the assembly pathways were induced in the same sequential order as they were required during flagellum assembly [13]. In a seminal work, Klipp et al. [14] used dynamic optimization to determine an optimal time hierarchy of enzymes along a pathway that minimizes the time to transform the substrate of the pathway into the product with a constraint on the total amount of available enzymes. In particular, they found that in contrast with a strategy in which all enzymes were activated at once, an induction in the order of their sequence in the pathway was optimal for minimizing the transition time required to synthesize the product [14]. Such a sequential activation of enzymes along a pathway is referred to as ‘Just-in-time-activation’ in reference to an industrial production strategy whereby components are produced just as they are needed. The utilization of such a sequential activation strategy was later confirmed experimentally to occur in several amino acid biosynthetic pathways of Escherichia coli after deprivation of the corresponding amino acid [11] as well as in many metabolic pathways in yeast [12]. Inspired by these findings, later work focused on the factors as well as different objectives of pathway activation, that determine optimal activation strategies (see Table 1). A common observation is that the sequential activation is a special case among optimal activation strategies and is strongly influenced by cellular protein biosynthetic capacity [15], pathway topology [16] and kinetic properties of enzymes [15,17,18]. In particular, Bartl et al. [15] showed that if all enzymes within a pathway are under transcriptional control, protein biosynthetic constraints can either lead to the optimality of a concomitant activation of all enzymes of a pathway, a partial sequential activation of parts of a pathway or a fully sequential activation of the individual enzymes of the pathway. Moreover, it was found that the catalytic efficiency of an enzyme has a strong influence on its
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Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
Table 1 Overview of studies using dynamic optimization to decipher principles behind the control of metabolism
Pathway activation
Reference
Objective
Klipp et al. [14]
Minimum of transition time
Zaslaver et al. [11]
Minimum of (1) activation time and (2) enzyme costs Minimum of (1) activation time and (2) enzyme costs Minimum of transition/ operating time (1) Maximum of product in finite time
Oyarzun et al. [17]
Bartl et al. [18] Bartl et al. [15]
de Hijas-Liste el al. [16]
Nimmegeers et al. [27]
Pathway regulation
Wessely et al. [20]
de Hijas-Liste et al. [21]
Resource allocation
Multiple objectives: transition time, enzyme costs, intermediate accumulation, survival time Minimum of (1) activation time and (2) enzyme costs
Minimum of (1) regulation and (2) enzyme costs Minimum of (1) regulation and (2) enzyme costs
Ewald et al. [22]
Minimum of (1) regulation and 2) enzyme costs
Molenaar et al. [42]
Maximum of growth rate
Giordano et al. [40]
Maximum of growth rate
Investigated determinants
Solution approach
Pathway length, central metabolism (yeast) Enzyme costs
Numerical: maximum number of switches, genetic algorithm
Enzyme costs, kinetics
Analytical: Pontryagin’s Minimum Principle Numerical: TOMLAB Analytical: Pontryagin’s Minimum Principle Numerical: extension of quasi-sequential approach [49] Numerical: hybrid of CVP and eSS scatter search solver
Kinetics Enzyme synthesis capacity Multiple objectives, branching topology, central metabolism (yeast)
Numerical: not specified, analytical: prove of hierarchy
Linear and glycolysis like pathways, multiple objectives, parametric uncertainty
Numerical: Pomodoro toolbox parametric uncertainty propagation: linearization, sigma points, polynomial chaos expansion
Enzyme costs
Numerical: extension of quasi-sequential approach [49] Numerical: CVP approach
Protein biosynthetic rates, feedback inhibition, complex pathway topologies Toxicity of intermediates, kinetics Alternative pathways for substrate acquisition, two substrate species Feedback mechanisms
Numerical: extension of quasi-sequential approach [49] Numerical: General Algebraic Modeling System (GAMS), KNITRO solver Analytical: infinite horizon maximum principle Numerical: Bocop toolbox
position in the activation sequence, with less efficient enzymes synthesized first and highly efficient enzymes synthesized later compared with neighboring enzymes [15], an observation that allowed to explain discrepancies in the activation sequence of specific metabolic pathways in earlier experimental observation [11]. Additionally, similar optimality principles were found in the assembly of protein complexes [19] pointing toward sequential activation strategies to be of importance for the regulation of functionally related proteins also beyond metabolism, as already suggested by earlier work on sequential activation strategies in flagellum assembly [13].
