Department of Mechanical Engineering, University of California, Santa Barbara, Engineering II Bldg., Santa Barbara, CA, 93106-5070, USA

Department of Anesthesia, Stanford University School of Medicine, Stanford, CA, 94305-5117, USA

Abstract

Background

Activation of the NF-κB transcription factor and its associated gene expression in microglia is a key component in the response to brain injury. Its activation is dynamic and is part of a network of biochemical species with multiple feedback regulatory mechanisms. Mathematical modeling, which has been instrumental for understanding the NF-κB response in other cell types, offers a valuable tool to investigate the regulation of NF-κB activation in microglia at a systems level.

Results

We quantify the dynamic response of NF-κB activation and activation of the upstream kinase IKK using ELISA measurements of a microglial cell line following treatment with the pro-inflammatory cytokine TNFα. A new mathematical model is developed based on these data sets using a modular procedure that exploits the feedback structure of the network. We show that the new model requires previously unmodeled dynamics involved in the stimulus-induced degradation of the inhibitor IκBα in order to properly describe microglial NF-κB activation in a statistically consistent manner. This suggests a more prominent role for the ubiquitin-proteasome system in regulating the activation of NF-κB to inflammatory stimuli. We also find that the introduction of nonlinearities in the kinetics of IKK activation and inactivation is essential for proper characterization of transient IKK activity and corresponds to known biological mechanisms. Numerical analyses of the model highlight key regulators of the microglial NF-κB response, as well as those governing IKK activation. Results illustrate the dynamic regulatory mechanisms and the robust yet fragile nature of the negative feedback regulated network.

Conclusions

We have developed a new mathematical model that incorporates previously unmodeled dynamics to characterize the dynamic response of the NF-κB signaling network in microglia. This model is the first of its kind for microglia and provides a tool for the quantitative, systems level study the dynamic cellular response to inflammatory stimuli.

Background

The nuclear factor-κB (NF-κB) transcription factor is ubiquitously expressed in mamallian cells and regulates the expression of many target genes. In the nervous system NF-κB is known to play a key role in the immune and injury responses and in governing normal brain function

Members of the NF-κB family of transcription factors are found in their inactive state as dimers bound to their IkB inhibitor proteins. Upon stimulation by a diverse set of stimuli, NF-κB is freed from its inhibitor to coordinate gene expression in a highly specific and tightly regulated manner. The IκBα inhibitor and p65(RelA):p50 NF-κB heterodimer are the most extensively studied members of their respective families, and their response to extracellular stimuli illustrates the canonical pathway of NF-κB activation (Figure

The canonical NF-κB activation pathway

**The canonical NF-κB activation pathway**. Binding of TNFα trimers to TNFR receptors initiates the canonical signaling pathway by activating the upstream kinase IKK. IKK phosphorylates the IκB inhibitor that is bound to NF-κB in the resting state. This targets IκB proteins for the ubiquitin-proteasome system, which leads to IκB destruction by the 26S proteasome and release of NF-κB. Free NF-κB enters the nucleus and activates gene expression of many target genes and induces negative feedback regulation by synthesizing IκB and A20. IκB proteins inhibit NF-κB activity by sequestering NF-κB from the nucleus to form an inner feedback loop, while A20 attenuates stimulus induced IKK kinase activation further upstream in an outer negative feedback loop.

In the canonical pathway, binding of extracellular TNFα trimers to TNFR1 receptors at the cell membrane initiates NF-κB activation. The ligand-receptor complex interacts with several adapter proteins, including TNF receptor-associated factor 2 (TRAF2) and receptor-interacting protein-1 (RIP1), which are essential for recruitment and activation of the IκB kinase complex (IKK)

NF-κB is released following proteasomal degradation of IκBα

Given the dynamic nature of NF-κB signaling and its regulation involving multiple feedback loops, it is necessary to consider the network as a whole when studying this system. The seminal work by Hoffmann and colleagues

Here we develop a mathematical model to describe NF-κB signaling in microglia. Beginning with a recently published model structure shown to be capable of predicting NF-κB signaling in other cell types

Results

TNFα stimulates dynamic NF-κB and IKK activation in BV2 microglia

To characterize the dynamics of canonical NF-κB activation in microglia, cells from the microglial cell line BV2 were cultured and treated with 10 ng/ml TNFα. Whole cell extracts were collected in triplicate over a time course following stimulation in five identical experiments conducted on different days. ELISA measurements of NF-κB p65 DNA binding activity show that NF-κB activation in BV2 microglia is strongly induced by TNFα (Figure

Dynamics of NF-κB and IKK activation in BV2 microglia treated with TNFα

**Dynamics of NF-κB and IKK activation in BV2 microglia treated with TNFα**. ELISA measurements of (A) NF-κB p65 DNA binding activity and (B) IKKβ kinase activity following continuous stimulation by 10 ng/ml TNFα. Data markers at each time point are sample averages from independent experiments performed on separate days. Error bars indicate one standard deviation of the samples.

