Propagation of kinetic uncertainties through a canonical topology of the TLR4 signaling network in different regions of biochemical reaction space
© Gutiérrez et al; licensee BioMed Central Ltd. 2010
Received: 17 November 2009
Accepted: 15 March 2010
Published: 15 March 2010
Signal transduction networks represent the information processing systems that dictate which dynamical regimes of biochemical activity can be accessible to a cell under certain circumstances. One of the major concerns in molecular systems biology is centered on the elucidation of the robustness properties and information processing capabilities of signal transduction networks. Achieving this goal requires the establishment of causal relations between the design principle of biochemical reaction systems and their emergent dynamical behaviors.
In this study, efforts were focused in the construction of a relatively well informed, deterministic, non-linear dynamic model, accounting for reaction mechanisms grounded on standard mass action and Hill saturation kinetics, of the canonical reaction topology underlying Toll-like receptor 4 (TLR4)-mediated signaling events. This signaling mechanism has been shown to be deployed in macrophages during a relatively short time window in response to lypopolysaccharyde (LPS) stimulation, which leads to a rapidly mounted innate immune response. An extensive computational exploration of the biochemical reaction space inhabited by this signal transduction network was performed via local and global perturbation strategies. Importantly, a broad spectrum of biologically plausible dynamical regimes accessible to the network in widely scattered regions of parameter space was reconstructed computationally. Additionally, experimentally reported transcriptional readouts of target pro-inflammatory genes, which are actively modulated by the network in response to LPS stimulation, were also simulated. This was done with the main goal of carrying out an unbiased statistical assessment of the intrinsic robustness properties of this canonical reaction topology.
Our simulation results provide convincing numerical evidence supporting the idea that a canonical reaction mechanism of the TLR4 signaling network is capable of performing information processing in a robust manner, a functional property that is independent of the signaling task required to be executed. Nevertheless, it was found that the robust performance of the network is not solely determined by its design principle (topology), but this may be heavily dependent on the network's current position in biochemical reaction space. Ultimately, our results enabled us the identification of key rate limiting steps which most effectively control the performance of the system under diverse dynamical regimes.
Overall, our in silico study suggests that biologically relevant and non-intuitive aspects on the general behavior of a complex biomolecular network can be elucidated only when taking into account a wide spectrum of dynamical regimes attainable by the system. Most importantly, this strategy provides the means for a suitable assessment of the inherent variational constraints imposed by the structure of the system when systematically probing its parameter space.
Normal and abnormal cellular states represent macroscopic behaviors emerging from intricate dynamical patterns (either transient or stationary) of biochemical activity. These are sustained by a complex web of reaction mechanisms that play the role of information processing systems, generically referred to as signal transduction networks [1–3]. In other words, these networks represent the dynamical systems that instruct cells to enter into specific regimes of biochemical activity, which ultimately determine the universe of functional states accessible to the cell, such as differentiation, apoptosis, cell division, etc. [1–3]. Operatively, functional regimes of biochemical activity within a cell are basically accomplished via direct protein-protein interactions and enzyme-catalyzed reactions (i.e. phosphorylation, RNA synthesis, etc.) triggered in response to either internal or external stimuli [3, 4].
The spectrum of functionalities that a signal transduction network can potentially perform is inherently constrained by its design principle [5, 6], which encapsulates a series of aggregated components involving diverse regulatory schemes and biochemical reaction rules modulated quantitatively via internal reaction parameters. This structure-function puzzle has motivated considerable research efforts in the last decade aimed at elucidating possible mechanistic bases of fundamental emergent properties such as robustness, evolvability and epistasis, of highly-modular regulatory systems [7–13]. Importantly, the investigation of the robustness properties of a signal transduction network requires heavy emphasis to be made on two fundamental aspects of the underlying reaction mechanism: an observable/quantifiable dynamical feature (either transient or stationary) of the system, and one or several perturbable parameters directly or indirectly involved in the development of the system's feature being studied. For instance, important quantitative dynamical features of signal transduction networks have been proposed as suitable targets for assessing their robustness properties in the face of random changes in internal reaction parameters [14, 15]. Sources of perturbations impinging upon such parameters may stem from environmental vicissitudes (temperature, pH, etc.), genotypic variation or intrinsic fluctuations (molecular noise) [16, 17].
