A model for shear stress sensing and transmission in vascular endothelial cells

Abstract

Arterial endothelial cell (EC) responsiveness to flow is essential for normal vascular function and plays a role in the development of atherosclerosis. EC flow responses may involve sensing of the mechanical stimulus at the cell surface with subsequent transmission via cytoskeleton to intracellular transduction sites. We had previously modeled flow-induced deformation of EC-surface flow sensors represented as viscoelastic materials with standard linear solid behavior (Kelvin bodies). In the present article, we extend the analysis to arbitrary networks of viscoelastic structures connected in series and/or parallel. Application of the model to a system of two Kelvin bodies in parallel reveals that flow induces an instantaneous deformation followed by creeping to the asymptotic response. The force divides equally between the two bodies when they have identical viscoelastic properties. When one body is stiffer than the other, a larger fraction of the applied force is directed to the stiffer body. We have also probed the impact of steady and oscillatory flow on simple sensor-cytoskeleton-nucleus networks. The results demonstrated that, consistent with the experimentally observed temporal chronology of EC flow responses, the flow sensor attains its peak deformation faster than intracellular structures and the nucleus deforms more rapidly than cytoskeletal elements. The results have also revealed that a 1-Hz oscillatory flow induces significantly smaller deformations than steady flow. These results may provide insight into the mechanisms behind the experimental observations that a number of EC responses induced by steady flow are not induced by oscillatory flow.

FIGURE 1

Schematic diagram of the working model for endothelial shear stress sensing and transmission. Cell-surface flow sensors (which may be discrete structures, cell membrane microdomains, or the entire cell membrane) detect a flow stimulus and transmit it directly via cytoskeleton to various intracellular transduction sites including the nucleus, cell-cell adhesion proteins, and focal adhesion sites on the abluminal cell surface.

FIGURE 2

Schematic diagram of n-Kelvin bodies coupled (A) in series and (B) in parallel. Each body consists of a linear spring with spring constant k₁ in parallel with a Maxwell body, which consists of a linear spring with spring constant k₂ in series with a dashpot with coefficient of viscosity μ.

FIGURE 3

Model networks for endothelial shear stress transmission. (A) Schematic and Kelvin body representation of a four-body network consisting of a flow sensor (body 1) connected to two actin stress fibers (bodies 2 and 3) that are in turn connected to the cell nucleus (body 4). (B) Schematic and Kelvin body representation of a six-body network consisting of a flow sensor (body 1) connected to two actin stress fibers (bodies 2 and 3) that are connected to the nucleus (body 6) via two microtubules (bodies 4 and 5).

FIGURE 4

(A) Time evolution of the deformation of two identical Kelvin bodies connected in parallel in response to steady and oscillatory flow. Because the evolution to the asymptotic response for the two types of flow occurs over different timescales, oscillatory flow evolution is shown in the inset. For both types of flow, a shear force F₀ is applied at t = 0. Because they are connected in parallel, the two bodies deform equally. The deformation exhibits an instantaneous jump (at t = 0) due to the elastic springs with subsequent creeping as the dashpot deforms. (B) Time evolution of the force in body 1 in response to steady and oscillatory (inset) flow. (C) Peak deformation of the bodies as a function of the applied shear force in response to steady and oscillatory flow.

FIGURE 5

Effect of the spring constant k₁₂ (spring constant of spring 1 in Kelvin body 2) on (A) peak deformation and (B) peak asymptotic force in body 1 in a system of two Kelvin bodies connected in parallel under both steady and oscillatory flow conditions. The remaining constants in the two-body system (k₁₁, k₂₁, μ₁, k₂₂, and μ₂) are assumed constant and are assigned the baseline values of actin (Table 1).

FIGURE 6

Effect of the spring constant k₂₂ (spring constant of spring 2 in Kelvin body 2) on (A) peak deformation and (B) peak asymptotic force in body 1 in a system of two Kelvin bodies connected in parallel under both steady and oscillatory flow conditions. The remaining constants in the two-body system (k₁₁, k₂₁, μ₁, k₁₂, and μ₂) are assumed constant and are assigned the baseline values of actin (Table 1).

FIGURE 7

Effect of the dashpot coefficient of viscosity μ₂ of Kelvin body 2 on (A) peak deformation and (B) peak asymptotic force in body 1 in a system of two Kelvin bodies connected in parallel under both steady and oscillatory flow conditions. The remaining constants in the two-body system (k₁₁, k₂₁, μ₁, k₁₂, and k₂₂) are assumed constant and are assigned the baseline values of actin (Table 1).

FIGURE 8

Time evolution of the force in body 1 in a system of two Kelvin bodies connected in parallel and subjected to steady flow. The evolution is shown for selected values of (A) k₁₂, (B) k₂₂, and (C) μ₂. Only the parameter shown is varied. All other parameters are maintained constant at the baseline values of actin (Table 1).

FIGURE 9

Dependence of (A) peak long-term deformation and (B) peak asymptotic force division on frequency in a system of two Kelvin bodies connected in parallel and subjected to oscillatory flow. The results are shown for different values of body 2 dashpot coefficient of viscosity μ₂. The deformations have been normalized by the steady state value for steady flow.

FIGURE 10

Time evolution of the deformation of each component of model endothelial networks under steady flow conditions. (A) Four-body network depicted in Fig. 3 A. (B) Six-body network depicted in Fig. 3 B.

FIGURE 11

Dependence of the peak asymptotic deformation of each component of the six-body model endothelial network depicted in Fig. 3 B on the frequency under oscillatory flow conditions.