Systems Analysis of the Role of Bone Morphogenic Protein 4 in Endothelial Inflammation

Abstract

Shear stress is an important factor in the onset and progression of atherosclerosis. High and unidirectional laminar stress is seen as protective, while low and oscillatory shear stress is considered pro-inflammatory and pro-atherogenic. The mechanosensitive response of endothelial cells is governed by a complex system of genes, proteins, and signals that operate at distinctly different time scales. We propose a dynamic mathematical model that quantitatively describes this mechanosensing system and permits novel insights into its functioning. The model, the first of its kind, is constructed within the guidelines of Biochemical Systems Theory and accounts for different time scales by means of approximated delays. Parameter values are obtained directly from biochemical observations in an ad hoc fashion. The model reflects most documented observations well and leads to a number of predictions and novel hypotheses. In particular, it demonstrates the crucial role of Bone Morphogenic Protein 4 and p47^phox-dependent NADPH oxidases in endothelial inflammation.

Appendix A

Parameter Estimation

Due to the paucity of available time-series and quantitative data, the rate constant and kinetic order parameter values (γ _i and g _i ) were estimated in an ad hoc fashion as described in “Methods” section (“Parameter Estimation” section). All constraints (both for kinetic orders and rate constants) derived from equations with two terms (see Eq. 3) are presented in Table A1.

TABLE A1 Parameter constraint equations

Given these constraints and assigning a default value (such as 0.5 for substrate dependency or activation and −0.5 for inhibition) to one of the two relevant kinetic orders yields the value for the other one. This procedure requires knowledge of the ratio $M_{{iS_{\text{O}} }}$ or $M_{{iS_{\text{L}} }} .$ Either one of the ratios of experimental measurements to the control can be used for the computation. When both ratios are experimentally available, the ratio measured under OSS condition was used. The available ratios that were recalculated from experimental measurements are listed in Table A2.

TABLE A2 Experimental ratios of steady-state values under OSS and LSS in relation to static control

Once the kinetic orders are set, the rate constants are secondarily deduced with the constraint equations in a similar fashion. While these methods determine a good portion of the needed parameter values, other parameters require default assumptions, experience, and additional efforts for fine tuning and validating, based on available biological observations. Examples are the parameters in the system equations in Eq. (3) that contain external inputs or more than two terms.

Simulation Settings

All simulations (with results displayed in Figs. 2 to 7 in the rext) use the same numerical sets of initial conditions (Table A3) and parameter values (Table A4), with the exception of parameters under investigation in a specific simulation, as described in the text and the corresponding figure legends.

TABLE A3 Numerical values of initial conditions (ICs) and independent variables

TABLE A4 Numerical values of rate constants, kinetic orders, and time delays

The signal of shear stress acting on the phosphorylation of p47 is represented by an exponentially decaying function of the form

$$ p = \left\{ {\begin{array}{*{20}c} {\rho \cdot \exp ( - \sigma \cdot (t - t_{0} )) + offset,} \hfill & {t \ge t_{0} } \hfill \\ {1,} \hfill & {t < t_{0} } \hfill \\ \end{array} } \right., $$

where ρ represents the intensity of the phosphorylation signal after shear stress is added to the system at time t = t ₀. The numerical values of ρ, σ, and offset are listed in Table A3.