Recently, several computational studies have yielded interesting numerical evidence supporting the idea that the robustness properties of highly-dimensional biochemical reaction networks may be strongly dependent on three fundamental aspects: i) the reaction topology (network architecture) [7–9], ii) the system's current position in parameter space [18–20], and iii) the dynamic nature of the trajectories displayed by the reaction species involved [13, 20–22]. The robustness properties of a biomolecular network are typically assessed by means of standard sensitivity analysis-based approaches implementing both local and global perturbation methods [18, 23–27]. Robustness is usually assessed with respect to either observable or hypothetical stationary states and transient dynamics of just few reaction species in the network [24, 28, 29]. However, a complementary quantitative approach to studying the robustness properties, as well as information processing capabilities, of a complex reaction network should provide the means for assessing the extent to which the full dynamical behavior of the system is reproducible under, for example, kinetic uncertainties. This is because a reaction network may be coupled dynamically in unexpected ways to other important subsystems not included in the model [11, 30], whereby biochemical information exchange among cellular processes can take place in parallel. Under these considerations, we thus believe that general properties of a canonical biomolecular network could be revealed under the following methodological strategies. Firstly, a large ensemble of disparate, but biologically plausible dynamical trajectories attainable by the network should be tested for general robustness properties in the face of random perturbations impinging upon the whole set of reaction parameters; that is to say, the overall robust performance of the network should be evaluated in widely scattered regions of its accessible parameter space. Secondly, the reproducibility of particular ouputs (i.e. experimentally reported wild-type transcriptional readouts) should be assessed in different regions of the accessible parameter space via both local and global perturbation strategies. Addressing these points would pave the way to gaining general insight into systems-level features of the complex reaction mechanisms endowing the cells with the potential to reach a wide spectrum of robust behaviors.
In this study, efforts were focused on a comprehensive and unbiased statistical assessment of the robustness properties and information processing capabilities of a canonical reaction topology underlying TLR4-mediated signaling events. This signaling network is temporally deployed in inflammatory cells (i.e. macrophages) in response to external stimuli. We constructed a deterministic, non-linear dynamic model of this reaction topology, using an informational basis retrieved from a series of previous computational studies and review papers providing important clues about mechanistic reaction steps involved in the process (see the Results and Discussion section below). We adopted this signaling network as our model system mainly because this functional module plays a crucial role in the development of innate immune cellular responses ([31–37]). For instance, Toll-like receptors recognize conserved pathogen-associated molecular patterns such as lipopolysaccharide (LPS), which results in the triggering of both microbial clearance and the induction of immunoregulatory chemokines and cytokines. Here, we centered our attention specifically on the immediate cellular response, in macrophages, triggered by the rapid activation of the canonical MyD88-dependent and TRIF-dependent reaction cascades upon LPS binding to TLR4. We probed the robustness properties and information processing capabilities of this canonical network in different points distributed across diverse regions of the biochemical reaction space. Importantly, the behavior of the network in a given region of the biochemical reaction space was selected so that it was congruent with a hypothetical, but biologically plausible dynamical regime of molecular activity (see below). Global (non-orthogonal) and local (orthogonal) perturbation strategies were implemented as a means of systematically exploring the biochemical reaction space inhabited by the network. Critically, reaction parameters were subjected to random perturbations without a priori knowledge on their relative importance for the network in the accomplishment of a given signaling task. Our extensive numerical analyses permitted us the identification of global and particular variational constraints in the network. This was achieved by means of a detailed characterization of some statistical regularities on the dynamical performance of the system under kinetic uncertainties (i.e. random fluctuations in internal reaction parameters). Overall, our simulation results provide convincing numerical evidence supporting the following idea: a canonical reaction mechanism underlying TLR4-mediated signaling events is endowed with the intrinsic capacity to perform information processing in a robust manner, which is remarkably independent of the signaling task required to be executed. Nevertheless, our statistical analysis indicate that the robust performance of the network is not solely determined by its architecture (topology), but this may be strongly conditioned by the network's current position in biochemical reaction space. Ultimately, our simulation results provide interesting mechanistic insigths into structure-function relationships in the TLR4 signal transduction network, which enabled the identification of plausible rate limiting steps that most effectively control the performance of the system under diverse dynamical regimes.
