Novel Analytic Model of von Willebrand Factor for Predicting Unfurling Time in Extensional Flow

The conformation of von Willebrand Factor (vWF), a long, multimeric biopolymer essential for initiating clot formation, plays a central role in its hemostatic function. In its compact, coiled state, vWF is inactive. However, under elongational flow, vWF can unfurl, exposing its A2 domains, which in turn unfold to reveal platelet-binding sites. Predicting not just whether, but how quickly vWF unfurls is critical for understanding thrombus formation under dynamic flow conditions. While critical shear and extension rates have been proposed as thresholds for activation, real physiological flows rarely maintain constant rates. We introduce a reduced analytical model to estimate vWF unfurling time in pure extensional flow. The model is based on a force balance applied to a ball-and-chain representation, where a compact globular domain is attached to a linear chain of beads. The resulting differential equations are solved analytically, yielding a closed-form expression for unfurling time as a function of the number of dimers N, the initial protrusion length m, and a dimensionless parameter P. We validate the model against high-fidelity simulations in a four-way pipe junction, where opposing inlets and outlets generate a central stagnation point with zero shear and finite extension rate. Simulations explore various N, flow rates (affecting P), and initial conformations (m). The analytical model reliably predicts unfurling time across these conditions, offering a fast and interpretable tool for estimating vWF activation under physiologically relevant flows.