Modeling of complex biochemical systems can be classified into two distinct methodologies: stoichiometric and kinetic. While stoichiometric models do not require any kinetic information, and are particularly attractive when steady-state assumptions can be justified [2], they cannot predict the temporal evolution of biochemical networks (as is needed for study of signaling evolution for instance). Kinetic models have been successfully applied to the analysis of a wide variety of biological systems, recent examples include neuronal signaling and the role of synaptic plasticity [3], phase sensitivity in circadian rhythms [4], and prediction of IL-2 response from T-cell receptor activation [5]. Unfortunately, accurate kinetic descriptions for specific protein, DNA, and metabolite interactions are typically difficult to determine in vivo, with reliable kinetic coefficient extraction being a non-trivial challenge [6].
Recent research has focused on applying the well characterized rule-based Boolean models to the challenging problem of characterizing biological networks [14]. In a Boolean model, the nodes of the network represent biological entities and the edges represent the interactions between them. The nodes can have a value of 0 or 1, representing an inactive or an active state respectively. The dynamics of the overall network are governed by Boolean rules for each node, that determine the state of the node at the next time-step based on the state of the upstream or effector nodes, and the update strategy that is employed. Rule-based Boolean network algorithms have been successfully used to aid in explaining the experimentally observed inherent robustness in cellular networks [7,9,10], even in the face of alteration in the network components and individual reaction rates [8].
In the context of Boolean networks, it has been shown that the topological organization of the regulatory network plays an invaluable role towards affecting the network dynamics [11]. Topological analysis provides an objective, rational method to analyze the sensitivity of a stimulated cell response to perturbation by delineating critical, dependent and independent pathways. However, merely topological analysis does not allow us to capture the effects of important biological features such as bi-directional signal propagation and synergistic effects [14]. Furthermore, network dynamics, while being immensely useful, only yield information about the phenotypic function as it relates to the network state. Network dynamics yield no information about the importance of particular nodes, critical nodes or hubs, in causing and maintaining the particular phenotype [14].
Various metrics have also been proposed to determine the critical nodes. Some of these metrics rely on topological properties (such as node degree, connectivity) and some on the network functionality, i.e., the effect of deleting a particular node on the overall network behavior [12]. In the context of metabolic networks the elementary flux mode analysis approach was employed by Stelling and co-workers to determine the effect of genetic knockouts [13]. Objective ranking of cellular processes was determined by a quantity known as the control effective flux (CEF), that takes both into account both the network efficiency and robustness to provide an objective measure of importance of an individual interaction [13]. While genetic knockout experiments do provide valuable information regarding gene functionality and network robustness, it does not a priori assign to each gene a measure of criticality with respect to the overall network function.
In this work, we have developed and applied a novel hybrid technique that combines the results from the Boolean pseudo-dynamics with an algorithm that a priori ranks the network nodes based on their importance in the efficient and robust operation of the network. In the context of this work, we also present a unique approach to enumerate the set of active pathways from the network state based purely on the topology and Boolean rules that govern the network behavior.
In order to demonstrate the utility of the proposed approach, and to compare our results with published literature we consider the guard cell signaling network in plant cells. This signaling network has been described in detail by Li and co-workers, and involves the abscisic acid (ABA) signal transduction which plays a role in ABA induced stomatal closure [14]. Two major secondary messengers involved in the closure of the stomata via ABA signal transduction are cytosolic calcium and the cytosolic pH. These two messengers are in turn regulated by a variety of other enzymes, secondary messengers, small molecules, and membrane channels so that the regulatory interactions between species are relatively complex. An integrated analysis of the network revealed different mechanisms resulting in stomatal closure in the presence and absence of abscisic acid. Furthermore, the results obtianed from the proposed approach showed excellent agreement with published results[14].
The proposed approach identifies the critical components a priori, eliminating the need for expensive knockout studies, providing significant computational savings (7X reduction). In addition, a quantitative ranking of the critical interactions is obtained, providing pathway context specific measure of relative importance. CEF values add to the available knowledge base, and can be used advantageously in the design of more experimental knockout studies, serving as an excellent basis for model refinement through hypothesis generation.
References: 1. D'haeseleer, P. et al., Genetic network inference: from co-expression clustering to reverse engineering, Bioinformatics, 16(8):707-726, (2000). 2. Varner, J. and D. Ramkrishna (1999). Mathematical models of metabolic pathways. Curr Opin Biotechnol. 10:146. 3. Ajay SM, Bhalla US. Synaptic plasticity in vitro and in silico: insights into an intracellular signaling maze. Physiology (Bethesda). 2006 Aug;21:289-96. 4. Gunawan R, Doyle FJ 3rd. Phase sensitivity analysis of circadian rhythm entrainment. J Biol Rhythms. 2007 Apr;22(2):180-94. 5. Kemp ML, Wille L, Lewis CL, Nicholson LB, Lauffenburger DA. Quantitative Network Signal Combinations Downstream of TCR Activation Can Predict IL-2 Production Response. J Immunol. 2007 Apr 15;178(8):4984-92. 6. Yao, K.Z., Shaw, B.M., Kou, B., McAuley, K.B., and Bacon, D.W. ‘Modeling ethylene/butene copolymerization with multi-site catalysts:parameter estimability and experimental design', Polym. React. Eng., 2003, 11, pp. 563–588. 7. Thomas, R., Boolean formalization of genetic control circuits, J. Theor. Biol., 42:563-585, (1973). 8. Chaves, M. et al., Robustness and fragility of Boolean models for genetic regulatory networks, J. Theor. Biol. 235:431-449, (2005). 9. Albert, R., et al., The topology of the regulatory interactions predicts the expression pattern of the Drosophila segment polarity genes, J. Theor. Biol., 223:1-18, (2003). 10. Kauffman, S., et al., Random Boolean network models and the yeast transcriptional network, Proc. Natl. Acad. Sci., 100:14796-14799, (2003). 11. Albert, R.,Scale-free networks in cell biology, J. of Cell Science, 118:4947-4957, (2005). 12. Mason, O. and Verwoerd, M.(2007). Graph Theory and Networks in Biology, IET Syst. Biol., 1(2):89-119. 13. J. Stelling, S. Klamt, K. Bettenbrock, S. Schuster, and E. D. Gilles, Metabolic network structure determines key aspects of functionality and regulation, Nature, 420, 190-193 (2002). 14. S. Li, S. Assmann, R Albert. Predicting Essential Components of Signal Transduction Networks: A Dynamic Model of Guard Cell Abscisic Acid Signaling. PLoS Biology 4:e312 (2006).