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Introduction

Intracellular biochemical networks are comprised of signaling, metabolic, and regulatory processes. (Note that here we use “regulation” to refer specifically to transcriptional regulatory and protein synthesis networks, and “signaling” to describe intracellular reactions that drive responses to the extracellular environment.) Until recently, computational analyses focused independently on signaling, metabolic, and regulatory networks. However, high-throughput experimental data coupled with computational systems analysis techniques have elucidated multifunctional components involved in fundamental disease processes [1]–[4]. For example, signaling cascades are triggered by the presence of extracellular stimuli and often result in activation of transcription factors. These transcription factors function in regulatory networks, regulating the transcription of associated genes and the synthesis of various proteins used in signal transduction and metabolism. Cellular metabolism is responsible for the production of energy in the form of adenosine triphosphate (ATP) and the synthesis of amino acids among other biomass precursors, all of which are used elsewhere in the cell. Consequently, a key challenge in the post-genomic era is to consider the interconnectedness of biochemical networks and how extracellular cues affect highly integrated intracellular processes to elicit cellular responses such as growth or differentiation.

Dynamic [5],[6] and structural analyses [7] have been employed to quantitatively analyze large-scale biochemical network modules. Typically, in dynamic analyses, a set of ordinary differential equations (ODEs) describing the mass (balance) of each species in the system is constructed. Despite its generality, the application of this type of mechanistic model at a genome-scale is largely considered impractical because it necessitates the consideration of many pathways for which detailed reactions and their kinetic parameters are not yet known. Structural analyses like flux balance analysis (FBA) can calculate phenotypic properties of a biological network like a steady-state flux (i.e., reaction rate) distribution without detailed kinetic information. FBA requires a physiologically relevant objective function (e.g., in the case of metabolism, maximizing the growth rate or maximizing ATP production), mass-balance constraints (i.e., the stoichiometry of the reactions), and constraints on reaction directions and thermodynamics. Since the physicochemical constraints are readily defined (e.g., from the annotated genome sequence and measured enzymatic capacities), FBA has been used effectively to study large-scale biochemical networks, particularly metabolic networks [8]. However, in general, the steady-state assumption of FBA prevents it from generating dynamic concentration profiles of intracellular species.

An additional challenge to the modeling of integrated systems is that time scales of intracellular biochemical networks generally span multiple orders of magnitude. Signaling and metabolic reactions typically occur rapidly. For example, kinase/phosphatase reactions, protein conformational changes, and most metabolic reactions occur on the order of fractions of a second to seconds [9]. By contrast, receptor internalization [10] and regulatory events [11],[12], as well as end-stage phenotypic properties such as cellular growth or differentiation [13] can take several minutes to hours. These multiple time-scales pose computational challenges for the quantitative analysis of integrated systems. For instance, kinetic model-based strategies suffer from a scarcity of values for kinetic parameters as well as poor accuracy of known kinetic parameters [14]. In addition, models of integrated systems are inherently “stiff,” i.e., they must include “fast” and “slow” reaction dynamics simultaneously [15], and they are consequently difficult to simulate and extremely sensitive to modeling errors [16]. Indeed, it is challenging to apply FBA to models of integrated systems because of the steady-state assumption intrinsic to FBA and the “fast” and “slow” reaction dynamics that coexist intracellularly. Due to these complexities, previous models and analyses have focused primarily on network modules rather than integrated systems. These include kinetic, stoichiometric, and causality analyses of modular signaling systems [17]–[20], metabolism [21]–[25], and regulation [26],[27].

