Deterministic model of dermal wound invasion incorporating receptor-mediated signal transduction and spatial gradient sensing

Abstract

During dermal wound healing, platelet-derived growth factor (PDGF) serves as both a chemoattractant and mitogen for fibroblasts, potently stimulating their invasion of the fibrin clot over a period of several days. A mathematical model of this process is presented, which accurately accounts for the sensitivity of PDGF gradient sensing through PDGF receptor/phosphoinositide 3-kinase-mediated signal transduction. Analysis of the model suggests that PDGF receptor-mediated endocytosis and degradation of PDGF allows a constant PDGF concentration profile to be maintained at the leading front of the fibroblast density profile as it propagates, at a constant rate, into the clot. Thus, the constant PDGF gradient can span the optimal concentration range for asymmetric phosphoinositide 3-kinase signaling and fibroblast chemotaxis, with near-maximal invasion rates elicited over a relatively broad range of PDGF secretion rates. A somewhat surprising finding was that extremely sharp PDGF gradients do not necessarily stimulate faster progression through the clot, because maintaining such a gradient through PDGF consumption is a potentially rate-limiting process.

FIGURE 1

Model schematic. (a) Single-cell model. The dimensionless PDGF concentration u controls activation of cell surface receptors locally; activated receptors bind the PI 3-kinase enzyme, drawing upon a homogeneous cytosolic pool, yielding a local increase in the density of 3′ PI lipid messengers in the plasma membrane. Random migration and chemotaxis are assumed to depend on the average 3′ PI density and the asymmetry in the 3′ PI profile, respectively, and are coupled in a consistent way to fibroblast proliferation and receptor-mediated PDGF clearance. (b) Macroscopic model. The PDGF concentration u and fibroblast density v evolve as a function of time and location in two tissue domains: a finite clot, where PDGF is produced by platelets and macrophages (normalized rate k_s), and an adjoining, semi-infinite dermis, where the fibroblasts are initially found at a constant, resting density (v = 1). A one-dimensional patch wound with clot thickness l is depicted here.

FIGURE 2

PDGF gradient sensing is optimized at intermediate PDGF concentrations. Analysis of the single-cell model (Eqs. 2–6) is presented, assuming α = 10, κ = 0.1, m₀ = 0.1. (a) Average receptor activation and 3′ PI messenger level, and messenger levels at the front (ξ = 1) and back (ξ = 0) of the cell, for PDGF gradients of varying midpoint PDGF concentration and a 30% gradient across the cell. (b) The difference between front and rear messenger levels [m(1) – m(0) = e(1) – e(0)] is plotted as a function of midpoint PDGF concentration with relative gradient δ = 0.02, 0.05, 0.1, 0.2, 0.5, or 1. The curves peak at u = u_opt ≈ 0.5.

FIGURE 3

Dynamic range of the PDGF concentration profile in the clot. The steady-state nullclines, in the absence of spatial gradients, describe conditions where PDGF synthesis and consumption are balanced (Eq. 12; dot-dashed curve, base-case parameters; dotted curve, consumption by fibroblasts only, with k_u = 0) or where there is zero net fibroblast growth (Eq. 13, solid curve). The intersection of these curves, (u^†,v^†), satisfies both criteria, bounding the PDGF concentration profile between u^† and u_max = k_s/k_u.

FIGURE 4

Progression of wound invasion. (a–d) Model calculations were performed assuming base-case parameters and a one-dimensional patch clot with thickness l = 3 mm. Profiles of PDGF concentration u ((a) dashed line, u = u^† = 0.424), fibroblast density v ((b) dashed line: v = v^† = 8.53), and chemotactic signaling Δm (c) at the indicated times in days are shown. (d) The fibroblast penetration depth, defined here as the maximum distance into the clot at which v = 0.5, increases linearly with time after a transient of ∼2 days. (e) Model calculations assuming a two-dimensional slash clot. The V-shaped clot has a depth of 10 mm and a width of 5 mm at the top; the dimensions of the tissue portion shown are 20 × 20 mm. Profiles of PDGF concentration (scale, 0–10), fibroblast density (scale, 0–10), and chemotactic signaling (scale, 0–0.1) are shown at t = 7 days. The magnitude of the latter incorporates both x- and y-components of the Δm vector, according to .

FIGURE 5

Sensitivity of fibroblast invasiveness to PDGF secretion rate. Calculations were performed as in Fig. 4, a–d, after adjustments to the normalized PDGF secretion rate, k_s. The dimensionless saturation constant for fibroblast proliferation was set at the base-case value (0.1, a and b; the vertical lines signify the base-case value of k_s, 1 h⁻¹) or 10-fold lower (c and d). Fibroblast penetration depth (determined as in Fig. 4 d) and population size () were assessed at t = 10 days (a and c). The peak values in chemotactic signaling, Δm, at the leading fibroblast front and clot-dermis interface at t = 7 days are shown for the same parameter values (b and d). The asterisks signify that the peak at the clot-dermis interface is progressively moved rearward into the dermis as k_s gets large.

FIGURE 6

Sensitivity of fibroblast invasiveness to other PDGF rate constant values. Calculations were performed as in Fig. 5 after adjustments to rate constants in the PDGF balance, Eq. 7. See accompanying descriptions in the text. (a) Fibroblast penetration depth and population size at t = 10 days, normalized by the base-case values, after a 10-fold reduction or increase in the indicated parameter values. (b) Peak values in chemotactic signaling, Δm, at the leading fibroblast front and clot-dermis interface at t = 7 days were determined for the same parameter values as in a. (c and d) Fibroblast density and PDGF concentration profiles at t = 7 days for the case of a 10-fold reduction in k_u (c) or 10-fold increases in both k_s and k_v (d); these parameter shifts yield the same u_max and nearly identical u^† values.

FIGURE 7

Variation of fibroblast invasiveness with maximum chemotactic cell speed. The peak value in chemotactic signaling (Δm) at the leading fibroblast front is a decreasing function of the maximum chemotactic cell speed, . The penetration depth at t = 10 days, evaluated as in Fig. 5, a and c, increases linearly with the change in the product of Δm(). Above the base-case value of 0.1 mm/h, the fibroblasts reach the exterior boundary of the 3 mm clot by 10 days, and so the comparable penetration depth was estimated by extrapolation of the constant propagation velocity regime (values denoted by asterisks).