Fig 1.
Bifurcation diagram of VWF conformational stability at different shear stresses.
Bold lines display branches corresponding to stable steady states and dashed lines display branches corresponding to unstable steady states. τ* is a bifurcation value of shear stress. The value τc is a value of shear stress under which VWF molecules must be unwound to their full length. Degree of unfolding (u) denotes a ratio of the tail length of the VWF molecule to the VWF contour length (u = x/L). Adapted from [35].
Fig 2.
Partially unfolded VWF molecule in shear flow.
r denotes the radius of the VWF globular part, x corresponds to the length of the unfolded part on the platelet surface (“tail”) of the molecule, d denotes the characteristic size of the VWF monomeric subunit, Ff denotes the force due to effective surface tension that tends to fold a VWF molecule, Fun is the force that unfolds a VWF molecule under the action of shear stress τ, η denotes blood viscosity, and ω reflects the rotational speed of the globular part of the molecule. The symbol dV/dy denotes a velocity gradient perpendicular to the direction of the flow (i.e. shear rate). Adapted from [35].
Fig 3.
Potential energy profile at various
values.
Three characteristic regions are shown. Any potential curve at ,
and
lies in the domain denoted in red, yellow and green, respectively. Points q = qs and q = qc are points of maximum and minimum potential energy
, respectively. Bold lines correspond to potential energy curves for
,
, and
and are define as
,
and
, respectively.
Fig 4.
Schematic view of the phase portrait at .
The value q = 0 corresponds to the unfolded state of the VWF molecule; and q = qm, to the folded state. q = qs is a saddle point, and the curve ASB denotes the separatrix.
Fig 5.
VWF molecule unfolding under the action of a rectangular shear stress profile with various durations.
(a) Shear stress impulse is shown in red,
is its amplitude, and
is its duration. (b) Phase trajectory of the solution (Eq (8)) is illustrated by the red line for the shear stress profile from Fig 5a; initial conditions: q = qm,
.
,
and
are time moments when the image point reaches points E, F and G, respectively. If the duration of the shear stress impulse
is greater than that of
, the trajectory crosses the part of the separatrix AS, and the molecule can unwind to its full length.
is the absolute value of the momentum at point A.
Fig 6.
Dependence of critical time duration on amplitude
of a rectangular shear stress impulse.
The solid curve corresponds to the dependence obtained in the framework of developed approach (S4 Text for details). The dotted line
is a vertical asymptote of the curve
. The curve
is asymptotically tending to the dashed curve with increasing
. The values of
are measured in units of characteristic unfolding time
(S4 Text). The value of CSS0 is given by Eq (16). All curves were built for the case of N = 36.