Some preliminary dynamic analyses of integrated systems have been completed. Integrated analyses of regulatory and metabolic networks revealed novel mechanisms in Saccharomyces cerevisiae and Escherichia coli [28]–[30]. Metabolic reactions were represented stoichiometrically, and regulatory reactions were captured by representing gene regulatory rules using a Boolean formalism. FBA was implemented assuming quasi-steady-state conditions, i.e., the typical time constant of metabolic transients was relatively faster than the simulation time step for temporal integration of phenotypic variables (e.g., biomass as a measure of cellular growth). Recently, a kinetic model accounting for signal transduction, metabolism, and regulation was constructed to describe the response of S. cerevisiae to osmotic shock [31]. This model connected specific outputs of one network (e.g., signaling) with the inputs of another network (e.g., metabolism) in a “sequential” fashion. The complete set of interactions among the biochemical networks, such as feedback and feed-forward of proteins expressed as a function of the regulatory network to signaling and metabolism, was not considered. Additionally, it required an exhaustive list of kinetic parameters (e.g., rate constants) for the reactions, and time-courses of individual modules (or collections of reactions) were evaluated separately. As reconstructions of large-scale signaling and metabolic networks are emerging, there is a growing need for the development of a framework to study these networks from an integrated perspective [9].

The purpose of this study was to develop a FBA-based computational framework, termed integrated dynamic Flux Balance Analysis (idFBA), for the quantitative, dynamic analysis of cellular behaviors arising from signaling, metabolic, and regulatory networks at the genome-scale. The idFBA framework requires an integrated stoichiometric reconstruction of signaling, metabolic, and regulatory processes. It assumes quasi-steady-state conditions for “fast” reactions and incorporates “slow” reactions in a time-delayed manner. To assess the efficacy of idFBA, we developed a prototypic integrated system with topological features characteristic of those observed in existing signaling, metabolic, and regulatory network reconstructions as well as kinetic parameters reported in literature. Additionally, we applied in a similar manner the idFBA framework to a representative module in S. cerevisiae as a validation of our approach. idFBA allowed for quantitative, dynamic analysis of systemic effects of extracellular cues on phenotypes of these systems and generated comparable time-course predictions when contrasted with kinetic models. Ultimately, we demonstrate how idFBA enables genome-scale quantitative, dynamic analysis of integrated systems.

Methods

The idFBA framework facilitates the dynamic analysis of cellular phenotypes on the genome scale arising from extracellular cues. The systems evaluated as part of this study, including an integrated prototype spanning signaling, metabolism, and regulation, and a representative module from yeast are described here. The implementation details of the framework are also delineated.

Biological Systems Evaluated: Prototypic Integrated System

In order to assess the efficacy of idFBA, a prototypic integrated system was constructed with characteristics typical of those observed in published reconstructions of signaling, metabolic, and regulatory networks (see Figures 1 and 2). Specifically, we generated representative reactions with stoichiometric relationships and estimated their associated rate constants from literature. Here we briefly describe the reactions that are considered in each network and their typical time scales. Detailed information on these reactions and associated kinetic parameters is provided in Text S1.

Figure 1. The prototypic integrated system.

The prototypic integrated system, comprised of integrated signaling, metabolism, and regulation, is illustrated. Solid boundary lines indicate the three functional network modules: signal transduction (upper left), metabolism (upper right), and transcriptional regulation (bottom). Dashed lines between the modules represent interactions spanning multiple modules, arising from compounds that simultaneously participate in reactions of different functional modules. The components and reactions within these networks are based on published network reconstructions of actual biological systems. Primary roles of network components are shaded by color: blue for signaling, red for metabolism, and green for regulation. Detailed reactions are presented in Text S1.

doi:10.1371/journal.pcbi.1000086.g001

Figure 2. The degree of interconnectivity across signaling, metabolism, and regulation in the prototypic integrated system.

Network components with overlapping functions across the three functional network modules of signal transduction, metabolism, and transcriptional regulation are illustrated. Primary roles of the network components are shaded by color: blue for signaling, red for metabolism, and green for regulation. (See Table 3 for a listing of the input/output relationships within the prototype.)

doi:10.1371/journal.pcbi.1000086.g002

Signal transduction.