Regulatory programs controlling product dilution Beyond the activation of a metabolic pathway also the fine-tuning of pathway flux due to changes in the required synthesis of pathway product or substrate availability is of importance. In such a context, rather than minimizing the time required to form a certain amount of product, the minimization of the change in protein concentrations required to achieve a change in pathway flux as well as the minimization of protein investment
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Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
for maintaining pathway flux are central objectives of optimization (see Table 1). Assuming a five-step linear metabolic pathway with randomized product dilutions modeled as time-varying outflow to simulate different growth phases or environmental changes, Wessely et al. [20] identified optimal time courses in enzyme concentrations for controlling product concentrations under non-steady-state conditions. It could be shown that depending on a balance between protein costs and the speed at which fluxes within a pathway need to be adjusted, two types of regulatory programs are employed — sparse transcriptional regulation and pervasive transcriptional regulation (see Figure 1). Sparse transcriptional regulation refers to the control of only key enzymes within a pathway (the remainder being constitutively expressed) that are sufficient to precisely control pathway flux in accordance with cellular requirements, while pervasive transcriptional regulation corresponds to a transcriptional control of all enzymes within a pathway. As a main result, it was found that the first and last enzymes of a pathway are the main targets of transcriptional regulation in a linear metabolic pathway with irreversible Michaelis–Menten kinetics (see Figure 1A). Sparse transcriptional control minimizes the time required to adjust the flux through a pathway since only the concentrations of key enzymes need to be adjusted to regulate pathway flux and late intermediates are accumulated to buffer product dilution [20]. However, the remaining enzymes of the pathway are constitutively expressed which increases their overall abundance in comparison with a case with pervasive control of all enzymes in a pathway [20]. In addition, more complex models of, for example, branched pathway topologies show that two pathways that converge via one intermediate entail the same regulatory program. Similar to linear pathways, the first enzymes of both branches and the last enzyme beyond the convergent reaction are controlled (see Figure 2A) [21]. For pathways with a branch into two pathways with separate products after a common intermediate, it was found that the control of the initial reaction of the entire pathway, the reactions after the branch and the terminal reaction is optimal (see Figure 2B) [21]. The consideration of feedback mechanisms by the product, like the inhibition of the first enzyme via post-translational modifications, engenders optimal programs where the first enzyme is less involved in the control of the flux through the pathway since product inhibition can substitute the control (see Figure 2C) [21]. An important feature of the regulatory programs that were identified included the accumulation of intermediates upstream of transcriptional-regulated proteins. While in previous studies [20,21] the unlimited accumulation to toxic levels was prevented through a constraint on total metabolite concentrations, differences in the toxicity of metabolic intermediates are not considered. Based on an extended model [22] of a linear metabolic pathway also accounting for differences in toxicity of intermediates, it was found that especially enzymes upstream of particularly toxic intermediates and enzymes with substrates with low toxicity were target of regulation (see Figure 2D) [22]. Moreover, accounting for differences in enzyme efficiency, it was observed that regulation often targets the most efficient enzyme in a pathway (in terms of high catalytic activity or high substrate affinity) (see Figure 2D) [22]. Especially, the latter observation questions the often quoted assumption that the least efficient enzyme within a pathway should be target of regulation [2]. Indeed, highly efficient enzymes are more suitable targets of regulation since they are usually present in smaller abundance and changes in flux require smaller (and hence faster) absolute changes in protein abundance compared with less efficient enzymes.
Recent approaches for solving dynamic optimization problems in systems biology While we are focusing on the application of dynamic optimization problems to solve optimal control problems in metabolism, we also briefly cover the variety of approaches for solving these problems. Due to timedependent variables and the non-linearity in modeling of enzyme kinetics and objective functions, analytical solutions could often not be derived. On the other hand, numerical approaches are computationally very expensive through the discretization of time-dependent variables. For dynamic pathway activation models, some analytical solutions were derived for initial or simplified models by limiting the numbers of enzymes and switches [14], or by using Pontryagin’s Maximum/Minimum Principle [17,18]. To tackle extended models with different topologies, kinetics and more complex objective functions, numerical approaches were used to solve the dynamic optimization problem (see Table 1). Numerical approaches have in common that they transform the continuous problem to a finite optimization problem by discretization (also called collocation or parameterization) of variables, and a variety of approaches and solvers can be used to determine optimal solutions (see http://plato.asu.edu/guide.html [23] for an overview
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Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
Figure 1. Trade-offs in the evolution of regulatory networks controlling metabolism. Balance between protein cost minimization and response time minimization. Response times are minimized by sparse transcriptional control while protein costs are minimized by pervasive transcriptional control. (A) Time course profiles of sparse transcriptional control show adjustments in enzyme concentrations (y-axis) in response to changes in required pathway output at time points 10 and 20. (B) In case of pervasive transcriptional control, protein biosynthetic capacities and the amount of protein to be produced determine whether there is a concomitant, partial sequential or fully sequential induction of enzymes along a pathway (from left to right). Graphs indicate optimal time course profiles of enzymes during the activation of a metabolic pathway for increasing amounts of protein to be synthesized. The corresponding optimal operon distribution is shown below the time course profiles. Abbreviations: a.u., arbitrary units.
