Phototropic bending of non-elongating and radially growing woody stems results from asymmetrical xylem formation

Active phototropic bending of non-elongating and radially growing portion of stems (woody stems) has not been previously documented, whereas negative gravitropic bending is well known. We found phototropic bending in woody stems and searched for the underlying mechanism. We inclined 1-year-old Quercus crispula Blume seedlings and unilaterally illuminated them from a horizontal direction perpendicular to ('normal' illumination) or parallel to ('parallel' illumination) the inclination azimuth. With normal illumination, active phototropic bending and xylem formation could be evaluated separately from the negative gravitropic response and vertical deflection resulting from the weight of the seedlings. One-year-old stems with normal illumination bent significantly, with asymmetrical xylem formation towards the illuminated upper surface and side of the stem, whereas those with parallel illumination showed non-significant lateral bending, with asymmetrical xylem formation only on the upper side. A mechanical model was built on the assumption that a bending moment resulted from the asymmetrical xylem formation during phototropic bending of the woody stems. The model fitted the relationship between the observed spatial distributions of the xylem and the observed lateral bending, and thus supported the hypothesis that phototropic bending of woody stems results from asymmetrical xylem formation, as such occurs during gravitropism.     

Oscillation frequencies of tapered plant stems

Free oscillations of upright plant stems, or in technical terms, slender tapered rods with one end free, can be described by considering the equilibrium between bending moments in the form of a differential equation with appropriate boundary conditions. For stems with apical loads, where the mass of the stem is negligible, Mathematica 4.0 returns solutions for tapering modes α = 0, 0.5, and 1. For other values of α, including cases where the modulus of elasticity varies over the length of the stem, approximations leading to an upper and a lower estimate of the frequency of oscillation can be derived. For the limiting case of ω = 0, the differential equation is identical with Greenhill's equation for the stability against Euler buckling of a top-loaded slender pole. For stems without top loads, Mathematica 4.0 returns solutions only for two limiting cases, zero gravity (realized approximately for oscillations in a horizontal orientation of the stem) and for ω = 0 (Greenhill's equation). Approximations can be derived for all other cases. As an example, the oscillation of an Arundo donax plant stem is described.     

An oscillation theory of handwriting

Handwriting production is viewed as a constrained modulation of an underlying oscillatory process. Coupled oscillations in horizontal and vertical directions produce letter forms, and when superimposed on a rightward constant velocity horizontal sweep result in spatially separated letters. Modulation of the vertical oscillation is responsible for control of letter height. Modulation of the horizontal oscillation is responsible for control of corner shape through altering phase or amplitude. The vertical velocity zero crossing in the velocity space diagram is important from the standpoint of control. Changing the horizontal velocity value at this zero crossing controls corner shape. Changing the slope at this zero crossing controls writing slant. The corner shape and slant constraints completely determine the amplitude and phase relations between the two oscillations. This theory applies generally to a number of acceleration oscillation patterns such as sinusoidal rectangular and trapezoidal oscillations. The oscillation theory also provides an explanation for how handwriting might degenerate with speed. An implementation of the theory in the context of the spring muscle model is developed. Here sinusoidal oscillations arise from a purely mechanical source; orthogonal antagonistic spring pairs generate particular cycloids depending on the initial conditions. Modulating between cycloids can be achieved by changing the spring zero settings at the appropriate times. Frequency can be modulated either by shifting between coactivation and alternating activation of the antagonistic springs or by presuming variable spring constant springs. An acceleration and position measuring apparatus was developed for measurements of human handwriting. Measurements of human writing are consistent with the oscillation theory.     