Signal transduction pathways govern a cell's response to extracellular stimuli, including, e.g., how a cell adapts its transcriptional regulatory program in response to specific environmental cues. The prototypic signaling network is comprised of a set of reactions that attempts to mimic what is typical of biological signaling pathways such as phosphorelay and kinase cascade modules. As shown in Figure 1 (top left), ligands (L₁, L₂, and L₃) bind to receptors (R₁, R₂, and R₃) to form ligand-receptor complexes (L₁R₁, L₂R₂, and L₃R₃). These complexes are subsequently either internalized or involved in phosphorylation events. Phosphorylation of signaling components takes place through a series of reactions involving ATP and other activated components. Any one signaling component can also activate multiple other signaling components; this activity represents the type of multi-functionality (e.g., crosstalk) that is often found in biological systems [32]. Ultimately, activated transcription factors (T₁p, T₂p, and T₃p), that are representative of phosphorylated proteins are the downstream effector molecules that result from the signaling pathways.

The model of signal transduction consists of a total of 45 reactions. As previously described, the rate constants for these reactions are based on values observed for similar signaling reactions in literature [10],[18],[31]. Most of the reactions in the prototypic signaling network are “fast” relative to transcriptional regulation; steady-state concentrations are achieved on the order of seconds. However, there are some “slow” reactions that take on the order of several minutes to hours to reach steady state. These include the internalization of ligand-receptor complexes and inhibition and hydrolysis of activated components (see Text S1). The typical order of magnitude of the concentrations of signaling components in this prototypic integrated system is micro-molar (µM) [18],[31].

Metabolism.

Metabolic pathways produce energy, amino acids, and other precursors required for the growth and maintenance of a cell. The metabolic reactions in the prototypic system comprise pathways representative of glycolysis and amino acid synthesis (see Figure 1, top right). The model contains 13 reactions, and the associated kinetic parameters were adapted from previous work [21],[22],[31]. The biosynthetic requirements for cellular growth (i.e., biomass production) were defined based on the prototypic metabolic reactions defined in [29] (see Equation 1), where H₁ and H₂ are representative of amino acids and F and G are representative of metabolites.
(1)
The maximum carbon utilization rate, S_u^max, was set to 10.5 mmol/(g(dry weight)•h) as in [24].

Most of the metabolic reactions in the model are “fast” and achieve steady states in several seconds. The growth of biomass is on the order of hours. The typical order of magnitude of metabolite concentrations is milli-molar (mM) [22].

Regulation.

Transcriptional regulatory networks control the transcription state of a genome. In general, they describe the connections between environmental cues and transcriptional responses [1]. Inputs to regulatory networks are environmental cues, including the presence and absence of extracellular metabolites, reaction fluxes, and specific conditions such as pH values. The internal reactions, often not known in chemical detail, are represented by regulatory rules that describe the activation or inhibition of gene transcription in response to these environmental cues. The outputs are the synthesized protein products that result through a combination of the signaling inputs acting upon the regulatory rules as well as consequent transcription and translation.

These networks have been mathematically described using a Boolean formalism, in which the state of a gene is represented as either transcribed or not transcribed in response to regulatory signals [1]. This formalism employs Boolean operators such as AND, OR, and NOT to describe the dependence of gene transcription upon extracellular metabolites and transcription factors as in [29]. Recently, a formalism that represents such regulatory rules in matrix form was developed, allowing for the systemic characterization of the properties of a transcriptional regulatory network and facilitating the computation of the transcriptional state of the genome under any given set of environmental conditions [1]. Furthermore, this “quasi-stoichiometric” matrix formalism enables regulatory networks to be represented alongside stoichiometric representations of signaling and metabolic networks: if a gene is repressed, fluxes of reactions involving the corresponding protein product are constrained to zero.

