From Transcript to Tissue: Multiscale Modeling from Cell Signaling to Matrix Remodeling

Abstract

Tissue-level biomechanical properties and function derive from underlying cell signaling, which regulates mass deposition, organization, and removal. Here, we couple two existing modeling frameworks to capture associated multiscale interactions-one for vessel-level growth and remodeling and one for cell-level signaling-and illustrate utility by simulating aortic remodeling. At the vessel level, we employ a constrained mixture model describing turnover of individual wall constituents (elastin, intramural cells, and collagen), which has proven useful in predicting diverse adaptations as well as disease progression using phenomenological constitutive relations. Nevertheless, we now seek an improved mechanistic understanding of these processes; we replace phenomenological relations in the mixture model with a logic-based signaling model, which yields a system of ordinary differential equations predicting changes in collagen synthesis, matrix metalloproteinases, and cell proliferation in response to altered intramural stress, wall shear stress, and exogenous angiotensin II. This coupled approach promises improved understanding of the role of cell signaling in achieving tissue homeostasis and allows us to model feedback between vessel mechanics and cell signaling. We verify our model predictions against data from the hypertensive murine infrarenal abdominal aorta as well as results from validated phenomenological models, and consider effects of noisy signaling and heterogeneous cell populations.

Keywords: Constrained mixtures; Growth and remodeling; Homeostasis; Logic-based modeling; Mechanobiology.

Figure 1:

Multiscale coupling between vessel-level growth and remodeling (G&R) processes and cell signaling. Under imposed changes in pressure, flow, and axial load (γP_o, εQ_o, f_z, where γ and ε denote fold-changes from original homeostatic values, subscript o), a constrained mixture G&R model calculates changes in intramural and wall shear stress, which depend on the wall geometry, properties, and applied loads. These changes in mechanical state feed into a logic-based network model, containing 52 species and 89 reactions (see Supplementary Material), to determine corresponding changes in cell signaling. Black solid lines denote activation, red dotted lines inhibition, and the ‘&’ symbol denotes the logical ‘AND’ operation. For clarity, inhibition is shown to affect a node directly; however, an ‘AND NOT’ logic operation is used with all incoming reactions to the node. Outputs from the network model directly affect matrix turnover and contractility, which can be incorporated into the G&R framework, providing (generally negative) feedback via the resulting changes in stresses at the vessel level. Network visualization was carried out using Cytoscape (Shannon, 2003) and Netflux (https://github.com/saucermanlab/Netflux).

Figure 2:

Experimental data (circle ± error bars) and coupled model predictions (solid lines) of the evolving infrarenal abdominal aorta geometry (inner radius, a/a_o, and wall thickness, h/h_o) and circumferential and axial stresses (σ_θθ, σ_zz) in response to a 68% increase in pressure over 28 days. Material parameters are given in Table 1 and network model parameters were fit to the referential mass densities of collagen and intramural cells (${\rho }_R^c/{\rho }_o^c$, ${\rho }_R^m/{\rho }_o^m$) at day 28 (open circles), resulting in best-fit values: y_Stress(0) = 0.163, y_Wss(0) = 0.582, y_AngIIin(0) = 0.113, y_Integrins(0) = 0.20, w = 0.70, n = 1.378, and EC₅₀ = 0.604 (with y_SACs(0) = 0.25).

Figure 3:

Timecourses for five levels of pressure increase (10–50%) from homeostatic, together with a flow increase of 5%. Solid lines indicate results from the coupled model; dashed lines represent results from the phenomenological model, for parameters in Table 2 with η = 1.25. The coupled model was fit only to end values of ${\rho }_R^c/{\rho }_o^c$ and ${\rho }_R^m/{\rho }_o^m$ that were generated by the phenomenological model with P/P_o = 1.3 and Q/Q_o = 1.05 (highlighted in green) yielding y_Stress(0) = 0.216, y_Wss(0) = 0.436, y_AngIIin(0) = 0.20, y_SACs(0) = 0.248, y_Integrins(0) = 0.251, w = 0.763, n = 1.954, and EC₅₀ = 0.621. For the inner radius, a, and wall thickness, h, asterisks indicate ideal adaptations, given by a/a_o → (Q/Q_o)^1/3 and h/h_o → (P/P_o)(Q/Q_o)^1/3, though homeostasis only requires that regulated variables return toward, not precisely to, original values.