Kinetics and mechanics of stem gravitropism in Coprinus cinereus

Using video recordings we have completed the first kinetic analysis of mushroom stem gravitropism. The stem became gravireceptive after completion of meiosis, beginning to bend within 30 minutes of being placed horizontal. Stem bending first occurred in the apical 15% of its length, then the position of the bend moved rapidly towards the base, traversing 40% of stem length in 2.5 h. Meanwhile, the stem elongated by 25%, mostly in its upper half but also in basal regions. If the apex was pinned horizontally the stem base was elevated but overshot the vertical, often curling through more than 300 degrees. When the base was pinned to the horizontal (considered analogous to the normal situation), 90% of the initial bend was compensated as the stem brought its apex accurately upright, rarely overshooting the vertical. The apex had to be free to move for this curvature compensation to occur. Stems transferred to a clinostat after some minutes gravistimulation showed curvature which increased with the length of initial gravistimulation, indicating that continued exposure to the unilateral gravity vector was necessary for continued bending. Such gravistimulated stems which bent on the clinostat subsequently relaxed back towards their original orientation. Reaction kinetics were unaffected by submergence in water, suggesting that mechanical events do not contribute, but submerged stems bent first at the base rather than apex. In air, the gravitropic bend appeared first near the apex and then moved towards the base, suggesting basipetal movement of a signal. In water, the pattern of initial bending was changed (from apex to base) without effect on kinetics. Taken together these results suggest that bending is induced by a diffusing chemical growth factor (whose extracellular propagation is enhanced under water) which emanates from the apical zone of the stem. The apex is also responsible for regulating compensation of the bend so as to bring the tip to the vertical. The nature of this latter stimulus is unknown but it is polarized (the apex must be free to move for the compensation to occur) and it may not require reference to the unilateral gravity vector.     

Gravitropism in Higher Plant Shoots : IV. Further Studies on Participation of Ethylene

Ethylene at 1.0 and 10.0 cubic centimeters per cubic meter decreased the rate of gravitropic bending in stems of cocklebur (Xanthium strumarium L.) and tomato (Lycopersicon esculentum Mill), but 0.1 cubic centimeter per cubic meter ethylene had little effect. Treating cocklebur plants with 1.0 millimolar aminoethoxyvinylglycine (AVG) (ethylene synthesis inhibitor) delayed stem bending compared with controls, but adding 0.1 cubic centimeter per cubic meter ethylene in the surrounding atmosphere (or applying 0.1% ethephon solution) partially restored the rate of bending of AVG-treated plants. Ethylene increases in bending stems, and AVG inhibits this. Virtually all newly synthesized ethylene appeared in bottom halves of horizontal stems, where ethylene concentrations were as much as 100 times those in upright stems or in top halves of horizontal stems. This was especially true when horizontal stems were physically restrained from bending. Ethylene might promote cell elongation in bottom tissues of a horizontal stem or indicate other factors there (e.g. a large amount of `functioning' auxin). Or top and bottom tissues may become differentially sensitive to ethylene. Auxin applied to one side of a vertical stem caused extreme bending away from that side; gibberellic acid, kinetin, and abscisic acid were without effect. Acidic ethephon solutions applied to one side of young seedlings of cocklebur, tomato, sunflower (Helianthus annuus L.), and soybean (Glycine max [L.] Merr.) caused bending away from that side, but neutral ethephon solutions did not cause bending. Buffered or unbuffered acid (HCl) caused similar bending. Neutral ethephon solutions produced typical ethylene symptoms (i.e. epinasty, inhibition of stem elongation). HCl or acidic ethephon applied to the top of horizontal stems caused downward bending, but these substances applied to the bottom of such stems inhibited growth and upward bending—an unexpected result.     