Studies on the dynamic behavior of regulation have involved constructing mass-balanced models of messenger RNA (mRNA) transcripts, ribosomes, and proteasomes in order to quantitatively predict protein synthesis [26],[31]. However, these approaches require estimation of rate constants that are difficult to measure experimentally. Furthermore, these descriptions of regulation are not complete because they do not account for the amino acids produced from metabolism and required for protein synthesis. In order to effectively couple regulation with other functional cellular modules, a more complete representation of the dynamic behavior of protein synthesis that facilitates balancing of input/output relationships across network modules is required.

The goal of the idFBA approach presented here, therefore, is to quantitatively account for the production and use of proteins throughout the cell. The transcriptional regulatory network is comprised of transcription factors that associate with specific genes, leading to the activation or inactivation of gene transcription. Activated genes yield proteins that participate in various intracellular signaling, metabolic, and regulatory reactions. Additionally, we considered amino acid requirements for protein synthesis: typically 30–80 moles of amino acids were required for every mole of protein, as shown in Table 1 [33]. Kinetics of protein synthesis were modeled as a second-order reaction between two amino acids H₁ and H₂ (k_poly[H₁][H₂]), and the kinetic parameter k_poly was estimated by considering a typical time constant for protein production based on [28]. The concentrations of mRNA transcripts, ribosomes, and proteasomes were assumed to be constant, and their effects on protein synthesis were captured by k_poly (for the purposes of our model, we assumed that only amino acids could contribute mass to protein production).

Table 1. Amino acid requirements (H₁ and H₂) for synthesis of protein (a_iH₁+b_iH₂ → Protein_i).

doi:10.1371/journal.pcbi.1000086.t001

The prototypic transcriptional regulatory network presented here is comprised of 18 genes (see Figure 1, bottom). Three transcription factors are inputs to the system, and 18 protein products with functions in metabolism and signaling are outputs of the network. Of the 18 genes, six are regulated by the presence or absence of the transcription factors. The remaining genes are defined to be constitutively active. The transcriptional regulatory rules for the six regulated genes are described using a Boolean formalism, as in [1] and [29]. For example, the regulatory rule in Equation 2 implies that Gene ER₃ is expressed only if both T₁p and T₂p are present.
(2)
The complete set of transcriptional regulatory rules for the prototypic integrated system is defined in Table 2. For simplicity, the amino acid requirements for protein synthesis are only considered for the proteins indicated in Table 1; however, similar requirements could be implemented as desired.

Table 2. Regulatory rules for the transcriptional regulatory network of the prototypic integrated system.

doi:10.1371/journal.pcbi.1000086.t002

Interactions and mixed-time scale model.

As previously described, a cellular phenotype ultimately arises from complex interactions of network components across signaling, metabolism, and regulation. The prototypic integrated system described above was designed to exhibit the interconnectedness seen in actual cellular systems, as illustrated by the input/output relationships between the three functional modules of signaling, metabolism, and regulation (see Figure 2 as well as Table 3). Furthermore, kinetic rate constants for the prototypic system were collected from representative reactions in literature (as reported in Text S1), and consequently the prototypic system exhibits dynamics across multiple time scales as seen in vivo.

Table 3. Input/output relationships in the prototypic integrated system.

doi:10.1371/journal.pcbi.1000086.t003

Biological Systems Evaluated: Representative Module in S. cerevisiae

To assess the applicability of idFBA to actual biological systems, a representative integrated module in S. cerevisiae, the prototypic single-cell eukaryote, was investigated. This module was comprised of key aspects of yeast osmoregulation, i.e., the active processes with which yeast cells monitor and adjust pressure and control their shape, turgor, and water content in response to extracellular conditions [34]. The signaling, metabolic, and regulatory activities included in this module are illustrated in Figure 3.

Figure 3. A representative integrated module in S. cerevisiae.