To better characterize the inflammatory response in microglia we additionally examined the activation of the upstream IκB kinase (IKK) experimentally. The time course of IKK activity was measured for the first 30 min following 10 ng/ml TNFα treatment in three identical experiments. IKK is rapidly activated, reaching peak levels near 5 min. By 10 min IKK activity sharply drops to below half-maximal levels and gradually declines to near basal levels over the next 20 min (Figure

Intermediate steps in the IKK-induced IκBα degradation pathway reconcile the mathematical model with NF-κB activation in microglia

Next we sought to quantitatively describe microglial NF-κB activation using a mathematical model. While a number of mathematical models for NF-κB have been published in recent years (reviewed in

We first attempted to identify parameters for the existing model structure to fit the experimental NF-κB and IKK activation profiles of microglia. An optimization-based parameter estimation algorithm was run using many randomly selected parameter values from the parameter space as initial guesses. However, no parameter sets were found that matched microglial IKK and NF-κB activity. In particular, the model was unable to qualitatively reproduce the rapid induction and attenuation of IKK activity observed in microglia for any of the parameter sets tested, and NF-κB activation was predicted to occur more rapidly than the 5 min delay observed in Figure

Sensitivity analyses were performed on the model to quantify the relative contributions of each of the system parameters to the concentration of free NF-κB during the first 10 min given the large mismatches between the model and data in this interval. Only seven of the original 26 system parameters have appreciable effects on NF-κB activity during this time based on their time-averaged sensitivity scores (Figure

New model structure required to characterize NF-κB activation in microglia

**New model structure required to characterize NF-κB activation in microglia**. (A) NF-κB activity during the first 10 minutes following stimulation was only highly sensitive to seven of the 26 rate parameters. (B) By using an IKK signal derived from experimental measurements as the model input, the outer feedback loop can be removed (indicated by gray lines), isolating the downstream NF-κB activation module with IκBα feedback. Similarly, once the concentration of nuclear NF-κB is known, this signal can be used to drive the upstream IKK activation network independently of the downstream module. (C) Model structure from the original model (top) and the new model (bottom). (D) Simulations with parameters estimated for the existing model (dashed line) and the new model (solid line) using the experimental IKK curve as input. The inset provides a detailed view of the model fits during the initial activation phase. (E) The results of 1980 randomly initialized parameter estimates for each model were checked for statistical consistency with the data using Fisher's Method (see Methods) and binned according to p-value. No estimated parameter sets with the original model achieved a P-value >10^{-7 }(red), while nearly half the estimated parameter sets with the new model (blue) had P > 0.01.

**Supplementary text and figures**. The pdf contains supplementary text describing development of the mathematical model; Tables S1-S3 which list the model species, reactions, and rate parameters; and Figures S1-S8 that provide more detailed simulation results.

Click here for file

To more easily explore these possibilities and to facilitate model development, we first considered the downstream network independently of the upstream IKK activation network. IKK interacts with the downstream module only through its enzymatic phosphorylation of IκBα and through feedback inhibition from A20 (Figure

With the IKK profile fixed as the model input, the least squares parameter estimation procedure was repeated with certain parameter values and biological features constrained by the literature (Additional file ^{-7 }(Figure

Taken together with the sensitivity results showing that very few system parameters significantly affect NF-κB activation during the first 10 min of activation, this strongly suggested there were likely unmodeled dynamics within the IKK-induced IκBα degradation pathway. We next investigated whether the model could be modified in a biologically meaningful way to incorporate missing dynamics and to better fit the data.

