Kinematics of Hypocotyl Curvature

Abstract

Time lapse photographs were analyzed for curvature, κ, of the plumular hook as a function of distance from the apex, s, of seedlings of lettuce. Lactuca sativa cv ‘Grand Rapids,‘ during hook maintenance (red light) and hook opening (white light). Curvature of the inner edge of the photographic projection of the hook increases from 0.14 mm–1 near the apex to 12.7 mm–1 at the hook bisector (about 1 mm from the apex) and decreases to approximately zero below the hook (2 mm from the apex). Using concepts from fluid dynamics we relate growth rates to curvature changes. For a material element of stem cross section located at s we predict that where M(s,o) and M(s,i) are the relative elemental growth rates at the outer and inner edges of the hypocotyl cross section, w is element width, t is time, and u(s) is velocity of departure of s from the apex. During hook maintenance ∂[ln(1 + κw)]/∂t is approximately zero if the apex is taken as origin. As each hypocotyl element is displaced from the apex it becomes increasingly curved; then, after displacement past the hook bisector, the element straightens. Growth rates, determined by measurements of the displacement of epidermal hairs, show the pattern expected from the equation: On the apical side of the hook, relative growth rates of the outer edge exceed growth of the inner edge, M(s,o) > M(s,i), while on the basal side of the hook. M(s,o) < M(s,i). The constraint for hook maintenance is that [L(s,o) – L(s,i)] /u(s) be constant in time, where L(s,o) and L(s,i) are the relative elemental growth rates at s. During hook opening ∂[ln(1 + κw)]/∂t becomes important.