A representative integrated module in S. cerevisiae, the high osmolarity glycerol (HOG) pathway, is illustrated. Signaling reactions appear in the upper left of the figure, metabolic reactions in the upper right, and regulatory reactions in the bottom. The components and reactions within these networks are based on published literature on the S. cerevisiae HOG pathway. Detailed components and reactions are presented in Text S2.

doi:10.1371/journal.pcbi.1000086.g003

Specifically, we reconstructed a portion of the high-osmolarity glycerol response (HOG) pathway, one of four major mitogen-activated protein (MAP) kinase cascades in S. cerevisiae, from existing literature. The HOG MAP kinase pathway plays a pivotal role in the adaptation of S. cerevisiae to conditions of high external osmolarity [35]. For example, yeast cells deficient in this pathway cannot proliferate on media containing high levels of osmotically active molecules [34]–[36]. Extensive genetic analysis has previously been performed, leading to experimental identification of many activating and inhibiting components of the HOG signaling pathway [37]. In general, yeast cells use the HOG pathway to accumulate glycerol under hyperosmotic conditions, to balance the osmotic pressure with the extracellular environment. Osmotic stress signals are communicated via the HOG signaling pathway, leading to the activation of Hot1 and other transcription factors. These transcription factors subsequently promote the expression of glycolytic enzymes, such as Stl1, Gpd1, and Gpp2, thereby catalyzing metabolic reactions leading to increased glycerol production.

As this model serves an illustrative purpose here, the HOG pathway was restricted to the key set of reactions necessary for its phenotypic function. Specifically, 26 reactions spanning 48 components were assimilated in stoichiometric matrix form, including 16 reactions across 33 components in signaling; a single transcription factor activating three regulated genes; and seven reactions across 12 components in metabolism. Inputs of this module included osmotic shock (signaling) and glucose (metabolism), and outputs included glycerol (metabolism). Key reactions connecting the underlying signaling, metabolic, and regulatory processes were the translocation of the kinase Hog1 into the nucleus for the activation of transcription factor Hot1 (signal transduction and metabolism), and the synthesis of metabolic enzymes Stl1, Gpd1, and Gpp2 for reactions involved in the conversion of glucose to glycerol (transcriptional regulation and metabolism). Other reactions in the HOG pathway as previously experimentally characterized (e.g., inhibition of Hog1 by phosphatases Ptp2, Ptp3, and Ptc1, thereby allowing the cell to keep the HOG pathway in check and maintain osmotic balance) were excluded from the reconstruction used here for simplicity.

As with the prototypic system, the representative integrated yeast module was implemented using the idFBA framework as well as a kinetic model similar to the one in [31], and the two approaches were contrasted for validation purposes. Rate constants describing the kinetics of the system were culled from available experimental data, notably [31]. For complete details of the reconstructed yeast HOG pathway, including listings of reactions, rate constants, and kinetic equations, see Text S2.

Flux Balance Analysis

One modeling technique for evaluating cellular phenotypes is called flux balance analysis (FBA). FBA is a constraints-based approach that attempts to derive a phenotype in the form of a steady-state flux distribution for the reactions in a given biological system. FBA is based on the principle that all expressed phenotypes of a given biological system must satisfy basic constraints that are imposed on the functions of all cells [8],[38],[39]. These constraints are physico-chemical (i.e., physical laws like conservation of mass and energy); topological (i.e., spatial restrictions on metabolites within cellular compartments); and environmental (i.e., nutrient availability, pH, and temperature, all of which vary over time and space) [8],[20],[38]. Because FBA yields fluxes rather than concentrations, limited kinetic information is required for its implementation.

FBA requires a stoichiometric reconstruction of the biochemical network of interest. An annotated genome cataloging which reactions specific enzymes catalyze is the basis for a detailed description of a network's components and interactions [40]. This biochemical network reconstruction can be represented in matrix form, S, where S is of size m components×n reactions and is comprised of stoichiometric coefficients that capture the underlying reactions of the biochemical network.