The original model structure describes IKK-dependent IκBα degradation in two steps: phosphorylation of IκBα catalyzed by IKK, and degradation of phosphorylated IκBα (Figure

With the new model structure in place, the parameters corresponding to the new stimulus-induced IκBα degradation reactions were estimated using the optimization algorithm while fixing all other parameters downstream of IκBα degradation to their previously estimated values. Remarkably, parameters were found to closely match microglial NF-κB activation, decreasing the data fitting error by nearly 67%, with over a 9-fold improvement during the first 20 min in particular (Additional file

These results provide strong evidence that the addition of dynamics roughly corresponding to the steps involving phosphorylated IκBα recognition and binding by the E3 ligase, polyubiquitination, and proteasomal degradation is sufficient to account for the slightly delayed NF-κB activation observed in microglia.

Nonlinearities in IKK activation and inactivation produce the rapid transient IKK activity in microglia

We next focused our attention on the upstream signaling pathway governing IKK activation in response to TNFα stimulation. The upstream signaling module was decoupled from the downstream model by using the concentration of free nuclear NF-κB produced by the downstream module as a fixed model input (Figure

The original upstream model in

Development of the upstream model and full model fits

**Development of the upstream model and full model fits**. (A) The upstream model with A20 inhibition at two points was modified to include nonlinear IKK activation and inactivation rates, indicated by dashed lines. (B) Simulations with the newly developed upstream model (solid) provided excellent agreement with the model (P = 0.854, SSE = 0.038), much better than what was possible with the existing model structure (dashed) (P = 0.0002, SSE = 0.661). With the newly developed upstream and downstream modules integrated into the full model, new parameter estimates were able to fit both NF-κB (C) and IKK (D) activity in microglia. (E) Model predictions for the total amount of IκBα (solid line) are compared with experimental measurements of total IκBα protein in BV2-cells following 10 ng/ml TNFα treatment.

Activation of the IKK complex at the biomolecular level involves the recruitment and assembly of a signaling complex following TNFα binding to its receptor, as well as numerous post-translational modifications to the complex subunits before IKK is activated by phosphorylation at two residues within its kinase domain

The quick attenuation of IKK activity following its induction is essential to proper signaling and the resulting biphasic NF-κB activity

Feedback from A20 in the published model was proposed to inhibit the transition of inactivated IKK back to its native state

Parameter estimation was performed using the newly developed upstream model with fixed nuclear NF-κB as the model input. Parameters were found for which the model produced excellent agreement with microglial IKK activation (Figure

The newly developed upstream and downstream signaling modules were integrated to form the full model characterizing both IKK and NF-κB activity in response to persistent TNFα stimulus (Additional file ^{-6}). Numerical investigation showed this more oscillatory behavior predicted by the integrated model was due to small changes in the later activation profile of IKK predicted by the upstream model, which had been assumed to remain at a constant, low level when developing the isolated downstream signaling module. After increasing the rate of IκBα nuclear import and re-estimating the A20 feedback and IKK recycling rates, the newly developed model was able to provide good agreement with the data, with fitting errors of only 0.34 for NF-κB (P = 0.013) and 0.43 for IKK (P = 0.291) (Figure

Model prediction validated experimentally

Given that the model was developed using a limited set of data from IKK and NF-κB activation, we next sought to test its ability to predict the dynamics of other model species for which no information was used during parameter estimation. The model was first simulated to obtain the levels of total cellular IκBα protein following TNFα stimulus (Figure

To test this prediction experimentally, BV2 cells were again treated with 10 ng/ml TNFα, and levels of total cellular IκBα were measured at several time points after treatment using ELISA. The results of the experiments were normalized with respect to the initial quantities and compared with the simulation predictions (Figure

Model analysis highlights robustness properties of the network and a dynamic role of feedback regulation in both NF-κB and IKK signaling

The model was next analyzed using sensitivity analysis to gain deeper insight into how the different components of the system interact to regulate the dynamic NF-κB response in microglia. Sensitivity analyses of the NF-κB regulatory network have been performed previously

The normalized sensitivity coefficients for NF-κB activation were solved and plotted as heat maps to illustrate the dynamic relationship between the signaling components and the system response (Figure

Model analysis of transient IKK and NF-κB activation in microglia

**Model analysis of transient IKK and NF-κB activation in microglia**. Sensitivity analysis of the model shows the dynamic NF-κB response (A) and IKK response (B) are regulated differently by different groups of parameters depending on the time interval of interest. See Additional file

The sensitivity results clearly show that the NF-κB response is nearly completely insensitive to variations in some rate parameters (rows of light green), but also moderately or highly sensitive to others (dark red and blue) (Figure

