Mathematical Modeling and Parameter Estimation of Intracellular Signaling Pathway: Application to LPS-induced NFκB Activation and TNFα Production in Macrophages

Abstract

Due to the intrinsic stochasticity, the signaling dynamics in a clonal population of cells exhibit cell-to-cell variability at the single-cell level, which is distinct from the population-average dynamics. Frequently, flow cytometry is widely used to acquire the single-cell level measurements by blocking cytokine secretion with reagents such as Golgiplug ™ . However, Golgiplug ™ can alter the signaling dynamics, causing measurements to be misleading. Hence, we developed a mathematical model to infer the average single-cell dynamics based on the flow cytometry measurements in the presence of Golgiplug ™ with lipopolysaccharide (LPS)-induced NFκB signaling as an example. First, a mathematical model was developed based on the prior knowledge. Then, average single-cell dynamics of two key molecules (TNFα and IκBα) in the NFκB signaling pathway were measured through flow cytometry in the presence of Golgiplug ™ to validate the model and maximize its prediction accuracy. Specifically, a parameter selection and estimation scheme selected key model parameters and estimated their values. Unsatisfactory results from the parameter estimation guided subsequent experiments and appropriate model improvements, and the refined model was calibrated again through the parameter estimation. The inferred model was able to make predictions that were consistent with the experimental measurements, which will be used to construct a semi-stochastic model in the future.

Identification of cell‐to‐cell heterogeneity through systems engineering approaches

“…To generate artificial experimental measurements, we assumed a 4 and b 3 to be log‐normally distributed and mutually independent. Then, the activity of C3a ( x 2 ) was measured every hour from 0 to 10 hr subject to the following TNFα stimulation: At each sampling point, t k , x 2 was measured from 5,000 cells, and the measurements were subsequently corrupted with noise (Equation ). Specifically, the proportional factor ( c ) and the measurement noise ( ν ) were log‐normally distributed, and their statistics were known a priori from the equipment calibration, where μ c , σ c , μ ν , and σ ν were 0, 0.1, −3, and 0.3, respectively.…”

“…Also, the proposed approach can be used as a method to develop a grey‐box model to improve the prediction accuracy, when there is a considerable amount of model uncertainty. In many circumstances, a first‐principle model available is partially accurate due to our limited understandings of an process . As a result, a model training via parameter estimation is unlikely to be successful due to the model mismatch.…”

“…In many circumstances, a first-principle model available is partially accurate due to our limited understandings of an process. 41,55 As a result, a model training via parameter estimation is unlikely to be successful due to the model mismatch. On the other hand, our proposed methodology can be used for modeling and estimating a partially known system by introducing PDFs of model parameters, which will take into account the uncertainty in a model structure.…”

“…At each sampling point, t k , x 2 was measured from 5,000 cells, 2,55 and the measurements were subsequently corrupted with noise Figure 5 indicates that the originally homogeneous cell population (i.e., t = 0) evolved into a heterogeneous one with two distinct subsets characterized by the low and high levels of C3a activities. Additionally, it also shows that the GMM smoother performed well to estimate the underlying PDF of the obtained population snapshot data, which justified the use of the GMM smoother for approximating the underlying PDFs.…”

Special Issue on “Biological Networks”

“…The paper by Lee et al [5] looks at the modeling of NF-κb signaling dynamics induced by lipopolysaccharide (LPS) in the presence of a cytokine secretion blocker. Parameter estimation based on average single-cell flow cytometry data points to a previously unidentified action of the cytokine secretion blocker, which was validated in additional experiments and subsequent model refinement.…”