After the network is reconstructed, fluxes are calculated by deriving a dynamic mass balance for all the components within the system [7],[38]. Specifically, at steady state, the change in the amount of a component C over time t across all reactions within the system must be zero. Consequently, mass balance is defined in terms of the flux through each reaction and the stoichiometry of that reaction, and a set of coupled ordinary differential equations relating the roles of reactions with components may be written in the form of Equation 3.
(3)
Here, S denotes the m×n matrix of stoichiometric coefficients and v denotes the vector of n reaction fluxes, with each element (row) of the n-row vector v corresponding to the flux in the associated reaction (column) in S. The vector C is a m-row vector defining the concentrations of the m components within the system. This mass balance represents the principal constraint in FBA and defines a feasible solution space for the set of fluxes. Additional constraints such as thermodynamics can be incorporated into FBA as well, further narrowing the possible distribution of fluxes [24],[41].

Equation 3 generally leads to an under-determined system because the number of components tends to be far fewer than the number of reactions. Even with additional constraints, FBA usually requires performing an optimization with linear programming (LP) to identify a particular flux distribution. In other words, FBA involves optimizing the set of fluxes such that the flux through a particular cellular reaction is maximized (or minimized). A cellular objective represents what a given biological system has optimized toward through evolutionary pressures [42]. It is defined as a linear equation (Equation 4), where c is the vector that defines the coefficients, or weights, for each of the fluxes in v [41].
(4)
This general representation of Z, wherein the elements of c can be easily manipulated, enables the formulation of many diverse objectives. Common choices for cellular objective functions in models of metabolic networks include biomass production [24],[43], energy [44], and byproduct production [45].

Ultimately, FBA attempts to solve the LP problem in Equation 5 to find a physiologically-relevant cellular phenotype in the form of a flux distribution v that optimizes Z while lying in the bounded solution space defined by a set of physio-chemical, topological, and environmental constraints.
(5)
Note that v_lb and v_ub are the lower and upper bounds on the reaction fluxes, respectively. For example, thermodynamic constraints or reaction directionalities can be incorporated by setting a given v_lb = 0.

Though the steady-state assumption of FBA precludes the calculation of dynamic concentrations of the network components, dynamic profiles of cellular phenotypes (e.g., cellular growth or differentiation) have been successfully predicted with a quasi-steady-state assumption [24],[25],[29]. This assumption involves discretizing the time domain into intervals, and (1) solving the LP problem contained within FBA at the beginning of each interval, and (2) based on the resultant flux data, solving a system of ODEs for concentrations over time within each interval.

Applications of FBA to dynamic simulations have focused on metabolic networks because time constants of metabolic transients are typically very rapid when contrasted with time constants characterizing whole-cell phenotypic changes. Exceptions include the incorporation of gene regulatory events, which are much slower than metabolic reactions, into FBA for time-course simulation of metabolic reactions [28],[29]. In these cases, the regulatory constraints were described as Boolean operators and imposed in a time-delayed manner. However, these examples are limited to metabolic and regulatory processes and do not consider changes in the mass balance (e.g., protein synthesis) arising from the interactions between metabolic and regulatory processes and signaling systems. Consequently, quantitative, dynamic analyses of integrated cellular systems have not been explored in detail, limiting the characterization of whole-cell function.

idFBA: An FBA-Based Approach for the Dynamic Simulation of Integrated Systems

As previously described, the stoichiometric reconstruction enforces explicit, chemically-consistent accounting of the components and reactions underlying a biochemical network, and facilitates the systematic analysis of fundamental network properties with FBA and associated analysis techniques [32]. The stoichiometric reconstruction and FBA are particularly applicable to large-scale networks, for which a lack of kinetic data (e.g., rate constants) makes kinetic-based approaches impractical. Indeed, stoichiometric reconstruction and FBA have been applied successfully to large-scale metabolic and signaling networks, elucidating characteristics of these networks [8],[9].

Therefore, integrating signaling reactions with metabolic and regulatory reactions using FBA can facilitate the dynamic analysis of cellular phenotypes arising from environmental cues and provide a complete snapshot of cellular sysems. However, as previously described, applying FBA directly to integrated networks is challenging. First, unlike metabolic systems in which objectives for the FBA formulation are often experimentally characterized (e.g., the production of biomass), objectives of signaling and regulatory systems are not well-defined. Second, integrated networks are comprised of reactions with mixed time scales (e.g., signaling reactions are generally much faster than regulatory reactions), and FBA has previously been applied only to fast reactions for which steady-state assumptions hold.

