# A Cybernetic Approach to Modeling Lipid Metabolism in Mammalian Cells

^{1}

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## Abstract

**:**

## 1. Introduction

_{2}-Lipid A (KLA) and activated with a purinergic P2X7 receptor agonist adenosine triphosphate (ATP) [18]. Despite their ability to effectively predict the metabolite levels, these models either do not incorporate biological regulatory mechanisms or only account for simple regulation, such as at the gene expression level [17,18,19,20]. Given that biological processes are regulated at many other stages, such as posttranslational protein modification and interaction with a protein or substrate molecule, we have used the cybernetic modeling framework to account for such regulations in the modeling of lipid metabolism in this work.

## 2. Materials and Methods

_{2}), and its subsequent conversion into downstream products, prostaglandin D2 (PGD

_{2}), prostaglandin E2 (PGE

_{2}), and prostaglandin F2α (PGF

_{2α}). In this simple network of PG formation, the main focus is on how the PGH

_{2}conversion into the three downstream PG products is regulated, which may represent a central decision point in the lipid metabolic system in the macrophage inflammatory response (Figure 1a). The behavior of this network is modeled in three separate conditions, a control, a treatment with ATP, and a combined treatment of ATP and KLA. Measurements were made at 0, 0.25, 0.5, 1, 2, 4, 8, and 20 h after ATP stimulation (Figure 1b). The data for all these conditions was taken from LIPID MAPS [28,29,30,31]. Details of the experimental procedure can be obtained from the LIPID MAPS website (protocol available at: www.lipidmaps.org/protocols/PP0000004702.pdf).

#### 2.1. Development of the Kinetic Model

_{2}using simple linear kinetics. The kinetics of this reaction is modeled as three separate mechanisms, including a basal rate of synthesis, generation due to ATP stimulation, and KLA priming of cells.

_{s}= 2 h

^{−1}) and decreases exponentially (k

_{d}= 17.2 h

^{−1}) following the initial half hour of the experiment. For the KLA treatment case, the same f(t) was used with a 4 h adjustment to account for the 4 h priming of KLA prior to the ATP stimulation.

#### 2.2. The Cybernetic Framework

_{2}to a PG product.

^{th}pathway is modeled as regulated by the control of the enzyme level and its activity, that is:

#### 2.3. Defining the Cybernetic Goal or Objective

_{2}, PGF

_{2α}, and PGD

_{2}, respectively). Weights were obtained from ${c}_{i}$ and $\overline{[P{G}_{i}]}$, the average concentration of $P{G}_{i}$ across time, via regression of $P{G}_{i}$ and TNFα data (Matlab

^{®}function ‘fmincon’, using the interior-point algorithm) using eight time points across ATP stimulated and control conditions; ${c}_{i}$ does not change with time. Of the three pathways modeled, there is a varying degree of inflammation that results from the generation of each $P{G}_{i}$ as described by the objective function. In this particular system, the ROI for each pathway is assumed to be the amount of TNFα that each unregulated pathway can yield at each instant in time, which is described by ${\rho}_{i}$.

#### 2.4. Estimation of the Kinetic Rate Parameters and Uncertainty Analysis

_{2}, PGF

_{2α}, and PGD

_{2}metabolites as an eight-point time series over a 20 h time window. PGH

_{2}is an unstable intermediate metabolite we could not measure experimentally; in our model, we constrained the maximum concentration of PGH

_{2}to be ~10 pmol/μg DNA based on the total amount of PGs produced. The magnitudes of the different metabolites varied from 0.001 to 10 pmol/μg of DNA. To fit the model to the data, a least squared fit error was computed from the scaled profiles of the lipid, with respect to its maximum value, to ensure that the varying magnitude of each PG’s level did not skew the parameters towards the sole fit of the PGs with higher magnitudes. The overall objective function for fitting the data was to minimize the fit-error between the experimental and the predicted metabolite concentrations [17], as follows:

_{0}is the set of initial conditions of the enzyme concentrations; ni is the number of time-points, 21, interpolated from 0 to 20 h (indexed as j) in order to provide equal weightage to later time points in the model fit; and nsp is the total number of species (indexed as i). The ODEs in the model were solved using ode15s for the stiff systems in MATLAB (version 9.4.0.813654, R2018a, Natick, MA, USA). The parameters (Table A1) were optimized using a two-step hybrid optimization procedure that started with a genetic algorithm seeded with random initial parameter values and evolved up to 100 generations in order to determine near optimal parameter values (Matlab

^{®}function ‘ga’). The results from the application of the genetic algorithm-based optimization were then further refined using a generalized constrained non-linear optimization employing a gradient search method (Matlab

^{®}function ‘fmincon’, using the interior-point algorithm).