Information processing and biochemical reaction space of the signal transduction network
Canonical reaction topology underlying TLR4-mediated signal transduction events
General robustness properties of the signal transduction network in different regions of the biochemical reaction space
Variability of key individual dynamical trajectories
Comparison of total parameter variation spectra
Robustness of particular input-output maps: effects of local and global perturbations at the level of individual transcriptional outputs
Upon extensive exploration and statistical characterization of general robustness properties inferred from hypothetical, but biologically plausible dynamical trajectories displayed by the network, we then focused on a detailed analysis of particular input-output maps embedded in the model reaction scheme.
Local perturbation analysis of transcriptional outputs
Statistics on overall state senstivities from local perturbation experiments for the transcriptional output Tnfα. Values shown were averaged over the ensemble of 10 reference parameter configurations evaluated. Mean-D (mean D Statistic); SD-D (standard deviation of D Statistic)
Statistics on Overall State Senstivities from Local Perturbations Experiments for the Transcriptional Output Cxcl10. Values shown were averaged over the ensemble of 10 reference parameter configurations evaluated. Mean-D (mean D Statistic); SD-D (standard deviation of D Statistic)
Revealing the global perturbation landscapes of transcriptional outputs
The main purpose of this in silico work was to explore whether important system-level attributes of a complex biomolecular network were strongly conditioned by the type of signaling tasks (i.e. particular dynamical regimes of molecular activity) simulated. Specifically, our computational approach permitted us an unbiased statistical assessment of the robustness properties, as well as the information processing capabilities, of the canonical reaction mechanism underlying TLR4-mediated signal transduction events. This was achieved by considering a broad spectrum of plausible dynamical behaviors displayed by the network (including wild type phenotypes), which are likely encountered in any cell lineage (i.e. macrophage) under diverse physiological conditions. This is the rationale behind our work, and we highlight that these considerations have been largely underappreciated in previous studies of network robustness. Recent investigations, however, have stressed the importance of assessing the spectrum of variational constraints (i.e. robustness, evolvability, epistasis, etc.) of complex developmental regulatory networks under different hypothetical and observable dynamical regimes [13, 56]. Our work thus differs considerably from recent computational studies wherein heavy emphasis have been placed on the characterization of robustness of particular intracellular networks under rather limited biological circumstances [17, 18, 27, 28].
To summarize, our numerical findings strongly suggest that the canonical TLR4 signaling network that drives crucial innate immune cellular responses in macrophages, should be operative in widely scattered regions of the biochemical reaction space; a robust property that allows the network to perform complex signaling tasks in a highly reproducible manner under rather different regimes of molecular activity, and when facing multiple kinetic uncertainties.
Deliberately, we have restricted our model signal transduction network to a simple biochemical reaction mechanism. Importantly, the design principle (topology) of the network was mathematically represented by means of basic reaction schemes defined in terms of mass action law and Hill saturation kinetics. Accordingly, information processing in our model network takes place only through the kinetic coupling of multiple, but rather simple, reaction rules accounting for ligand-receptor interaction, association and dissociation events between single or multiple reaction species, import/export fluxes between cellular compartments, enzyme-catalyzed reactions, and transcriptional control. Elaborated regulatory schemes, such as inhibitory reactions or feedback control, were not accounted for in our modeling framework. This is because within our narrow temporal window, in which immediate immune cellular responses are elicited, signal propagation is thought to be controlled in its entirety by the intrinsic crosstalking of MyD88-dependent and TRIF-dependent reaction cascades (see [46, 47] and references therein). Therefore, within our simulated time window, emphasis was not placed on the complex negative feedback control arising within the NFκ B regulatory module, which is triggered by a wide spectrum of pro-inflammatory stimuli . The many possible roles of negative feedback control deployed by the NFκ B regulatory module under different cellular contexts have been a central theme of investigation in intracellular signaling ([51, 57]); this issue, however, was beyond the scope of our study. Nevertheless, we acknowledge that our results on the robustness properties and information processing capabilities of the TLR4 signaling network are expected to differ considerably under a different mathematical representation of the reaction topology, wherein positive/negative feedback regulation taking place at any point along the signaling cascade were accounted for. This should come as no surprise, since the crucial role of such elaborated regulatory schemes in any signal transduction system has been well documented (see for example [16, 58], and references therein).