Here we describe the idFBA framework, including how we address these challenges. We use the prototypic integrated system as the basis for this discussion.

FBA-based representation of signaling networks.

As previously described, we represent signaling networks using a stoichiometric formalism, and we calculate a flux distribution with FBA (see Equation 5). Transcription factors activate transcriptional regulatory programs in response to extracellular cues. Consequently, one choice for the objective of a signaling network is maximizing the activation of transcription factors. However, as illustrated for the prototypic signaling network depicted in Figure 4, this objective by itself fails to generalize to a feasible flux distribution. Instead, maximizing the activation of the transcription factor T₁p consistently yields zero fluxes for the key pathway reactions denoted by dashed lines, including receptor internalization, pathway inhibition, and transcription factor degradation.

Figure 4. A representative signaling pathway in the prototypic integrated system.

One of the pathways of the prototypic signaling network is illustrated. Here, solid black lines represent reactions that have non-zero fluxes, while dotted red lines represent reactions that have zero fluxes when the production of the activated transcription factor T₁p is maximized as the pathway objective.

doi:10.1371/journal.pcbi.1000086.g004

To address this challenge, we model the objective of a signaling network by introducing a binary parameter, represented as the matrix I(R_i, t). I(R_i, t) indicates whether reaction R_i is to be included in the system at time t, given an underlying network objective. It is constructed on the basis of a set of rules and other parameters (e.g., the time that is required for receptor internalization or protein synthesis and degradation, etc.) that a user specifies as consistent with data about the given system. For a given reaction R_i at time t, I(R_i, t) is multiplied by the upper bound of the associated reaction flux v_i. If a particular reaction is included in the network at time t based on the user-defined rules and parameters (i.e., it has a non-zero flux at that time), the binary parameters I(R_i, t) for the reactions sharing components with the included reaction are set to zero at that time point and/or at future time points, depending on the specified time delays, indicating that they are not included in the network. Multiplying these zero-values by the upper bounds of the associated reaction fluxes nets a new upper bound of zero. As a consequence, fluxes through the reactions sharing components with an included reaction are set to zero at specific times in order to drive all the flux through the included reaction. In this way, the hypothesized network objective is maximized, all the while ensuring that flux is driven through all “active” reactions. For example, in Figure 4, including the receptor internalization reaction (L₁R₁ → L₁R_1,int) implies that the binary variables for the reactions (L₁R₁+S₁ → L₁R₁ · S₁) and (L₁R₁ → L₁+R₁) are set to zero. In this manner, a feasible flux distribution for a signaling network is obtained by maximizing for the activation of transcription factors. Importantly, the binary parameter I(R_i, t) can take into account time delays associated with “slow” reactions, as described below.

Incorporation of slow reactions into FBA.

In addition, to characterize mixed time-scale phenomena using FBA, we implement idFBA by assuming quasi-steady-state conditions for “fast” reactions and incorporating “slow” reactions into the stoichiometric matrix in a time-delayed manner as in [29]. In other words, we approximate continuous phenomena occurring over long time as instantaneous events at particular time points. Two parameters are used to implement this approach: time-delay (τ_delay), indicating after what time a “slow” reaction is considered an “active” steady-state constraint in the stoichiometric matrix; and reaction duration (τ_duration), indicating how long the “slow” reaction remains as the effective constraint once it is activated. In the prototypic integrated system, “slow” reactions include protein degradation, pathway inhibition, and receptor internalization in the signaling network; the uptake of a carbon source and production of biomass in the metabolic network; and the synthesis of proteins in the transcriptional regulatory network.

Dynamic simulation of integrated systems.

