Sequential Metabolic Phases as a Means to Optimize Cellular Output in a Constant Environment

Temporal changes of gene expression are a well-known regulatory feature of all cells, which is commonly perceived as a strategy to adapt the proteome to varying external conditions. However, temporal (rhythmic and non-rhythmic) changes of gene expression are also observed under virtually constant external conditions. Here we hypothesize that such changes are a means to render the synthesis of the metabolic output more efficient than under conditions of constant gene activities. In order to substantiate this hypothesis, we used a flux-balance model of the cellular metabolism. The total time span spent on the production of a given set of target metabolites was split into a series of shorter time intervals (metabolic phases) during which only selected groups of metabolic genes are active. The related flux distributions were calculated under the constraint that genes can be either active or inactive whereby the amount of protein related to an active gene is only controlled by the number of active genes: the lower the number of active genes the more protein can be allocated to the enzymes carrying non-zero fluxes. This concept of a predominantly protein-limited efficiency of gene expression clearly differs from other concepts resting on the assumption of an optimal gene regulation capable of allocating to all enzymes and transporters just that fraction of protein necessary to prevent rate limitation. Applying this concept to a simplified metabolic network of the central carbon metabolism with glucose or lactate as alternative substrates, we demonstrate that switching between optimally chosen stationary flux modes comprising different sets of active genes allows producing a demanded amount of target metabolites in a significantly shorter time than by a single optimal flux mode at fixed gene activities. Our model-based findings suggest that temporal expression of metabolic genes can be advantageous even under conditions of constant external substrate supply

Inferring causal molecular networks: empirical assessment through a community-based effort

Inferring molecular networks is a central challenge in computational biology. However, it has remained unclear whether causal, rather than merely correlational, relationships can be effectively inferred in complex biological settings. Here we describe the HPN-DREAM network inference challenge that focused on learning causal influences in signaling networks. We used phosphoprotein data from cancer cell lines as well as in silico data from a nonlinear dynamical model. Using the phosphoprotein data, we scored more than 2,000 networks submitted by challenge participants. The networks spanned 32 biological contexts and were scored in terms of causal validity with respect to unseen interventional data. A number of approaches were effective and incorporating known biology was generally advantageous. Additional sub-challenges considered time-course prediction and visualization. Our results constitute the most comprehensive assessment of causal network inference in a mammalian setting carried out to date and suggest that learning causal relationships may be feasible in complex settings such as disease states. Furthermore, our scoring approach provides a practical way to empirically assess the causal validity of inferred molecular networks

Inferring causal molecular networks: empirical assessment through a community-based effort

It remains unclear whether causal, rather than merely correlational, relationships in molecular networks can be inferred in complex biological settings. Here we describe the HPN-DREAM network inference challenge, which focused on learning causal influences in signaling networks. We used phosphoprotein data from cancer cell lines as well as in silico data from a nonlinear dynamical model. Using the phosphoprotein data, we scored more than 2,000 networks submitted by challenge participants. The networks spanned 32 biological contexts and were scored in terms of causal validity with respect to unseen interventional data. A number of approaches were effective, and incorporating known biology was generally advantageous. Additional sub-challenges considered time-course prediction and visualization. Our results suggest that learning causal relationships may be feasible in complex settings such as disease states. Furthermore, our scoring approach provides a practical way to empirically assess inferred molecular networks in a causal sense