of available solvers). Commonly, the resulting optimization problems in systems biology are solved with direct and deterministic methods by nonlinear programming (NLP) solvers in an either sequential, simultaneous or quasi-sequential approach. Those approaches are characterized depending on whether control variables are discretized and then state variables are simulated in two layers (sequential [24]), control and state variables are both discretized simultaneously (simultaneous, one layer [25]) or as mixed form to combine advantages of both approaches (quasi-sequential [26]). For the described pathway activation problems (section ‘Optimal activation patterns of metabolic pathways’ and Table 1) at first simple solving strategies were used as in the work of Klipp et al. [14], where the control variable is discretized by only a few number of switches and an otherwise constant expression of enzymes. The resulting optimization problems have been solved for linear pathways with an unspecified gradient-based method, and for a skeleton model of the central metabolism of yeast, a genetic algorithm was used to determine the optimal time course of switching points and expression values of enzymes. More recent work like Bartl et al. [15] used quasi-sequential approaches for a much more fine-grained determination of optimal control variables and the investigation of many parameterizations of the activation models. While those deterministic and gradient-based methods enable a fast convergence to local optimal solutions of NLPs, global and stochastic methods are more robust to identify the global optimum since randomization of variables broadly covers the solution space but increases computation time. The work of de Hijas-Liste et al. [16] showed that hybrid models of local and global methods can be useful for dynamic optimization problems to increase the
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Figure 2. Positions of key regulated enzymes. (A–C) Key (regulated) enzymes for different pathway topologies.Key enzymes indicated by red arrows. Lines ending in bars indicate feedback inhibition. (D) Further factors influencing the establishment of transcriptional control over an enzyme in a metabolic pathway. Lines ending in bars indicate antagonistic factors, whereas arrows indicate promoting factors.
performance and to enable multiobjective dynamic optimization by running multiple single-objective optimizations. In a recent study by Nimmegeers et al. [27], methods of parametric uncertainty propagation were used to identify strategies for robust metabolic control of biological networks, which is especially important in the industrial use of microorganisms. Further developments in the solution of dynamic optimization problems can be expected by approaches like multiple shooting where time intervals are separately optimized and parallel computing can easily be implemented to increase the performance of solvers [28]. Also, the usability of toolboxes is increased by public available frameworks like AMIGO2 [29], APMonitor [30] or GAMS [31], as well as a guide for solver selection [23], and makes dynamic optimization applicable for nonexpert users in the field of systems biology.
Biological implications of optimality principles The above-described optimality principles for the regulation of metabolic pathways have manifold implications for the evolution of regulatory networks controlling metabolism both concerning optimal control points and the genomic organization of metabolic pathways. Regarding optimal control points, evolution of regulatory networks controlling metabolic pathways is centered on enzymes that provide the best control of pathway flux [32]. This observation has primarily been used to validate optimality principles by showing that enzymes catalyzing reactions that correspond to optimal points of control are more often targeted by transcription factors and hence have longer promoter regions [21,22]. In particular, time hierarchies in the induction of enzymes belonging to a metabolic pathway have been suggested to strongly influence the genomic organization of enzymes catalyzing metabolic pathways [15] and protein complexes [19]. While the concomitant activation of all enzymes within a pathway is optimally achieved by a placement of all enzymes belonging to a pathway in the same operon, a time-hierarchy in induction requires the distribution across several operons. The protein biosynthetic capacity of the cell as well as the abundance of the individual enzymes represent important physiological constraints influencing the optimal operonic distribution [15]. Since protein biosynthetic capacity is strongly influenced by endogeneous
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Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
(e.g. ribosomal RNA copy number) as well as exogenous factors (e.g. nutrient availability), these factors contribute to the optimality of distinct operonic distributions of enzymes in the course of evolution of an organism. Intriguingly, while many advantages of the formation of operons such as reduction of expression noise or efficient co-regulation of pathways have been proposed [33], the requirement to precisely time the sequential induction of enzymes along a pathway represents an evolutionary advantage of the dissolution of operons rather than just random drift as the major causative factor in operon decline [34]. This is apparent for protein complexes which require their subunits in a precisely defined stoichiometry to build up functional complexes, thus making a distribution in a single operon optimal to avoid incomplete complexes due to expression noise. However, also for protein complexes, a hierarchic induction of subunits can be optimal to maximize the amount of available functional complex: due to the pathway-like assembly of the protein complex, similar induction strategies as for pathways are optimal [19]. This observation provides an explanation why for some very abundant protein complexes, such as RNA polymerase, a distribution across several operons is optimal. The observation that time hierarchies are important solutions for inducing metabolic pathways as well as the production of protein complexes suggests that similar principles might be applicable to the induction of any kind of functionally related proteins required during the response to an environmental stimulus. Optimality principles in the regulation of metabolism are not only relevant for evolutionary phenomena, but also for understanding pathogens and devising effective treatments against them. The knowledge about the interplay of pathway regulation and genomic organization such as promoter lengths or operon structure can be used in a reverse approach to identify those enzymes within a pathogen that are crucial points of pathway control and whose deregulation could open up new avenues to countering pathogenesis. An example is the observation of the link between highly regulated enzymes and toxic intermediates, which suggests an overexpression of the upstream controlling enzymes or down-regulation of the downstream enzymes to induce a cytotoxic effect by accumulation of toxic intermediates [22]. Especially in fungi, where therapeutic agents are limited due to their similarity to higher organisms, this provides new avenues for identifying drug targets. An example is the ergosterol biosynthesis where not only the disruption of synthesis is discussed but also a potential deregulation by an overexpression of key enzymes to accumulate toxic intermediates [22]. Other fields of application of dynamic optimization approaches are metabolic engineering and synthetic biology. In metabolic engineering, which also involves the introduction of foreign metabolic pathways into model organisms, identifying optimal programs of pathway regulation can help to identify regulatory programs that can be employed to increase the yield or minimize production time while preventing the accumulation of potentially toxic intermediates [35–37]. Furthermore, in synthetic biology, optimal control programs are identified to design synthetic circuits for the control of gene expression and to use cells as biological machines [38,39].
