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We present a new mathematical approach to metabolic pathway analysis, characterizing a metabolic network by minimal metabolic behaviors and the reversible metabolic space. Our method uses an outer description of the steady state flux cone, based on sets of irreversible reactions. This is different from existing approaches, such as elementary flux modes or extreme pathways, which use an inner description, based on sets of generating vectors. The resulting description of the flux cone is much more compact. By focussing on the reversible and irreversible reactions, our approach provides a different view of the network, which may also lead to new biological insights.
Relating Attractors and Singular Steady States in the Logical Analysis of Bioregulatory Networks
(2007)
In 1973 R. Thomas introduced a logical approach to modeling and analysis of
bioregulatory networks. Given a set of Boolean functions describing the
regulatory interactions, a state transition graph is constructed that captures
the dynamics of the system. In the late eighties, Snoussi and Thomas extended
the original framework by including singular values corresponding to
interaction thresholds. They showed that these are needed for a refined
understanding of the network dynamics.
In this paper, we study systematically singular steady states, which are
characteristic of feedback circuits in the interaction graph, and relate them
to the type, number and cardinality of attractors in the state transition
graph. In particular, we derive sufficient conditions for regulatory networks
to exhibit multistationarity or oscillatory behavior, thus giving a partial
converse to the well-known Thomas conjectures.
Flux coupling analysis is a method to identify blocked and coupled reactions
in a metabolic network at steady state. We present a new approach to
flux coupling analysis, which uses a minimum set of generators of the steady state
flux cone. Our method does not require to reconfigure the network by splitting
reversible reactions into forward and backward reactions.
By distinguishing different types of reactions (irreversible, pseudo-irreversible,
fully reversible), we show that reaction coupling relationships can only hold
between certain reaction types. Based on this mathematical analysis, we propose
a new algorithm for flux coupling analysis.
Based on the logical description of gene regulatory networks developed by R. Thomas, we introduce an enhanced modelling approach that uses timed automata. It yields a refined qualitative description of the dynamics of the system incorporating information not only on ratios of kinetic constants related to synthesis and decay, but also on the time delays occurring in the operations of the system. We demonstrate the potential of our approach by analysing an illustrative gene regulatory network of bacteriophage lambda.
Logical modeling of biological regulatory networks gives rise to a representation of the system's dynamics as a so-called state transition graph. Analysis of such a graph in its entirety allows for a comprehensive understanding of the functionalities and behavior of the modeled system. However, the size of the vertex set of the graph is exponential in the number of the network components making analysis costly, motivating development of reduction methods. In this paper, we present results allowing for a complete description of an asynchronous state transition graph of a Thomas network solely based on the analysis of the subgraph induced by certain extremal states. Utilizing this notion, we compare the behavior of a simple multi-valued network and a corresponding Boolean network and analyze the conservation of dynamical properties between them. Understanding the relation between such coarser and finer models is a necessary step towards meaningful network reduction as well as model refinement methods.
Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models.
We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model.
Flux variability analysis (FVA) is an important tool to further analyze the results obtained by flux balance analysis (FBA) on genome-scale metabolic networks. Standard FVA may predict unbounded fluxes through some reactions in the network even if the nutrient uptake rate is bounded. These fluxes violate the second law of thermodynamics. They may be eliminated by extending flux variability analysis with thermodynamic constraints.
We present a new algorithm for efficient flux variability (and flux balance) analysis with thermodynamic constraints, suitable for analyzing genome-scale metabolic networks. We first show that flux balance analysis with thermodynamic constraints is NP-hard. Then we derive a theoretical tractability result, which can be applied to metabolic networks in practice. We use this result to develop a new constraint programming algorithm Fast-tFVA for fast flux variability analysis with thermodynamic constraints (tFVA). Computational comparisons with previous methods demonstrate the efficiency of the new method. For tFVA, a speed-up of factor 30-300 is achieved.
In an analysis of genome-scale metabolic networks in the BioModels database, we found that in 485 out of 716 networks additional irreversible or fixed reactions could be detected.
For modeling and analyzing regulatory networks based on qualitative information and possibly additional temporal constraints, approaches using hybrid automata can be very helpful. The formalism focussed on in this paper starts from the logical description developed by R. Thomas to capture network structure and qualitative behavior of a system. Using the framework of timed automata, the analysis of the dynamics can be refined by adding a continuous time evolution. This allows for the incorporation of data on time delays associated with specific processes. In general, structural aspects such as character and strength of interactions as well as time delays are context sensitive in the sense that they depend on the current state of the system. We propose an enhancement of the approach described above, integrating both structural and temporal context sensitivity.
Starting from the logical description of gene regulatory networks developed by R.~Thomas, we introduce an enhanced modelling approach
based on timed automata. We obtain a refined qualitative description of the dynamical behaviour by exploiting not only information on ratios of kinetic parameters related to synthesis and decay, but also constraints on the time delays associated with the operations of the system. We develop a formal framework for handling such temporal constraints using timed automata, discuss the relationship with the original Thomas formalism, and demonstrate the potential of our approach by analysing an illustrative gene regulatory network of bacteriophage~$\lambda$.