Most systems biology studies centered on the structural and functional organization of highly-dimensional biomolecular systems point to the general idea that signal transduction networks should display the inherent capacity of accomplishing specific biological tasks in a robust manner (see  and references therein). Robustness seems to be a natural property stemming from the evolved design principle of biomolecular networks [7–9], which allow them to inhabit sloppy parameter spaces wherein system's behavior turn out to be highly sensitive to variation along a few stiff directions, while being remarkably insensitive to variation along a large number of sloppy axes in parameter space [11, 30]. Notably, accurate computational reconstructions of experimentally reported dynamical behaviors of many signal transduction networks have been successfully achieved [20, 51, 55, 57]. Interestingly, standard mathematical representations of the reaction topology of most signaling network models are typically founded on highly non-linear, but relatively simple, biochemical reaction rules, which despite being an abvious simplification of the underlying biochemistry have proven successful at providing mechanistic insight [20, 51, 55, 57]. This is an intriguing observation from an evolutionary standpoint. This suggests, for example, that the underlying mathematical structure of most signal transduction networks that has been favored over evolution to process in an efficient and robust manner the biochemical information arising in the cell, might simply rely on basic dynamic rules ([59, 60]). Intuitively, the most variable component affecting the temporal variation in the activity of the molecular species involved in a certain signaling event would be the number of the contributing reaction velocities to a particular flux. Following this line of arguments, it is tempting to speculate on the possibility that the deterministic component of the dynamical trajectories displayed by most signaling networks might have been the result of selection for simple biochemical reaction rules built, for example, upon mass action and Hill-like saturation kinetics.
Models and computational framework
The mathematical representation of the canonical reaction network retrieved from the literature, and the whole set of numerical experiments that are described below were implemented in Mathematica® 6.0.
Mathematical formulation of the signal transduction network in the language of dynamical systems
- 1A network with n reaction species is represented by the state vector:(1)
In our case, the TLR4 signaling network model incorporates n = 76 reaction species, including receptors, adapters, kinases, transcription factors and mRNAs.
- 2The biochemical reaction parameters controlling the signaling flux through the network and transcriptional processes of target genes are incorporated in the reaction vector(2)
With m = 116 internal reaction coefficients for the TLR4 network. Here, Θ encompasses a wide spectrum of parameters of different biochemical nature, ranging from transition rates between receptor states (susceptible ⇌ activated), production and degradation rates of receptors, association/dissociation rates among intracelular molecular species, phospho/dephosphorylation rates, nuclear import/export rates, maximal transcriptional rates, transcriptional efficiencies, Michaeles-Menten constants, cooperative coefficients, and mRNA degradation rates (see Additional file 1 for a detailed description of these parameters and their assigned range of values).
Where f(·) defines a non-linear state transition function accounting for reaction velocities or fluxes (see Additional file 1 for a detailed description of the dynamical system), which can be grounded on mass action laws and/or Hill kinetics, according to the reaction mechanisms modeled (receptor activation kinetics, binding and enzymatic reactions, or transcriptional dynamics). Y0 represents the vector of initial concentrations for the reaction species at time t0; g(·) defines a measurement function which is solved numerically; whereas X ∈ ℜ n gives the measurement output vector representing the concentration of the reaction species at a given point in time. In our case, the TLR4 signaling network model accounts for 76 dynamic variables (reaction species, Y j ∀ j ∈ (1, 2, 3, ....,76)), 32 of which were assigned zero initial conditions (see Additional file 1).
Multiparametric sensitivity analysis (MPSA): a combination of uncertainty and sensitivity analyses
Selection of reference parameter configurations (vectors) to be perturbed.