Perspectives: optimality principles on the whole-cell level A short-coming of dynamic optimization approaches is the restriction to rather small model systems, often just comprising individual pathways. However, environmental transitions often require a precisely timed readjustment of a large number of cellular processes that can only be captured in limited detail by models that are amenable to dynamic optimization. Thus, similar to the consideration of abstract metabolic pathways, simplified models of cells have been used for the study of optimal control problems on a organism-wide scale [40–42]. While these models are still limited in scope, they have already provided important insights into optimality principles underlying growth-associated phenomena including the interdependency between growth rate, ribosomal content and cell size [42], the growth rate-dependent partitioning of the proteome into different functional categories [41] as well as the influence of environment-dependent protein costs on the optimal choice between alternative pathways for nutrient catabolism [43,44]. In particular, the recent availability of large-scale quantitative proteomic data [45] is expected to provide an important cornerstone for these approaches since this kind of data allows, for the first time, to precisely quantify the actual proteomic cost of specific cellular operations which is at the heart of many of the above-described optimality principles. These approaches are complemented by optimization approaches that aim to model cells at various molecular levels either through use of uniform modeling principles [46,47] or by integration of different modeling approaches into a coherent framework [48]. We expect that together with drastically simplified models of cells as described above, these models will drastically expand our knowledge of how optimality principles have shaped the evolution of biological networks and how they influence the different levels of regulatory interactions in cells.
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Abbreviations CVP, control vector parameterization; MCA, metabolic control analysis; NLP, nonlinear programming.
Funding We acknowledge support by the Deutsche Forschungsgemeinschaft to C.K. [KA 3541/3, CRC/TR 124 FungiNet B2 and Excellence Cluster ‘Inflammation at Interfaces’, EXC306].
Competing Interests The Authors declare that there are no competing interests associated with the manuscript.
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Review Article
Deciphering the regulation of metabolism with dynamic optimization: an overview of recent advances Jan Ewald1, Martin Bartl1 and Christoph Kaleta2 1
Research Group Theoretical Systems Biology, Department of Bioinformatics, Friedrich-Schiller-Universität Jena, Jena, Germany and 2Research Group Medical Systems Biology, Christian-Albrechts-Universität zu Kiel, Kiel, Germany Correspondence: Jan Ewald ( [email protected])
Understanding optimality principles shaping the evolution of regulatory networks controlling metabolism is crucial for deriving a holistic picture of how metabolism is integrated into key cellular processes such as growth, adaptation and pathogenicity. While in the past the focus of research in pathway regulation was mainly based on stationary states, more recently dynamic optimization has proved to be an ideal tool to decipher regulatory strategies for metabolic pathways in response to environmental cues. In this short review, we summarize recent advances in the elucidation of optimal regulatory strategies and identification of optimal control points in metabolic pathways. We discuss biological implications of the discovered optimality principles on genome organization and provide examples how the derived knowledge can be used to identify new treatment strategies against pathogens. Furthermore, we briefly discuss the variety of approaches for solving dynamic optimization problems and emphasize whole-cell resource allocation models as an important emerging area of research that will allow us to study the regulation of metabolism on the whole-cell level.
Development of approaches to understand the regulation of metabolic pathways
Received: 21 April 2017 Revised: 21 June 2017 Accepted: 29 June 2017 Version of Record published: 28 July 2017
A precise control of metabolism in response to external and internal cues is of central importance for growth, development and survival of unicellular and multicellular organisms. However, the complexity of the metabolic networks as well as the myriad of potential environmental conditions most organisms are facing make the determination of optimal targets of pathway regulation a challenging task. Since metabolic pathways are often considered as chemical reaction chains, the first ideas about optimal points of control in metabolic pathways focused on the slowest reaction as the rate-limiting or -determining step, an idea originating from chemistry [1], and formulated in the concept of pacemaker enzymes catalyzing the corresponding reactions [2,3]. In a similar manner, other qualitative approaches to pathway regulation like the first committed step, which is the first irreversible reaction in a pathway, consider specific enzyme traits that determine flux controlling steps [3,4]. In contrast with this, metabolic control analysis (MCA) was developed to quantify the control of each enzyme across a pathway on the flux and discovered that, in many biological systems, multiple enzymes control pathway flux [5,6]. While this approach allows a much more fine-grained analysis of the distribution of control throughout a metabolic pathway, it is limited in the sense that it is focusing on steady-state conditions neglecting the dynamics of varying conditions. Additionally, MCA has been developed to deterministically investigate the effect of small changes and therefore fails to estimate the effect of large deviations, for example, after a change in environmental conditions that requires large adjustments in pathway flux [7].