Genome-scale metabolic networks are useful tools for achieving a system-level understanding of metabolism. However, due to their large size, analysis of such networks may be difficult and algorithms can be very slow. Therefore, some authors have suggested to analyze subsystems instead of the original genome-scale models.
Flux coupling analysis (FCA) is a well-known method for detecting functionally related reactions in metabolic networks.
In this paper, we study how flux coupling relations may change if we analyze a subsystem instead of the original network. We show mathematically that a pair of fully, partially or directionally coupled reactions may be detected as uncoupled in certain subsystems. Interestingly, this behavior is the opposite of the flux coupling changes that may occur due to missing reactions, or equivalently, deletion of reactions.
Computational experiments suggest that the analysis of plastid (but not mitochondrial) subsystems may significantly influence the results of FCA. Therefore, the results of FCA for subsystems, especially plastid subsystems, should be interpreted with care.
Elementary flux mode (EM) analysis is an important tool in the constraint-based analysis of genome-scale metabolic networks. Due to the combinatorial complexity of these networks, as well as the advances in the level of detail to which they can be reconstructed, an exhaustive enumeration of all EMs is often not practical. Therefore, in recent years interest has shifted towards searching EMs with specific properties. We present a novel method that allows computing EMs containing a given set of target reactions. This generalizes previous algorithms where the set of target reactions consists of a single reaction. In the one-reaction case, our method compares favorably to the previous approaches. In addition, we present several applications of our algorithm for computing EMs containing two target reactions in genome-scale metabolic networks.
Constraint-based analysis of metabolic networks has become a widely used approach in computational systems biology.
In the simplest form, a metabolic network is represented by a stoichiometric matrix and thermodynamic information on the irreversibility of certain reactions.
Then one studies the set of all steady-state flux vectors satisfying these stoichiometric and thermodynamic constraints.
We introduce a new lattice-theoretic framework for the computational analysis of metabolic networks, which focuses on the support of the flux vectors,
i.e., we consider only the qualitative information whether or not a certain reaction is active, but not its specific flux rate.
Our lattice-theoretic view includes classical metabolic pathway analysis as a special case, but turns out to be much more flexible and general,
with a wide range of possible applications.
We show how important concepts from metabolic pathway analysis, such as blocked reactions, flux coupling, or elementary modes, can be generalized to arbitrary lattice-based models. We develop corresponding general algorithms and present a number of computational results.
The huge number of elementary flux modes (EFM) in genome-scale metabolic networks makes analysis based on elementary flux modes intrinsically difficult. However, it has been shown that the elementary flux modes with optimal yield often contain highly redundant information. The set of optimal-yield elementary flux modes can be compressed using modules. Up to now, this compression was only possible by first enumerating the whole set of all optimal-yield elementary flux modes.
We present a direct method for computing modules of the thermodynamically constrained optimal flux space of a metabolic network. This method can be used to decompose the set of optimal-yield elementary flux modes in a modular way and to speed up their computation. In addition, it provides a new form of coupling information that is not obtained by classical flux coupling analysis. We illustrate our approach on a set of model organisms.
Flux coupling analysis (FCA) has become a useful tool for aiding metabolic reconstructions and guiding genetic manipulations. Originally, it was introduced for constraint-based models of metabolic networks that are based on the steady-state assumption. Recently, we have shown that the steady-state assumption can be replaced by a much weaker lattice-theoretic property related to the supports of metabolic fluxes. In this paper, we further extend our approach and delevelop an efficient algorithm for general qualitative flux coupling analysis (QFCA). We illustrate our method by thermodynamic flux coupling analysis (tFCA), which allows studying steady-state metabolic models with loop-law thermodynamic constraints. These models do not satisfy the lattice-theoretic properties required in our previous work. For a selection of genome-scale metabolic network reconstructions, we discuss both theoretically and practically, how thermodynamic constraints strengthen the coupling results that can be obtained with classical FCA.
Constraint-based modeling of genome-scale metabolic network reconstructions has become a widely used approach in computational biology. Flux coupling analysis is a constraint-based method that analyses the impact of single reaction knockouts on other reactions in the network.We present an extension of flux coupling analysis for double and multiple gene or reaction knockouts, and develop corresponding algorithms for an in silico simulation. To evaluate our method, we perform a full single and double knockout analysis on a selection of genome-scale metabolic network reconstructions and compare the results.
Asymptotic behavior is often of particular interest when analyzing asynchronous Boolean networks representing biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called symbolic steady states, can be exploited to investigate attractor properties as well as for model reduction techniques conserving attractors. In this paper, we propose a novel optimization-based method for computing all maximal symbolic steady states and motivate their use. In particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a short study of a MAPK pathway model.