Set a relatively large range of variation for each model parameter in order to account for a wide spectrum of biologically plausible perturbations (i.e. single or combined mutations, thermal fluctuations, etc.). In our case, a perturbation variable, ρ, was sampled in this way: ρ ~ U(-1, 1); a perturbation function was then applied over a reference parameter i, θi, ref, in order to obtain a newly perturbed parameter θi, pert= 10 ρ * θi, ref
- 3Under this perturbation strategy we initially generated seeds of LHS matrices with the basic structure as illustrated in Table 3, wherein a row vector stands for a perturbed parameter configuration. In this way, the matrix shown represents a set of input vectors distributed in parameter space in the vicinity (either immediate or distant) of a previously defined reference point; this matrix is assembled via the LHS strategy, and is designed for the systematic evaluation of the model's output. Note that in our case 5000 parameter configurations were generated from a previously defined reference point in parameter space. We constructed ensembles of 100 LHS matrices for evaluating the robust information processing capabilities of the network in different regions of biochemical reaction space. We also constructed an ensemble of 10 LHS matrices in order to test for the robustness properties of the two experimentally reported transcriptional outputs of the network. Before simulating each previously assembled seed LHS matrix, however, we permuted the elements of each column of a matrix as illustrated in Table 4. In this way, permuting the matrices permitted us to avoid any kind of bias in model evaluations.Table 3
A seed LHS matrix. IPPC stands for initial perturbed parameter configuration. θ i indicates any reference parameter value i to be perturbed systematically
θ 1, ref
θ 2, ref
θ 116, ref
A permuted version of the seed LHS matrix. NAPC stands for newly assembled parameter configuration. θ i indicates any reference parameter value i to be perturbed systematically
- 4Each LHS matrix was then simulated, and the corresponding discrepancy function evaluated, which is of the form:(5)
With J denoting a reference dynamical trajectory with associated parameter configuration Θ ref , and Θ pert being a perturbed version of it obtained from a LHS matrix; with h ∈ (1, ..., 5000) being any evaluation of the discrepancy function.
- 5This step implied determining whether a given perturbed parameter configuration was acceptable (robust) or unacceptable (sensitive) by comparing the discrepancy function value to a given threshold. If the discrepancy function value was found to be below the threshold value the evaluated parameter configuration was then classified as acceptable; otherwhise it was classified as unacceptable. Previous computational works suggest that results from MPSA should not be affected considerably with the choice of a given discrepancy function [25, 53, 63]. Here we implemented the average of the discrepancy function as our threshold value, defined as follows:(6)
- 6After systematic evaluation of the discrepancy function for each LHS matrix, statistical assessment followed in order to determine whether a given parameter configuration was deemed either acceptable or unacceptable. To do this, we applied the KS test to assess the global sensitivity of the system's output with respect to perturbations targeting individual parameters. The KS test provides the means for evaluating the cumulative frequency of the observations (parameter values) as a function of class, and calculate the maximum vertical distance between cumulative frequency distribution curves for m acceptable and n unacceptable cases of any given parameter θ j . This is obtained by calculating the D statistic, which is defined in this way:(7)
Where S(θ j ) and represent the cumulative frequency functions corresponding to acceptable and unacceptable cases, respectively, with θ j being any reaction parameter of the signal transduction network. Importantly, this estimator provides a robust quantitative notion for the sensitivity/robustness of the network model to random perturbations of the reaction parameters. The higher the D-value the more sensitive is the dynamical behavior of the network model with respect to variation of a given parameter, when the remaining parameters (i.e. biochemical background of the network) are also varied.
Total parameter variation
We calculated all T values for those perturbed parameter configurations that were deemed either robust or fragile obtained from a reference parameter configuration, Θ ref , exhibiting a global dynamical trajectory J (∀ J ∈ (1, ......., 100))
Local and global perturbation analysis of input-output maps
Given in arbitrary units of discrepancy, this threshold was selected upon a detailed analysis of both the qualitative and quantitative effects of the perturbations on the temporal dynamics of the transcriptional readouts.
We are greatly indebted to Drs. Masa Tsuchiya and Kumar Selvarajoo for helpful disccusions and critical comments on earlier versions of the manuscript. We also thank anonymous reviewers for insightful comments and helpful suggestions. Special thanks to Koichi Matsuo for kindly providing experimental data on gene expression. JG wishes to acknowledge Grupo dé Física y Astrofísica Computacional (FACom) for computing facilities, and Boris A. Rodríguez for important suggestions and constant technical support. This work was funded in part by Grupo de Inmunovirología, SIU, Universidad de Antioquia.
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