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Especially, the consideration of dynamic conditions is of high relevance since fluxes through metabolic pathways often need to be readjusted following a change in conditions or cyclic events like the cell cycle or circadian processes. The time frame for this environmental changes and the subsequent metabolic adjustments can be within seconds or minutes (e.g. heat-shock) or within hours or days (e.g. cell cycle). Depending on the time frame, the cell has to optimize the different levels of regulation like transcriptional regulation (rather slow) or post-translational modification (rather fast) to minimize its response time, which is of major evolutionary advantage especially in fluctuating environments [8]. It is important to note that this way of reasoning is distinct from approaches such as flux balance analysis [9] that uses linear optimization to identify the most likely flux through a metabolic network based on an a priori formulated evolutionary objective [9] and thus is not primarily concerned with pathway control. Complementary to MCA, approaches based on dynamic optimization problems, which have their roots in solving engineering problems [10], have also been used to identify optimal programs of metabolic pathway regulation. These approaches are used to investigate optimization problems in which a time-dependent behavior needs to be optimized after a perturbation, for instance, by the minimization of response times or the minimization of deviation of a system from a preferred target state. The interest in these approaches has particularly been emphasized by studies of short-term gene expression responses to environmental perturbations in which it has been found that after such perturbations, often very complex regulatory programs are induced like in metabolic pathways [11,12] or flagellum assembly pathways [13]. This shows that time-dependent principles in pathway regulation are of high importance in conditions in which cells are not in steady state. In this short review, we summarize studies that have used dynamic optimization to elucidate the optimal timing patterns in the control of metabolic networks. While we focus mainly on regulatory strategies for pathway activation and control of product dilution, we further discuss their functional implications such as the genomic organization of genes. Beyond this, we briefly elaborate solution approaches for dynamic optimization problems and an outlook on models investigating optimality principles on the whole-cell level.
Determinants of optimal regulatory programs controlling metabolism defined by dynamic optimization Optimal activation patterns of metabolic pathways In principle, the optimal solution during the response to an environmental stimulus is to produce all proteins that are required for this response at once. However, this approach is often not feasible due to physiological constraints such as the immediate availability of building blocks for protein biosynthesis, a limited cellular volume, general constraints in protein biosynthetic capacity and other physiological constraints. The principle that functionally related proteins are induced in a precisely timed manner was shown for the first time for the flagellum assembly pathway [13]. In this experimental study, it was shown that the proteins along the assembly pathways were induced in the same sequential order as they were required during flagellum assembly [13]. In a seminal work, Klipp et al. [14] used dynamic optimization to determine an optimal time hierarchy of enzymes along a pathway that minimizes the time to transform the substrate of the pathway into the product with a constraint on the total amount of available enzymes. In particular, they found that in contrast with a strategy in which all enzymes were activated at once, an induction in the order of their sequence in the pathway was optimal for minimizing the transition time required to synthesize the product [14]. Such a sequential activation of enzymes along a pathway is referred to as ‘Just-in-time-activation’ in reference to an industrial production strategy whereby components are produced just as they are needed. The utilization of such a sequential activation strategy was later confirmed experimentally to occur in several amino acid biosynthetic pathways of Escherichia coli after deprivation of the corresponding amino acid [11] as well as in many metabolic pathways in yeast [12]. Inspired by these findings, later work focused on the factors as well as different objectives of pathway activation, that determine optimal activation strategies (see Table 1). A common observation is that the sequential activation is a special case among optimal activation strategies and is strongly influenced by cellular protein biosynthetic capacity [15], pathway topology [16] and kinetic properties of enzymes [15,17,18]. In particular, Bartl et al. [15] showed that if all enzymes within a pathway are under transcriptional control, protein biosynthetic constraints can either lead to the optimality of a concomitant activation of all enzymes of a pathway, a partial sequential activation of parts of a pathway or a fully sequential activation of the individual enzymes of the pathway. Moreover, it was found that the catalytic efficiency of an enzyme has a strong influence on its
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Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
Table 1 Overview of studies using dynamic optimization to decipher principles behind the control of metabolism
Pathway activation
Reference
Objective
Klipp et al. [14]
Minimum of transition time
Zaslaver et al. [11]
Minimum of (1) activation time and (2) enzyme costs Minimum of (1) activation time and (2) enzyme costs Minimum of transition/ operating time (1) Maximum of product in finite time
Oyarzun et al. [17]
Bartl et al. [18] Bartl et al. [15]
de Hijas-Liste el al. [16]
Nimmegeers et al. [27]
Pathway regulation
Wessely et al. [20]
de Hijas-Liste et al. [21]
Resource allocation
Multiple objectives: transition time, enzyme costs, intermediate accumulation, survival time Minimum of (1) activation time and (2) enzyme costs
Minimum of (1) regulation and (2) enzyme costs Minimum of (1) regulation and (2) enzyme costs
Ewald et al. [22]
Minimum of (1) regulation and 2) enzyme costs
Molenaar et al. [42]
Maximum of growth rate