However, the system shows extreme sensitivity to rates controlling the inner and outer feedback loops. The system is very senstive to the rates for induced IκBα synthesis and its association with NF-κB during a time period coinciding with the decline of the first peak, with synthesis and binding rates negatively affecting NF-κB activation. The rate of conversion of inactivated IKK back to native IKK (

The NF-κB response is also highly sensitive to the outer A20 feedback loop in a time-dependent manner. The rates for IKK inactivation by A20 significantly affect the termination of initial NF-κB activity as well as the second phase of activity. This effect is actuated through inhibition of the activation of IKK that has recently been converted from the inactive form and made available for activation; feedback from A20 inhibition of IKK activation has a less substantial role on the dynamics in the model. The outer feedback parameters governing A20 act in opposition to the IKK recycling rate (

Although many features of the NF-κB response have been studied previously using sensitivity analysis, little attention has been paid to the dynamic sensitivities of IKK. We therefore assessed parameter sensitivities of IKK activation in the same way as just described for NF-κB (Figure

While sensitivity analysis with respect to small variations is informative, the nonlinear nature of the system makes it possible that the results may be different when large magnitude changes to the parameters are considered

While the system response is robust to large changes in many of the parameter values, the system is much more responsive to changes in the reaction rates involved in both the inner IκBα and outer A20 feedback loops. In particular, the NF-κB activation profile changes significantly when the rates of induced transcription or translation are changed only a small amount, as indicated by the large distance between the nominal and perturbed trajectories at these values (Figure

Discussion

Our quantitative experimental studies show that microglia share many general features of canonical NF-κB activation observed in many other cell types

Despite the general similarities in NF-κB and IKK activation between microglia and other cell types, a recently published mathematical model of the signaling network

The new model was developed in a modular fashion, which was made possible by collecting ELISA-based measurements of IKK in addition to measurements of NF-κB activity and by exploiting the multiple feedback structure of the network. First the IKK data set from microglia was used to develop the downstream signaling module independently of the outer feedback loop, then the upstream signaling pathway was modified to fit IKK activation data, and finally the two modules were integrated to form the full model for which the parameter estimates were refined. The novel downstream signaling pathway includes additional reactions preceding stimulus-induced IκBα degradation, which are sufficient to capture the delayed onset of NF-κB activity observed in microglia (Figure

Ubiquitination of IκBα is typically thought to occur almost instantaneously following its phosphorylation by IKK

Besides potentially less efficient recognition of IκBα by βTrCP, another possibility is that the normally rapid polyubiquitination of IκBα occurs less efficiently in microglia due to smaller quantities of Nedd8-ylated Cul-1 in the SCF complex. Conjugation of only a small fraction of Cul-1 with Nedd8 greatly increases the efficiency of ubiquitination of IκBα without affecting the association between βTrCP and phosphorylated IκBα

The new model structure indicates a more prominent role of the ubiquitin-proteasome system in regulating NF-κB activation dynamics, which merits consideration of what are its functional implications on how microglia respond to inflammatory stimuli. Analyses of the model show that the ubiquitin-related parameters have large effects on the initial activation of NF-κB and a relatively smaller role in regulating later dynamics (Figure

Substantial modifications to the upstream signaling pathway are required to fit the new model to the microglial IKK activation data. The TNFα-induced IKK activation and inactivation reaction kinetics are changed from first order linear mass-action rates to nonlinear Hill equations in the new model. We note that the new model differs from

In contrast to the parameters governing initial transient IKK activity, our model analyses indicate that the signaling components which regulate later phase IKK activation also exert significant control over NF-κB activation (Figure

The sensitivity analyses of the new model presented here provide new insights into how this signaling pathway is regulated. In particular, we show by examining the temporal evolution of the sensitivities that there is a strong temporal component to system regulation (Figure ^{2}-norm of the dynamic sensitivities

It is important to note that our particular model, which is developed to reproduce population average measurements of IKK and NF-κB activity in microglia, is not unique and other models are capable of producing the same dynamics. It may be desirable in different contexts to extend or otherwise modify this model to explore aspects not considered here. For instance, delayed negative feedback from the IκBε isoform may also contribute substantially to later phase NF-κB signaling dynamics

The analysis from this model for microglial NF-κB activation clearly portrays the canonical NF-κB response on one hand as very robust: cells are able to parse extracellular signals into transient IKK activation to produce a quick and dynamic rise in NF-κB activity, even in the face of uncertainty in many of the reaction rates in both the upstream and downstream pathways. This finding is consistent with sensitivity analysis of related models, in which the response was found to be largely insensitive to the majority of the rate parameters