_{1}= $\left(2\times nt\right)$ and df

_{2}= $\left(2\times nt\times \left(nr-1\right)\right)$. F statistic values smaller than F

_{0.95}(16, 32) = 1.97 indicate statistically equal variance in simulated (fitted) and experimental data; whereas, F values smaller than F

_{0.05}(16, 32) = 0.4580 indicate the fit-error is statistically smaller than the experimental error.

## 3. Results

#### 3.1. Development of the Kinetic Model for the COX Pathway

_{2}, and its subsequent conversion into downstream prostaglandin products, PGE

_{2}, PGF

_{2α}, and PGD

_{2}. In this simple network of PG formation, the primary intent is on the regulation of PGH

_{2}conversion into the three downstream PG products. To address the latter, cybernetic regulation (implementation of ${u}_{i}$ and ${v}_{i}$ variables) was used at this branch point. The model for the COX pathway was described by 7 ODEs and 18 kinetic parameters (Table A1) in total; these 18 rate constants were estimated using a hybrid optimization approach (Materials and Methods). Using the optimized parameters, the eicosanoid profiles for the control and ATP stimulated cases were simulated (Figure 2). For most time points, the difference between the simulated and experimental data in both the treatment and control conditions fell within the standard error of the mean. The goodness of fit for the model was further examined by performing the F-test, indicating that the fit-error was less than the experimental measurement error (Table 1).

_{2}, PGE

_{2}, and PGF

_{2α}or in the KLA parameter is seen in response to metabolite changes. This is especially relevant to note given that the data set in which the parameter set was optimized for simulation was not treated with KLA and, consequently, would not have a dependence on this parameter. Based on these results, our model of eicosanoid metabolism is shown to be robust with respect to parametric perturbations.

#### 3.2. Prediction of the Eicosanoid Profile in KLA Primed ATP Stimulated Macrophages

#### 3.3. Understanding the Role of Regulation in the Cybernetic Variables

_{2α}branch is a non-enzymatically regulated process and does not have an associated gene for comparison with the corresponding cybernetic variable.

_{i}/e

_{i,max}) to visualize a clear comparison of the dynamic trends. These comparisons are made for both the ATP and the combined KLA primed ATP stimulated treatment conditions. Overall, the scaled predicted enzyme profiles in solid green (ATP stimulated case) and magenta (KLA primed ATP stimulated case) match the general behavior of their corresponding genes, identified in Table 2, which are denoted by dashed black lines (Figure 5). There are some discrepancies such as in the Ptges profile for KLA primed and ATP stimulated BMDM, as well as, the Hpgds profile for ATP stimulated BMDM. This could be attributed to the different types of data used in the comparison plot of the cybernetic variables, which represent the modeled proteomic levels of the enzymes with the available transcriptomic data.

## 4. Discussion

_{2}in macrophages is not detectable and is determined to be less than 1 nmol/min/mg of total protein, which is consistent with our model value, 1 × 10

^{−5}nmol/min/mg of total protein [46]. For the flux of PGE

_{2}, the reported literature value of 0.4 pmol/min/mg of the total protein is of the same order as our computed value, 0.1 pmol/min/mg of protein [43]. Our computed values are consistent with those reported in the literature and further validate our computational model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Reaction parameters were estimated for the eicosanoid metabolism model. The simulated and predicted columns refer to the parameters optimized for ATP stimulated BMDM cells and KLA primed ATP stimulated BMDM cells, respectively. The predicted parameters were further optimized from the simulated parameters within 30% variability.

PARAMETER | SIMULATED | PREDICTED |
---|---|---|

K_{PGH2} | 0.0022 | 0.0016 |

K_{PGE2} | 0.0044 | 0.0031 |

K_{PGF2α} | 0.0326 | 0.0339 |

K_{PGD2} | 0.0533 | 0.0585 |

𝛾_{PGE2} | 0.0062 | 0.0044 |

𝛾_{PGF2}_{α} | 0.0205 | 0.0197 |

𝛾_{PGD2} | 0.1275 | 0.0893 |

K_{KLA} | 17.3923 | 0.0001 |

K_{ATP} | 11.9112 | 8.3379 |

K_{E,PGE2} | 8.0801 | 10.4215 |

K_{E,PGF2α} | 0.2078 | 0.1478 |

K_{E,PGD2} | 0.2243 | 0.157 |

𝛾_{PGH2} | 0.2603 | 0.3384 |

α | 0.2244 | 0.2918 |

β | 0.7757 | 1.0082 |

E_{0,PGE2} | 0.3974 | 0.5094 |

E_{0,PGF2α} | 0.0133 | 0.0105 |

E_{0,PGD2} | 0.2601 | 0.3379 |

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**Figure 1.**(

**a**) The arachidonic acid metabolic pathway map for the breakdown of arachidonic acid into respective prostaglandin products via prostaglandin H2 (PGH

_{2}) is shown: (rectangles) enzymes, (ellipses) lipid metabolites, (shaded) measured metabolites, (arrows) enzymatic and non-enzymatic reactions; (

**b**) bone marrow derived macrophages (BMDM) were pretreated with or without KLA for 4 h and then stimulated with or without ATP. The media and cells were collected for lipidomic, tumor necrosis factor alpha (TNFα) and transcriptomic analysis at 0, 0.25, 0.5, 1, 2, 4, 8, and 20 h after ATP stimulation; (

**c**) depiction of the simplified system network used for kinetic modeling illustrates PGH

_{2}as a control point and e

_{1}, e

_{2}, and e

_{3}as cybernetic enzymes regulated via cybernetic variables for the regulation of PGD

_{2}, PGE

_{2}, and PGF

_{2α}fluxes.