Giordano et al. [40]
Maximum of growth rate
Investigated determinants
Solution approach
Pathway length, central metabolism (yeast) Enzyme costs
Numerical: maximum number of switches, genetic algorithm
Enzyme costs, kinetics
Analytical: Pontryagin’s Minimum Principle Numerical: TOMLAB Analytical: Pontryagin’s Minimum Principle Numerical: extension of quasi-sequential approach [49] Numerical: hybrid of CVP and eSS scatter search solver
Kinetics Enzyme synthesis capacity Multiple objectives, branching topology, central metabolism (yeast)
Numerical: not specified, analytical: prove of hierarchy
Linear and glycolysis like pathways, multiple objectives, parametric uncertainty
Numerical: Pomodoro toolbox parametric uncertainty propagation: linearization, sigma points, polynomial chaos expansion
Enzyme costs
Numerical: extension of quasi-sequential approach [49] Numerical: CVP approach
Protein biosynthetic rates, feedback inhibition, complex pathway topologies Toxicity of intermediates, kinetics Alternative pathways for substrate acquisition, two substrate species Feedback mechanisms
Numerical: extension of quasi-sequential approach [49] Numerical: General Algebraic Modeling System (GAMS), KNITRO solver Analytical: infinite horizon maximum principle Numerical: Bocop toolbox
position in the activation sequence, with less efficient enzymes synthesized first and highly efficient enzymes synthesized later compared with neighboring enzymes [15], an observation that allowed to explain discrepancies in the activation sequence of specific metabolic pathways in earlier experimental observation [11]. Additionally, similar optimality principles were found in the assembly of protein complexes [19] pointing toward sequential activation strategies to be of importance for the regulation of functionally related proteins also beyond metabolism, as already suggested by earlier work on sequential activation strategies in flagellum assembly [13].
Regulatory programs controlling product dilution Beyond the activation of a metabolic pathway also the fine-tuning of pathway flux due to changes in the required synthesis of pathway product or substrate availability is of importance. In such a context, rather than minimizing the time required to form a certain amount of product, the minimization of the change in protein concentrations required to achieve a change in pathway flux as well as the minimization of protein investment
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for maintaining pathway flux are central objectives of optimization (see Table 1). Assuming a five-step linear metabolic pathway with randomized product dilutions modeled as time-varying outflow to simulate different growth phases or environmental changes, Wessely et al. [20] identified optimal time courses in enzyme concentrations for controlling product concentrations under non-steady-state conditions. It could be shown that depending on a balance between protein costs and the speed at which fluxes within a pathway need to be adjusted, two types of regulatory programs are employed — sparse transcriptional regulation and pervasive transcriptional regulation (see Figure 1). Sparse transcriptional regulation refers to the control of only key enzymes within a pathway (the remainder being constitutively expressed) that are sufficient to precisely control pathway flux in accordance with cellular requirements, while pervasive transcriptional regulation corresponds to a transcriptional control of all enzymes within a pathway. As a main result, it was found that the first and last enzymes of a pathway are the main targets of transcriptional regulation in a linear metabolic pathway with irreversible Michaelis–Menten kinetics (see Figure 1A). Sparse transcriptional control minimizes the time required to adjust the flux through a pathway since only the concentrations of key enzymes need to be adjusted to regulate pathway flux and late intermediates are accumulated to buffer product dilution [20]. However, the remaining enzymes of the pathway are constitutively expressed which increases their overall abundance in comparison with a case with pervasive control of all enzymes in a pathway [20]. In addition, more complex models of, for example, branched pathway topologies show that two pathways that converge via one intermediate entail the same regulatory program. Similar to linear pathways, the first enzymes of both branches and the last enzyme beyond the convergent reaction are controlled (see Figure 2A) [21]. For pathways with a branch into two pathways with separate products after a common intermediate, it was found that the control of the initial reaction of the entire pathway, the reactions after the branch and the terminal reaction is optimal (see Figure 2B) [21]. The consideration of feedback mechanisms by the product, like the inhibition of the first enzyme via post-translational modifications, engenders optimal programs where the first enzyme is less involved in the control of the flux through the pathway since product inhibition can substitute the control (see Figure 2C) [21]. An important feature of the regulatory programs that were identified included the accumulation of intermediates upstream of transcriptional-regulated proteins. While in previous studies [20,21] the unlimited accumulation to toxic levels was prevented through a constraint on total metabolite concentrations, differences in the toxicity of metabolic intermediates are not considered. Based on an extended model [22] of a linear metabolic pathway also accounting for differences in toxicity of intermediates, it was found that especially enzymes upstream of particularly toxic intermediates and enzymes with substrates with low toxicity were target of regulation (see Figure 2D) [22]. Moreover, accounting for differences in enzyme efficiency, it was observed that regulation often targets the most efficient enzyme in a pathway (in terms of high catalytic activity or high substrate affinity) (see Figure 2D) [22]. Especially, the latter observation questions the often quoted assumption that the least efficient enzyme within a pathway should be target of regulation [2]. Indeed, highly efficient enzymes are more suitable targets of regulation since they are usually present in smaller abundance and changes in flux require smaller (and hence faster) absolute changes in protein abundance compared with less efficient enzymes.