The optimized flux distribution that results from FBA is used to predict the time-course of phenotypic variables. The time-scale separation between “slow” and “fast” reactions is determined by the discretization of the time domain. Specifically, a reaction that reaches steady state or that produces a product at a specified threshold concentration within a single time step is considered “fast.” “Slow” reactions are those that take longer than the unit time interval to attain steady state.

Ultimately, the implementation of the idFBA framework can be described as a seven-step process (see Figure 5):

Figure 5. The idFBA framework.

The key steps in the idFBA framework are summarized. Specifically, the time window is discretized into small steps, Δt. FBA is used to calculate a flux distribution through the network (1), phenotypic variables such as cell density at timepoint t_current are evaluated by integrating the resultant flux values over the time step Δt (2), and the fluxes and phenotypic variables are used to update constraints for the next time step (3). Part of step 3 involves updating an incidence matrix (I) denoting which reactions participate during which time steps of the simulation.

doi:10.1371/journal.pcbi.1000086.g005

1. Discretize the time window into small steps, Δt. For example, in the case of the prototypic integrated system, the time step was specified as 0.1 h as described in “Results” below.

2. Initialize a R_s×t_N incidence matrix (I) denoting which reactions participate during which time steps (Equation 6). Here R_s represents the number of reactions within the system and t_N the number of time intervals (see Equation 6).
   (6)
   Each row of I(R_i, t) denotes a reaction R_i, and each column denotes a time step Δt. The coefficients of I, at the intersection of reactions and time steps, are binary parameters indicating whether a given reaction participates during a given time step. A “0” denotes that a given reaction does not participate in the system at the specified time step, whereas a “1” denotes that the reaction does participate in the system at that time step. Although I is difficult to generate for an actual biological system given the limitations of available experimental technologies, it facilitates a best-guess of the system dynamics based on available literature. For example, for any given system, I can be derived from experimental data and assumptions inputted into the idFBA framework.

3. For each reaction in the system R_i, multiply the corresponding coefficient I(R_i, t) by the flux bounds of the reaction. By specifying I(R_i, t) = 0 for excluded reactions, the fluxes of these reactions are set to zero when a “slow” reaction is included. For example, consider the signaling network shown in Figure 4. If the internalization reaction [L₁R₁] → [L₁R_1,int] is included and “active” and the objective of the network is maximizing the production of the activated transcription factor T₁p, fluxes of the excluded reactions [L₁R₁]+[S₁] → [L₁R₁ · S₁] and [L₁R₁] → [L₁]+[R₁] are set to zero at the associated time steps.

4. Solve Equation 5 for the optimized flux vector, v, with the updated constraints, for the start of the current time step, t_current.

5. Given the optimized flux vector for t_current, integrate the phenotype variable, X_p, over the time step Δt (Equation 7).
   (7)
   Here we consider two phenotype variables, namely cell density (X) and substrate concentration (S_c). These terms are given by Equations 8 and 9, where μ is a specific growth rate, and S_u is the uptake rate for the carbon source.
   (8)
   
   (9)
   

6. Update I based on v at the current time step t_current given the time-delay and reaction duration parameters (τ_delay and τ_duration, respectively) (Equation 10).
   (10)
   Specifically, as previously described, the dynamic parameters τ_delay and τ_duration approximate the progression of “slow” reactions as steady-state constraints in the idFBA framework. The parameter τ_delay describes when a particular “slow” reaction appears as a steady-state constraint in the stoichiometric matrix and instantaneously becomes an “active” reaction. The parameter τ_duration indicates how long the “slow” reaction is kept as a constraint in the stoichiometric matrix and maintained active. For example, in the transcriptional regulatory network, if the reaction flux of a transcription factor exceeds a specified threshold, the transcription of its target gene is incorporated into the matrix after a defined time (τ_delay) (here τ_delay mimics the delay for protein synthesis, including transcription and translation), and the protein is assumed to remain in the system until it degrades (a period of time captured by τ_duration).