**Figure 2.**The computational simulation of the eicosanoid profile is generated using the cybernetic model in ATP stimulated BMDM. The mean experimental data (circles) with associated standard error of the mean (SEM) from three replicate experiments (n = 3) for the ATP stimulated (green) and control (red) cases are taken from the mass spectrometry measurements of lipids. The simulation results are shown for the treatment and control cases (solid green and red curves, respectively).

**Figure 3.**The slope of the sensitivity curves of the arachidonic acid (AA) metabolism are shown as a heat map. For example, the changes in the parameter associated with a conversion of AA into prostaglandin H2 (PGH

_{2}) resulted in an increase in all of the metabolites; whereas, changes in the degradation of PGH

_{2}resulted in a decrease in all of the metabolites. This is expected, given that PGH

_{2}is in the upper part of the network, so the changes associated with these parameters will result in an impact on all of the corresponding downstream metabolites.

**Figure 4.**The computational prediction of the eicosanoid profile is generated using the cybernetic model in KLA primed and ATP stimulated BMDM. The mean experimental data (circles) with associated standard error of the mean (SEM) from three replicate experiments (n = 3) for KLA primed ATP-treated (magenta) and control (red) cases are taken from the mass spectrometry measurements of the lipids. The prediction results are shown for the treatment and control cases (solid magenta and red curves, respectively).

**Figure 5.**The behavior of the scaled cybernetic model enzyme level simulations (green in ATP stimulated case and magenta in KLA primed, followed by ATP stimulated case) generally match the trends of the scaled gene expression values (black dashed lines) for Ptges and Hpgds/Ptgds2 pathways in (left) ATP and (right) combined KLA primed ATP stimulated treatments.

**Table 1.**Goodness of fit, F-test, for simulated/optimized (adenosine triphosphate (ATP) stimulated data) and predicted (Kdo2-Lipid A (KLA) primed and ATP stimulated) cases. F values smaller than F

_{0.05}(16, 32) = 0.4580 indicate that the fit-error is statistically smaller than the experimental error; whereas the F values smaller than F

_{0.95}(16, 32) = 1.97 indicate that the fit-error is statistically comparable to the experimental error. PGD

_{2}—prostaglandin D2; PGE2—prostaglandin E2; PGF

_{2α}—prostaglandin F2α.

Metabolite | Model Fit to ATP Data | Model Fit to KLA and ATP Data |
---|---|---|

PGE_{2} | 0.0312 | 0.2421 |

PGF_{2α} | 0.0470 | 0.0342 |

PGD_{2} | 0.2636 | 0.1192 |

**Table 2.**Enzymes were identified from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database and other selected resources for each pathway downstream of prostaglandin H2 (PGH

_{2}) in prostaglandin synthesis. There is not a specific enzyme associated with the regulation of PGH

_{2}into PGF

_{2α}.

Entrez ID | Pathway | Gene Symbol | Name |
---|---|---|---|

64292 | PGH_{2} → PGE_{2} | Ptges | prostaglandin E synthase |

54486 | PGH_{2} → PGD_{2} | Hpgds/Ptgds2 | hematopoietic prostaglandin D2 synthase |

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**MDPI and ACS Style**

Aboulmouna, L.; Gupta, S.; Maurya, M.R.; DeVilbiss, F.T.; Subramaniam, S.; Ramkrishna, D. A Cybernetic Approach to Modeling Lipid Metabolism in Mammalian Cells. *Processes* **2018**, *6*, 126.
https://doi.org/10.3390/pr6080126

**AMA Style**

Aboulmouna L, Gupta S, Maurya MR, DeVilbiss FT, Subramaniam S, Ramkrishna D. A Cybernetic Approach to Modeling Lipid Metabolism in Mammalian Cells. *Processes*. 2018; 6(8):126.
https://doi.org/10.3390/pr6080126

**Chicago/Turabian Style**

Aboulmouna, Lina, Shakti Gupta, Mano R. Maurya, Frank T. DeVilbiss, Shankar Subramaniam, and Doraiswami Ramkrishna. 2018. "A Cybernetic Approach to Modeling Lipid Metabolism in Mammalian Cells" *Processes* 6, no. 8: 126.
https://doi.org/10.3390/pr6080126