Recent approaches for solving dynamic optimization problems in systems biology While we are focusing on the application of dynamic optimization problems to solve optimal control problems in metabolism, we also briefly cover the variety of approaches for solving these problems. Due to timedependent variables and the non-linearity in modeling of enzyme kinetics and objective functions, analytical solutions could often not be derived. On the other hand, numerical approaches are computationally very expensive through the discretization of time-dependent variables. For dynamic pathway activation models, some analytical solutions were derived for initial or simplified models by limiting the numbers of enzymes and switches [14], or by using Pontryagin’s Maximum/Minimum Principle [17,18]. To tackle extended models with different topologies, kinetics and more complex objective functions, numerical approaches were used to solve the dynamic optimization problem (see Table 1). Numerical approaches have in common that they transform the continuous problem to a finite optimization problem by discretization (also called collocation or parameterization) of variables, and a variety of approaches and solvers can be used to determine optimal solutions (see http://plato.asu.edu/guide.html [23] for an overview
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Biochemical Society Transactions (2017) 45 1035–1043 DOI: 10.1042/BST20170137
Figure 1. Trade-offs in the evolution of regulatory networks controlling metabolism. Balance between protein cost minimization and response time minimization. Response times are minimized by sparse transcriptional control while protein costs are minimized by pervasive transcriptional control. (A) Time course profiles of sparse transcriptional control show adjustments in enzyme concentrations (y-axis) in response to changes in required pathway output at time points 10 and 20. (B) In case of pervasive transcriptional control, protein biosynthetic capacities and the amount of protein to be produced determine whether there is a concomitant, partial sequential or fully sequential induction of enzymes along a pathway (from left to right). Graphs indicate optimal time course profiles of enzymes during the activation of a metabolic pathway for increasing amounts of protein to be synthesized. The corresponding optimal operon distribution is shown below the time course profiles. Abbreviations: a.u., arbitrary units.
of available solvers). Commonly, the resulting optimization problems in systems biology are solved with direct and deterministic methods by nonlinear programming (NLP) solvers in an either sequential, simultaneous or quasi-sequential approach. Those approaches are characterized depending on whether control variables are discretized and then state variables are simulated in two layers (sequential [24]), control and state variables are both discretized simultaneously (simultaneous, one layer [25]) or as mixed form to combine advantages of both approaches (quasi-sequential [26]). For the described pathway activation problems (section ‘Optimal activation patterns of metabolic pathways’ and Table 1) at first simple solving strategies were used as in the work of Klipp et al. [14], where the control variable is discretized by only a few number of switches and an otherwise constant expression of enzymes. The resulting optimization problems have been solved for linear pathways with an unspecified gradient-based method, and for a skeleton model of the central metabolism of yeast, a genetic algorithm was used to determine the optimal time course of switching points and expression values of enzymes. More recent work like Bartl et al. [15] used quasi-sequential approaches for a much more fine-grained determination of optimal control variables and the investigation of many parameterizations of the activation models. While those deterministic and gradient-based methods enable a fast convergence to local optimal solutions of NLPs, global and stochastic methods are more robust to identify the global optimum since randomization of variables broadly covers the solution space but increases computation time. The work of de Hijas-Liste et al. [16] showed that hybrid models of local and global methods can be useful for dynamic optimization problems to increase the
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Figure 2. Positions of key regulated enzymes. (A–C) Key (regulated) enzymes for different pathway topologies.Key enzymes indicated by red arrows. Lines ending in bars indicate feedback inhibition. (D) Further factors influencing the establishment of transcriptional control over an enzyme in a metabolic pathway. Lines ending in bars indicate antagonistic factors, whereas arrows indicate promoting factors.
performance and to enable multiobjective dynamic optimization by running multiple single-objective optimizations. In a recent study by Nimmegeers et al. [27], methods of parametric uncertainty propagation were used to identify strategies for robust metabolic control of biological networks, which is especially important in the industrial use of microorganisms. Further developments in the solution of dynamic optimization problems can be expected by approaches like multiple shooting where time intervals are separately optimized and parallel computing can easily be implemented to increase the performance of solvers [28]. Also, the usability of toolboxes is increased by public available frameworks like AMIGO2 [29], APMonitor [30] or GAMS [31], as well as a guide for solver selection [23], and makes dynamic optimization applicable for nonexpert users in the field of systems biology.
