Bistable Mathematical Model of Neutrophil Migratory Patterns After LPS-Induced Epigenetic Reprogramming

Abstract

The highly controlled migration of neutrophils toward the site of an infection can be altered when they are trained with lipopolysaccharides (LPS), with high dose LPS enhancing neutrophil migratory pattern toward the bacterial derived source signal and super-low dose LPS inducing either migration toward an intermediary signal or dysregulation and oscillatory movement. Empirical studies that use microfluidic chemotaxis-chip devices with two opposing chemoattractants showed differential neutrophil migration after challenge with different LPS doses. The epigenetic alterations responsible for changes in neutrophil migratory behavior are unknown. We developed two mathematical models that evaluate the mechanistic interactions responsible for neutrophil migratory decision-making when exposed to competing chemoattractants and challenged with LPS. The first model, which considers the interactions between the receptor densities of two competing chemoattractants, their kinases, and LPS, displayed bistability between high and low ratios of primary to intermediary chemoattractant receptor densities. In particular, at equilibrium, we observe equal receptor densities for low LPS (< 15ng/mL); and dominance of receptors for the primary chemoattractant for high LPS (> 15ng/mL). The second model, which included additional interactions with an extracellular signal-regulated kinase in both phosphorylated and non-phosphorylated forms, has an additional dynamic outcome, oscillatory dynamics for both receptors, as seen in the data. In particular, it found equal receptor densities in the absence of oscillation for super-low and high LPS challenge (< 0.4 and 1.1 <LPS< 375 ng/mL); equal receptor densities with oscillatory receptor dynamics for super-low LPS (0.5 < LPS< 1.1ng/mL); and dominance of receptors for the primary chemoattractant for super-high LPS (>376 ng/mL). Predicting the mechanisms and the type of external LPS challenge responsible for neutrophils migration toward pro-inflammatory chemoattractants, migration toward pro-tolerant chemoattractants, or oscillatory movement is necessary knowledge in designing interventions against immune diseases, such as sepsis.

Keywords: bistability; cellular decision-making; lipopolysaccharide (LPS); mathematical model; neutrophil migration.

Figure 1

Schematic representation of the GRK2 and GRK5 mutual inhibition.

Figure 2

Network diagram for model (1).

Figure 3

Empirical data: (A) Ratio of fMLP/LTB4 cell migration, and (B) number of cells that oscillate (change direction at least three times) while migrating toward fMLP (green) and LTB4 (red) vs. LPS concentration in ng/mL. Data reproduced from Boribong et al. (2019).

Figure 4

Theoretical results: Dynamics of (A) [FPR1] (green) and [BLT1] (red) and, (B) [GRK2] (purple) and [GRK5] (pink) for [LPS](0) = 1 ng/mL (solid lines) and [LPS](0) = 100 ng/mL (dashed lines) as given by model (1). Parameters and initial conditions are given in Table 1.

Figure 5

FPR1/BLT1 at time t = 5 h, as given by model (1) vs. initial LPS dose.

Figure 6

FPR1/BLT1 at time t = 5 h, as given by model (1), for c_w = 28, $b_{wf}=5\times 10^{-4}$ (black stars); c_w = 15, b_wf = 0.13 (blue stars); and c_w = 5, b_wf = 0.245 (red stars).

Figure 7

Diagram for model (4).

Figure 8

Theoretical results: Dynamics of [ERK] and [ERKp] (A,D,G); [GRK2] and [GRK5] (B,E,H); and [FPR1] and [BLT1] (C,F,I), as given by model (4) for [LPS](0) = 1 ng/mL (A–C); [LPS](0) = 100 ng/mL (D–F); and [LPS](0) = 400 ng/mL (G–I).

Figure 9

FPR1/BLT1 at time t = 5 h, as given by model (1), for (Left) c_w = 10 (blue stars), c_w = 15 (black stars), c_w = 20 (red stars); (Middle) n = 1 (blue stars), n = 3 (black stars), n = 5 (red stars); and (Right) a_f = 0.1 (blue stars), a_f = 1 (black stars), a_f = 3 (red stars).