Figure 4:

Network-informed stimuli, Δψ_c (collagen), Δψ_p (cell proliferation), Δψ_m (MMPs) (Eqs 35, 48, and 39, respectively) for five levels of pressure increase (10–50%) from homeostatic, together with a flow increase of 5%, and evolution of three network species: AngII, NO and ET1. The coupled model uses y_Stress(0) = 0.216, y_Wss(0) = 0.436, y_AngIIin(0) = 0.20, y_SACs(0) = 0.248, y_Integrins(0) = 0.251, w = 0.763, n = 1.954, and EC₅₀ = 0.621. Exogenous AngII was applied via a sustained input y_AngIIin(s > 0), which is a free parameter in the fitting process. Note the mild transient increase in ET1 (when the wall distends elastically, thus reducing flow-induced shear stress) and complementary decrease in NO, as expected. There are transient (stress-driven) increases in AngII in addition to the sustained increase due to exogenous AngII.

Figure 5:

Steady state values for five levels of pressure elevation (10%, 20%, 30%, 40%, 50%) and three levels of flow increase (0%, 5%, 10%) relative to homeostatic for the coupled (grey surface) and phenomenological (red mesh) models, for parameters in Table 2 with η = 1.25. The coupled model was fit only to end values of ${\rho }_R^c/{\rho }_o^c$ and ${\rho }_R^m/{\rho }_o^m$ that were generated by the phenomenological model with P/P_o = 1.3 and Q/Q_o = 1.05, yielding y_Stress(0) = 0.216, y_Wss(0) = 0.436, y_AngIIin(0) = 0.20, y_SACs(0) = 0.248, y_Integrins(0) = 0.251, w = 0.763, n = 1.954, and EC₅₀ = 0.621.

Figure 6:

Sensitivity analysis of a single cell signaling network. (a) Parameter ranges, where parameters vary from their best-fit values by uniformly distributed noise of up to ±10% and (b) Overlayed timecourses for 30% pressure and 5% flow increases from homeostatic, where 100 parameter sets were sampled from the ranges shown in (a). Baseline parameters are y_Stress(0) = 0.216, y_Wss(0) = 0.436, y_AngIIin(0) = 0.20, y_SACs (0) = 0.248, y_Integrins (0) = 0.251, w = 0.763, n = 1.954, and EC₅₀ = 0.621.

Figure 7:

Sensitivity analysis of a heterogeneous population of cells with collective behavior. (a) Overlayed collagen and intramural cell mass production stimuli, given by Δψ_c and Δψ_p, respectively (Eqs 35, 48), for 100 cells which each contribute equally to collective stimuli (Eq 52), and (b) Resulting timecourses for 30% pressure and 5% flow increases from homeostatic. The 100 parameter sets for individual cells are the same as those used in Fig 6, which vary by up to ±10% from baseline values: y_AngIIin(0) = 0.20, y_SACs(0) = 0.248, y_Integrins(0) = 0.251, w = 0.763, n = 1.954, and EC₅₀ = 0.621. Additionally, y_Stress(0) = 0.216 and y_Wss(0) = 0.436. Asterisks indicate the values obtained under baseline conditions (no perturbations).

Figure 8:

Sensitivity analysis of a single cell signaling network and a heterogeneous population of cells with collective behavior. (a) Parameter ranges, where parameters vary from their best-fit values by uniformly distributed noise of up to ±20%, and (b) Overlayed timecourses for 30% pressure and 5% flow increases from homeostatic, where 100 parameter sets were sampled and applied to the network. We then apply these parameter sets to individual cells within a heterogeneous population, so that they respond collectively, and show (c) Overlayed collagen and intramural cell mass production stimuli, given by Δψ_c and Δψ_p, respectively (Eqs 35, 48), for 100 cells that contribute to collective stimuli (Eq 52), and (d) Resulting timecourses for 30% pressure and 5% flow increases from homeostatic. Parameters vary by up to ±20% from baseline values: y_AngIIin(0) = 0.20, y_SACs(0) = 0.248, y_Integrins(0) = 0.251, w = 0.763, n = 1.954, and EC₅₀ = 0.621. Additionally, y_Stress(0) = 0.216 and y_Wss(0) = 0.436. Asterisks in (d) indicate the values obtained under baseline conditions (no perturbations).