Biological implications of optimality principles The above-described optimality principles for the regulation of metabolic pathways have manifold implications for the evolution of regulatory networks controlling metabolism both concerning optimal control points and the genomic organization of metabolic pathways. Regarding optimal control points, evolution of regulatory networks controlling metabolic pathways is centered on enzymes that provide the best control of pathway flux [32]. This observation has primarily been used to validate optimality principles by showing that enzymes catalyzing reactions that correspond to optimal points of control are more often targeted by transcription factors and hence have longer promoter regions [21,22]. In particular, time hierarchies in the induction of enzymes belonging to a metabolic pathway have been suggested to strongly influence the genomic organization of enzymes catalyzing metabolic pathways [15] and protein complexes [19]. While the concomitant activation of all enzymes within a pathway is optimally achieved by a placement of all enzymes belonging to a pathway in the same operon, a time-hierarchy in induction requires the distribution across several operons. The protein biosynthetic capacity of the cell as well as the abundance of the individual enzymes represent important physiological constraints influencing the optimal operonic distribution [15]. Since protein biosynthetic capacity is strongly influenced by endogeneous
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(e.g. ribosomal RNA copy number) as well as exogenous factors (e.g. nutrient availability), these factors contribute to the optimality of distinct operonic distributions of enzymes in the course of evolution of an organism. Intriguingly, while many advantages of the formation of operons such as reduction of expression noise or efficient co-regulation of pathways have been proposed [33], the requirement to precisely time the sequential induction of enzymes along a pathway represents an evolutionary advantage of the dissolution of operons rather than just random drift as the major causative factor in operon decline [34]. This is apparent for protein complexes which require their subunits in a precisely defined stoichiometry to build up functional complexes, thus making a distribution in a single operon optimal to avoid incomplete complexes due to expression noise. However, also for protein complexes, a hierarchic induction of subunits can be optimal to maximize the amount of available functional complex: due to the pathway-like assembly of the protein complex, similar induction strategies as for pathways are optimal [19]. This observation provides an explanation why for some very abundant protein complexes, such as RNA polymerase, a distribution across several operons is optimal. The observation that time hierarchies are important solutions for inducing metabolic pathways as well as the production of protein complexes suggests that similar principles might be applicable to the induction of any kind of functionally related proteins required during the response to an environmental stimulus. Optimality principles in the regulation of metabolism are not only relevant for evolutionary phenomena, but also for understanding pathogens and devising effective treatments against them. The knowledge about the interplay of pathway regulation and genomic organization such as promoter lengths or operon structure can be used in a reverse approach to identify those enzymes within a pathogen that are crucial points of pathway control and whose deregulation could open up new avenues to countering pathogenesis. An example is the observation of the link between highly regulated enzymes and toxic intermediates, which suggests an overexpression of the upstream controlling enzymes or down-regulation of the downstream enzymes to induce a cytotoxic effect by accumulation of toxic intermediates [22]. Especially in fungi, where therapeutic agents are limited due to their similarity to higher organisms, this provides new avenues for identifying drug targets. An example is the ergosterol biosynthesis where not only the disruption of synthesis is discussed but also a potential deregulation by an overexpression of key enzymes to accumulate toxic intermediates [22]. Other fields of application of dynamic optimization approaches are metabolic engineering and synthetic biology. In metabolic engineering, which also involves the introduction of foreign metabolic pathways into model organisms, identifying optimal programs of pathway regulation can help to identify regulatory programs that can be employed to increase the yield or minimize production time while preventing the accumulation of potentially toxic intermediates [35–37]. Furthermore, in synthetic biology, optimal control programs are identified to design synthetic circuits for the control of gene expression and to use cells as biological machines [38,39].
Perspectives: optimality principles on the whole-cell level A short-coming of dynamic optimization approaches is the restriction to rather small model systems, often just comprising individual pathways. However, environmental transitions often require a precisely timed readjustment of a large number of cellular processes that can only be captured in limited detail by models that are amenable to dynamic optimization. Thus, similar to the consideration of abstract metabolic pathways, simplified models of cells have been used for the study of optimal control problems on a organism-wide scale [40–42]. While these models are still limited in scope, they have already provided important insights into optimality principles underlying growth-associated phenomena including the interdependency between growth rate, ribosomal content and cell size [42], the growth rate-dependent partitioning of the proteome into different functional categories [41] as well as the influence of environment-dependent protein costs on the optimal choice between alternative pathways for nutrient catabolism [43,44]. In particular, the recent availability of large-scale quantitative proteomic data [45] is expected to provide an important cornerstone for these approaches since this kind of data allows, for the first time, to precisely quantify the actual proteomic cost of specific cellular operations which is at the heart of many of the above-described optimality principles. These approaches are complemented by optimization approaches that aim to model cells at various molecular levels either through use of uniform modeling principles [46,47] or by integration of different modeling approaches into a coherent framework [48]. We expect that together with drastically simplified models of cells as described above, these models will drastically expand our knowledge of how optimality principles have shaped the evolution of biological networks and how they influence the different levels of regulatory interactions in cells.
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Abbreviations CVP, control vector parameterization; MCA, metabolic control analysis; NLP, nonlinear programming.
Funding We acknowledge support by the Deutsche Forschungsgemeinschaft to C.K. [KA 3541/3, CRC/TR 124 FungiNet B2 and Excellence Cluster ‘Inflammation at Interfaces’, EXC306].
Competing Interests The Authors declare that there are no competing interests associated with the manuscript.
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