Integrierte Modelle metabolischer und regulatorischer Netzwerke

Two cellular subsystems are the metabolic network and the gene regulatory network. In systems biology they have mostly been modelled in isolation with ordinary differential equations (ODEs) or with tailored formalisms as e.g. constraint-based methods for metabolism or logical networks for gene regulation. In reality the two systems are strongly interdependent. For mathematical modelling the integration is a challenge and a variety of different approaches has been proposed. Long term alterations in metabolism result from changes in gene expression, which determines the production of enzymes. This transcriptional control can adjust the metabolic network to changes in the environment or the requirements of the cell. In fact, the cell cycle is connected to cyclic changes in metabolism, so-called metabolic cycling, but alterations are also observed in non-proliferating cells in a constant environment. A mathematical model to describe and explain alterations in metabolism will be proposed here. At first, a resource allocation model for the enzymes in a metabolic network is developed and integrated into a constraint-based model of metabolism in Chap. 3. The reaction rates are bounded depending on the availability of enzymes, which in turn is determined by the overall distribution of the limited resources. In Chap. 4, this model is used to test the hypothesis that metabolic alterations are a means of the cell to achieve the required production of metabolic output most efficiently. First a toy model is analysed and then the method is applied to a core metabolic network of the central carbon metabolism. The tasks of this metabolic network are the production of biomass precursors as well as constantly providing a minimum of energy and anti-oxidants. The mathematical model gives a mixed integer linear optimisation problem with a few quadratic constraints and a quadratic objective function. Instead of searching for a single flux distribution, a feasible solution corresponds here to a sequence of several flux distributions together with the time that is spent in each of them. The consecutive usage of these flux distributions during the associated time spans yields the required output. The objective is the minimisation of the total time needed. The computations demonstrate that switching between several flux distributions allows producing the output in a significantly shorter time span, compared to an optimal single flux distribution. In a toy model we could identify the relationship between the model parameters and the results concerning the efficiency of static versus sequential flux distributions. Such a comprehensive analysis is not possible for the large number of parameters in our core metabolic network. To make sure that the confirmation of the hypothesis is not restricted to a minor region in the parameter space of the resource allocation model, we perturbed the parameters randomly and repeated all computations. This empirical analysis showed that the significant gain in performance is a robust feature of the model. From the mathematical point of view the proposed resource allocation model defines for each gene expression state a flux space from which a flux distribution can be chosen. This flux space is in general not linear and not convex, which turns out to depend on the space of all possible gene expression states. In our model the genes regulate the enzyme concentrations in an on-off manner, only determining the active and inactive parts of metabolism. Furthermore, certain groups of genes are regulated together as functional units. As a consequence, the enzyme concentrations cannot be perfectly adjusted to a given flux distribution in this model and it is for this reason that switching can increase the efficiency. A simpler model of resource allocation, which is solely based on molecular crowding, has been proposed before in the literature. It allows distributing the resources to perfectly match any given flux distribution and switching is then not necessary to obtain the minimal production time. In contrast to such a resource allocation model, our modelling assumptions and computational results suggest a design principle, where the optimal adjustment to given conditions and requirements is not achieved by fine-tuning of enzyme concentrations, but by switching between different flux distributions, which are only roughly determined by transcriptional control and which do not perfectly match one certain condition or requirement. In terms of geometry, the difference lies in the convexity of the flux space. If it is convex, minimal production time can always be achieved with a single flux distribution. To characterise a set of flux distributions sufficient to constitute an optimal sequence, the flux space of the network without the resource allocation model is considered in Chap. 3. The corresponding polytope allows characterising a finite subset of the flux space in terms of decomposability, a notion which is closely related to elementary modes. For any output requirements, an optimal sequence can be constituted from this finite set of flux distributions. In practice, solving the optimisation problem that was derived from the modelling approach as well as computing the sufficient finite subset, is not tractable for large networks. Also divide and conquer strategies are not promising to obtain optimal solutions in general, a counterexample is given in Chap. 6. Alternative computational methods to obtain optimal or approximative optimal solutions are then presented. The gene regulatory network behind the metabolic genes is not fully considered in the resource allocation model of Chap. 3. Only some constraints are added in the application to the core metabolic network in order to exclude unrealistic patterns of gene expression. Incorporating more information about the gene regulation into the computational model is in fact improving the tractability, because the search space is reduced. A sufficiently small search space of gene expression sequences gives the possibility to perform a more precise and extensive analysis using an alternative computational approach. In Chap. 5, the perturbations of model parameters, as applied to the core metabolic network to verify the robustness, are considered in general. From the mathematical point of view, the linear constraints that bound the flux space are perturbed. The consequences on the geometry of the flux space and on the objective value of an optimisation problem over this flux space are analysed and an effect is discovered, which is surprising at first