In the NF-κB signaling network, feedback from IκBα-induced transcription allows the system to respond robustly to stimuli to control gene expression, but at the same time makes the system sensitive to changes in feedback parameters. The highly responsive nature of the system makes it particularly susceptible to network perturbations affecting the feedback molecules IκBα and A20, perhaps as might be seen with severe injury such as stroke. However this feature also provides great opportunities for targeted treatment or intervention to modulate the response. Mathematical modeling and analysis may prove indispensible for future exploration of the NF-κB response and drug targeting in microglia, especially when considering crosstalk among multiple pathways that are simultaneously activated by brain injury.

Conclusions

Mathematical modeling has been used extensively in recent years to provide a detailed view into the activation of NF-κB, helping to make sense of the multiple layers of feedback and to provide a much deeper understanding of how the system functions as a whole. Here we present the development of a mathematical model that quantitatively describes canonical IKK and NF-κB activation in a novel cell type: microglia. The approach we used in model development exploits the multiple feedback structure of the network, and allows the model to be developed in multiple stages by breaking individual feedback loops and developing the modules using the appropriate experimental data. This approach may also prove useful for modeling other biological systems with feedback regulation.

This mathematical model differs significantly from existing NF-κB signaling models in two regards. First, it introduces nonlinearities into the activation and inactivation rates for IKK, which are necessary to reproduce the quantitative IKK profile obtained experimentally and correspond with known biological mechanisms. Secondly, the model includes intermediate dynamics in the induced IκBα degradation pathway. We showed these additional dynamics are essential to characterize NF-κB signaling observed in microglia in a statistically significant manner and are likely due to reactions involved in the ubiquitination and proteasomal degradation of IκBα, suggesting a more prominent role for this system in modulating the NF-κB response.

The mathematical model developed here is the first of its kind for microglia and offers a valuable new tool to study inflammatory signaling in this cell type, permitting rapid numerical simulation and analysis. Our numerical analyses emphasize the highly dynamic nature of regulation of the NF-κB network in response to TNFα stimulus, an aspect which has received relatively little attention in prior analyses. While several key parameters play a significant role in modulating the response throughout the entire duration, many others only regulate the response during specific time intervals, such as during the initial activation phase or the oscillatory later phase. The analysis further provides insight into the robustness properties of the system, indicating high sensitivity to feedback parameters, which we note is analogous to the operation of negative feedback systems in engineering.

Methods

Cell culture

BV2 cells, a mouse microglia cell line and kind gift from Dr. K. Andreasson at Stanford University, were cultured in Dulbecco's Modification of Eagle's medium (DMEM, GIBCO, cat# 11995) supplemented with 8% Fetal Bovine Serum (Hyclone, cat#SV30014.03), Penicillin (100 U/ml, GIBCO, cat#15140), and Streptomycin (100 μg/ml, GIBCO, cat#15140). Cells were passaged every four days and were used between passages 10-20.

Measurement of activated NF-κB p65

BV2 cells were seeded at 4 × 10^{5 }cells per well in six well plates 36 hrs prior to treatment with 10 ng/ml recombinant mouse TNFα (R&D systems, cat# 410-MT). Cells were then harvested for protein at the indicated time points with Phosphosafe Extraction buffer (Novagen, cat#71296) supplemented with 0.01 volume Protease Inhibitor cocktail (Sigma, cat# p8340) and 5 mM DTT before use. Protein concentration was measured using the Coomassie Plus assay (Pierce, cat#23236). 25 μg total protein from each sample was transferred to a pre-chilled Eppendorf tube and brought to 25 μl with complete lysis buffer. These aliquots were stored at -80°C until use for activated NF-κB p65 measurement. Active NF-κB was measured using the Trans AM NFκB p65 Transcription Factor Assay Kit (Active Motif, cat#40096) according to the manufacturer's instructions. 20 μg total protein was used for each sample. Three cultures were assayed for each group. Standards were prepared from recombinant p65 (Active Motif, cat#31102).

IKK measurements

IKK activity was measured by immunoprecipitation of IKK trimers, followed by a kinase assay/ELISA using a modification of the K-LISA IKKβ Inhibitor Screening Kit (Calbiochem, cat# is CBA044). A total of 400 μg protein from each sample was incubated at 4°C 5 hrs with 5 μg goat anti-IKKγ antibody M18 (Santa Cruz Biotechnology, Cat# is SC8256) with shaking, followed by overnight incubation with shaking with 50 μl 2 × diluted Protein G-Sepharose (Sigma, Cat# is P3296) previously washed in complete lysis buffer. Beads were then centrifuged for 5 min at 13,000 rpm 4°C, the post-immunoprecipitation supernatant removed, and beads were washed in the 1 × kinase assay buffer from the K-LISA kit. Beads were then incubated with shaking in an incubator for 1 h at 30°C in 75 μl 1 × kinase assay buffer containing 150 ng GST-IκBα and 1 × ATP/MgCl_{2 }mix from the kit. Beads were then centrifuged at 13,000 rpm for 5 min at 4°C, and 60 μl of supernatant was transferred to a well of the glutathione coated 96-well plate provided with the K-LISA kit. Two-fold serial dilutions of the recombinant IKKβ provided with the kit were run as standards according to the kit instructions, but omitting IKK inhibitor. In addition the post-immunoprecipitation supernatant was concentrated 20 × and run to demonstrate that all IKK activity was depleted from the supernatant. In all cases this sample showed no IKK activity. The plate was incubated 30 min at 30°C to allow the GST-IκBα to bind, and subsequent processing was done according to the vendor's instructions. Final concentrations measured were normalized to the total amount of protein used in a given experiment.

Total IκBα measurement

Total IκBα measurements from TNFα treated BV2 cells were performed using the PathScan Total IκBα Sandwich ELISA kit from Cell Signaling (#7360). BV2 cells from passage 14-18 were seeded at 4 × 10^{5 }cells/ml on day one and treated with 10 ng/ml TNFα on day three. Cell lysates were prepared and ELISA analysis performed following the manufacturer's instructions. Total protein concentrations were measured using the BCA method; 275 μg total protein was used to measure total IκBα at each time point. The experiments were repeated 3 times.

Analysis of experimental data

Data from each experiment for NF-κB and IKK was normalized relative to the maximum mean level of activity during that particular experiment to account for variations in optical absorbance readings between experiments. The normalized data were then averaged to produce the ensemble average data set used for data fitting.

Mathematical modeling and simulation

The model, based on the ordinary differential equation two-feedback model in

**SBML model of microglial NF-κB activation**. This .xml file is an SBML translation of the mathematical model originally developed in Matlab (MathWorks). Simulations correspond to the response of microglia cells following treatment with 10 ng/ml TNFα.

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Statistical evaluation of model simulations

The agreement between model simulations and experimental data was assessed using an approach based on Fisher's combined probability test ^{5}), and so it is assumed that the sample average from this large collection of cells is normally distributed with mean equal to the population average, but that the standard deviation can vary with time. Individual samples are assumed to be independent across experiment replicates and identically distributed with regard to their respective time points. This is justified since all samples are collected from independent cell populations.

Under these assumptions, a two-sided one sample t-test can be used to compare the population mean from the model simulations corresponding to a specific set of parameters, _{i }
_{i}
_{i}

Fisher's method combines the information from the individual test results to test the shared null hypothesis that all the _{i }

where log denotes the natural logarithm, and _{i }
^{2}
_{
F
}follows a chi-square distribution with 2_{F }confidence if _{F}< α, thus giving a statistical basis upon which a candidate parameter set can be rejected or retained.

Parameter estimation and sensitivity analysis

Parameter values were estimated by minimizing the a cost function based on the goodness of fit between model and data. Two objective functions were used: one which computed the normalized sum of squares error (SSE),

between the model simulations at parameter set _{i},θ), _{obs}(t_{i})^{2}
_{F}), an adaptation of the moment-matching algorithm proposed in

The normalized first order sensitivity coefficients of the system,

where _{i }
_{j }is the

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

MK, RGG and PWS contributed to the design of experiments, PWS developed the mathematical model and performed the simulations, JFE and XS perfomed the experiments with BV2 cells, PWS, RGG and MK contributed to writing the manuscript. All authors read and approved the final manuscript.

Acknowledgements

We thank Dr. Katrin Andreasson for BV2 cells and Gabriele Lillacci for assistance with the statistical methods and for reviewing the manuscript. This work was supported by NIH-R01 GM04983 to RGG and MK and by an NSF IGERT Traineeship DGE02-21715 to PWS.