sight. If the bounds on the reaction rates are perturbed individually, without a bias for increase or decrease, the expected objective value of a given linear optimisation problem is decreased in expectation. This effect emerges from the representation of the flux space. In particular redundancy of the constraints plays a crucial role. The modelling and the analysis of the dynamics of gene regulatory networks with so-called logical networks is a common discrete approach. Logical networks are often represented by logical functions, which have the advantage of being mathematical objects that can be given in a natural and easily understandable format, namely Boolean expressions. In Chap. 7, a method is presented to obtain a short and well readable representation of a given logical function. It is based on the minimisation of Boolean expressions, but is designed for multi-valued logical functions in particular. All possible dynamics of a logical network can be represented in the so-called state transition graph. Simply by assigning rates to all edges, which represent the transitions between different states, this directed graph becomes a continuous time Markov chain (CTMC) which we call a stochastic logical network. This modelling approach opens new possibilities for the analysis of quantitative dynamical properties as shown in Chap. 8. In contrast to this abstract model, detailed mechanistic and stochastic models of biochemical reaction systems can be formulated with the chemical master equation, which also defines a CTMC. In fact, these two formalisms can be combined, so that distinct components of the biological system are modelled in much detail by the master equation and other parts on a higher abstraction level as a stochastic logical network. The combined model can focus on certain aspects, capturing related quantitative and stochastic effects, while keeping the overall complexity to a minimum. Finally, Chap. 9 discusses the feedback regulation from metabolism to gene regulation. In an integrated dynamic model of gene regulation and metabolism, this aspect should not be missing. Since constraint-based models neglect the concentrations of metabolites, it is difficult to determine the regulatory feedback to the genes. This problem can be circumvented by only inferring metabolic mediated interactions between genes, in the sense that a switch in gene expression leads to an alteration in the metabolic network, which in turn gives a new regulatory input to the gene network. To this end, a constraint- based approach is proposed and compared to a method from the literature, which is based on metabolic sensitivity analysis. Furthermore, a strategy to derive concentration changes from changes in flux rates and enzyme activities is shortly presented.In der vorliegenden Dissertation wird die Regulation des zellulären Stoffwechsels durch Gene mit mathematischen Modellen beschrieben. Veränderungen in der Expression von Genen, die für die Produktion von Enzymen verantwortlich sind, bewirken längerfristige Umstellungen im Stoffwechsel. Auf diese Weise wird auf der Transkriptionsebene z.B. die Anpassung des Stoffwechsels an äußere Bedingungen oder sich ändernde Bedürfnisse der Zelle gesteuert. Ein Modell der Verteilung von Ressourcen für die Produktion von Enzymen wird in Kap. 3 entwickelt und in ein constraintbasiertes Stoffwechselmodell integriert. Die Flussraten der einzelnen Reaktionen werden dabei von der Menge an verfügbaren Enzymen beschränkt, welche wiederum von der Verteilung der Ressourcen auf das ganze Stoffwechselnetzwerk abhängt. In Kap. 4 wird dieses Modell dann verwendet, um der Hypothese nachzugehen, dass Umstellungen des Stoffwechsels der Zelle dazu dienen könnten, verschiedene benötigte Metaboliten mit höchstmöglicher Effizienz zu produzieren. Zuerst analysieren wir die Effizienz an einem Spielmodell, bevor dann ein Netzwerk des zentralen Kohlenstoffwechsels untersucht wird. In diesem Modell betrachten wir die Produktion von einigen Bausteinen der Biomasse. Zusätzlich wird die permanente Bereitstellung von genügend Energie und Antioxidantien gefordert. Das Ziel ist dabei, die geforderte Produktion von Metaboliten in möglichst kurzer Zeit zu erfüllen. Das mathematische Modell ist ein Optimierungsproblem mit gemischt-ganzzahligen Variablen, linearen und wenigen quadratischen Nebenbedingungen sowie einer quadratischen Zielfunktion. Eine Lösung entspricht einer Abfolge von Flussverteilungen und deren Dauer. Die Berechnungen in Kap. 4 zeigen, dass das Umschalten zwischen verschiedenen Flussverteilungen des Stoffwechsels es ermöglicht, die Biomasse in einer signifikant kürzeren Zeit zu produzieren als es eine einzelne Flussverteilung erlauben würde. Die Robustheit dieser Ergebnisse bezgl. der Parameterwahl wurde empirisch bestätigt. Die mathematischen Eigenschaften des Ressourcenverteilungsmodells werden in Kap. 3 analysiert. Unser Modell geht von einer groben Steuerung der Enzymkonzentrationen auf der Transkriptionsebene aus, was durch binäre Genexpression modelliert wird, die nur festlegt, welche Stoffwechselpfade aktiviert sind und welche nicht. Weiterhin gibt Kap. 3 eine Charakterisierung von bestimmten besonders effizienten Flussverteilungen. Aus dieser endlichen Menge kann immer eine Abfolge zusammengestellt werden kann, die optimal die gegebenen Anforderungen erfüllt, d.h. eine optimale Lösung unseres Optimierungsproblems ist. Für große Netzwerke sind die Optimierungsprobleme, die das Ressourcenverteilungsmodell formuliert, numerisch nicht lösbar. Daher werden in Kap. 6 verschiedene alternative Berechnungsmethoden vorgestellt. In Kap. 5 werden stochastische Störungen der Nebenbedingungen, die den Flussraum beschränken, untersucht. Es zeigt sich, dass hier ein unerwarteter Effekt auftritt. Er wird durch die im mathematischen Sinne nicht eindeutige Darstellung des Stoffwechselmodells durch lineare Nebenbedingungen bestimmt. Zur Modellierung und Untersuchung der Dynamik von genregulatorischen Netzwerken wird oft der Formalismus der sogenannten logischen Netzwerke verwendet. In Kap. 7 wird ein Algorithmus vorgeschlagen, der eine kurze und gut lesbare Darstellung der benötigten logischen Funktionen liefert. Eine Erweiterung von logischen Netzwerken zu einem Markov-Prozess wird in Kap. 8 vorgeschlagen, um stochastische und quantitative Aspekte darzustellen. Exemplarisch wird gezeigt, wie sich dieser Formalismus direkt mit einer Mastergleichung für bestimmte Reaktionen oder regulatorische Interaktionen kombinieren lässt. In einem Ausblick wird in Kap. 9 das Problem der Feedback-Regulation des genregulatorischen Netzwerkes durch den Stoffwechsel behandelt

Dependence of the minimal biomass production time in the base condition (all genes active) on the magnitude of the flux through the maintenance reactions.

<p>The fluxes through the maintenance reactions were increased. GSHox flux was increased up to 50-fold and ATPase flux 4-fold of their normal values. The surface starts at the bottom with the minimal production time of 8h with all genes active and maintenance demand of 0.002, 5 mmol/gDW/h, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118347#pone.0118347.t001" target="_blank">Table 1</a>. Only if ATP consumption by the maintenance reactions is increased by a factor ≥ 2.5, the minimal production time is prolonged. This is due to the fact that fulfillment of the metabolic objectives requires ATP production in all metabolic phases. As long as the responsible reactions are not rate limiting, i.e., their catalyzing enzymes do not operate at the upper flux bound, the rate of ATP synthesis can be increased to balance the additional ATP demand of the maintenance reactions up to an increase to the 2.5-fold of the normal. Below this threshold, only GSSG reduction acts as a bottleneck for biomass production. With 15 mmol/gDW/h of ATP consumption the time for production becomes 13.1 h and if the consumption rate tends towards 18.7 mmol/gDW/h, the total available amount of amino acids has to be allocated to the ATP-producing flux mode, and <i>de novo</i> production of biomass is not possible anymore. In contrast, owing to its smallness, variations of the flux through the GSH oxidase reaction have only little impact on the minimal production time. An even 5-fold higher rate of GSH oxidation prolongs the minimal production time by only 0.06 h.</p

Minimal production times in 1, 2, 3 and 4 different phases.

<p>Minimal production times in 1, 2, 3 and 4 different phases.</p

Simplistic metabolic network with two target fluxes.

<p>Strategy A: All genes are constantly active, the demanded metabolic output is generated during the time interval <i>τ</i>₀ by a single flux mode composed of the two MinModes <i>w</i>¹ and <i>w</i>² (= reference case). Strategy B: The two minimal gene sets are separately active, during the first time interval <i>τ</i>₁ only the demanded amount of product <i>P</i>₁ is generated, whereas in the second time interval <i>τ</i>₂ only the demanded amount of <i>P</i>₂ is produced. Strategy C: During the initial time interval <i>τ</i>₁ only the minimal gene set <i>χ</i>₁ is active and only a certain fraction <i>α</i> < 1 of the demand for <i>P</i>₁ is produced. Thereafter, the second minimal gene set is additionally activated so that the products <i>P</i>₁ and <i>P</i>₂ are produced simultaneously. Strategy D: During the initial time interval <i>τ</i>₁ only the minimal gene set <i>χ</i>₂ is active, thereafter the second minimal gene set is additionally activated so that the products <i>P</i>₁ and <i>P</i>₂ are produced simultaneously. The gray-shaded panels illustrate the proportions in which the demanded amounts Γ₁ and Γ₂ of the two output metabolites are produced in strategies A-D.</p

Robustness of optimal solutions.

<p>Possible variation ranges of the gain obtained with different numbers of metabolic phases and glucose and lactate as allowed substrates. (A) Genes can be switched on and off, (B) Genes can only be switched on. The red mark represents the base condition (<i>η</i>_<i>j</i> = 1 for <i>j</i> = 1, …, <i>n</i>).</p

Flux rates through the biomass producing target reactions within various phases for the solution of the optimization problem with <i>l</i> = 4 different phases.

<p>The size of the colored areas correspond to the amount of biomass component produced in the respective time interval. (A) Glucose and lactate are available substrates, (B) Glucose is the only substrate, (C) Lactate is the only substrate. For the dividing cell, where we assume that genes are only progressively activated (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118347#pone.0118347.t004" target="_blank">Tab. 4</a>), the resulting production profiles are very similar (not shown).</p

Minimal production times if genes can only be switched on.

<p>Minimal production times if genes can only be switched on.</p