Oscillations of plants' stems and their damping: theory and experimentation

Free oscillations of upright plants' stems, or in technical terms slender tapered rods with one end free, can be modelled by considering the equilibrium between bending moments and moments resulting from inertia. For stems with apical loads and negligible mass of the stem and for stems with finite mass but without top loading, analytical solutions of the differential equations with appropriate boundary conditions are available for a finite number of cases. For other cases approximations leading to an upper and a lower estimate of the frequency of oscillation omega can be derived. For the limiting case of omega = 0, the differential equations are identical with Greenhill's equations for the stability against Euler buckling of slender poles. To illustrate, the oscillation frequencies of 25 spruce trees (Picea sitchensis (Bong.) Carr.) were compared with those calculated on the basis of their morphology, their density and their static elasticity modulus. For Arundo donax L. and Cyperus alternifolius L. the observed oscillation frequency was used in turn to calculate the dynamic elasticity modulus, which was compared with that determined in three-point bending. Oscillation damping was observed for A. donax and C. alternifolius for plants' stems with and without leaves or inflorescence. In C. alternifolius the difference can be attributed to the aerodynamic resistance of the leaves, whereas in A. donax structural damping in addition plays a major role.     

Gravitropism in Higher Plant Shoots : II. Dimensional and Pressure Changes during Stem Bending

Dimensional changes during gravitropic bending of cocklebur (Xanthium strumarium L.) dicot stems were measured using techniques of stereo photogrammetry. The differential growth is from an increased growth rate on the bottom of the stem and a stopping or contraction of the top.

Contraction of the top was especially evident upon release and immediate bending of horizontal stems that had been restrained between stiff wires for 36 hours. The energy for this could have been stored in both the top and bottom, since the bottom elongated, and the top contracted.

Forces developed during bending were measured by fastening a stem tip to the end of a bar with attached strain gauges and recording electrical output from the strain gauges. Restrained mature cocklebur stems continued to accumulate potential energy for bending for about 120 hours, after which the recorded force reached a maximum.

Pressures within castor bean (Ricinus communis L.) stems were also measured with 3.5-millimeter diameter pressure transducers. As expected, the pressure on the bottom of the restrained plants increased with time; pressures decreased in vertical controls, tops of restrained stems, and bottoms of free-bending stems. Pressures increased in tops of free-bending stems. When restrained plants were released, pressure on the bottom decreased and pressure on the top increased. Results suggest a possible role for cell contraction in the top of stems bending upward in response to gravity.

     

Gravitropism in Higher Plant Shoots : III. Cell Dimensions during Gravitropic Bending; Perception of Gravity

Cross and longitudinal sections were prepared for light microscopy from vertical control plants (Xanthium strumarium L. Chicago strain), free-bending horizontal stems, plants restrained 48 hours in a horizontal position, and plants restrained 48 hours and then released, bending immediately about 130°. Top cells of free-bending stems shrink or elongate little; bottom cells continue to elongate. In restrained stems, bottom cells elongate some and increase in diameter; top cells elongate about as much but decrease in diameter. Upon release, bottom cells elongate more and decrease in diameter, while top cells shorten and increase in diameter, accounting for the bend. During restraint, bottom cells take up water while tissue pressures increase; top cells fail to take up water although tissue pressures are decreasing.

Settling of amyloplasts was observed in cells of the starch sheath.

Removal of different amounts of stem (Xanthium; Lycopersicon esculentum Miller, cv Bonny Best; Ricinus communis L. cv Yolo Wonder) showed that perception of gravity occurs in the bending (elongation) zone, although bending of fourth and fifth internodes from the top was less than in uncut controls. Uniform application of 1% indoleacetic acid in lanolin to cut stem surfaces partially restored bending. Reversing the gradient in tension/compression in horizontal stems (top under compression, bottom under tension) did not affect gravitropic bending.

     

Dynamic behaviour of inflorescence-bearing Triticale and Triticum stems

The mechanical response of cereal plant shoots to loads caused by wind and gravity in the field is swaying in flexure around the vertical or near vertical transient equilibrium position determined by the stationary component of the wind pressure. The aim of this work was to characterise the kinematic and dynamic attributes and their interrelations in freely swaying inflorescence-bearing stems of wheat (Triticum aestivum L.) and Triticale. The fundamental natural frequency of the stems appeared to be considerably lower than predicted from the theory of vibration using the model of a cantilever beam oscillator and assuming the spring constant to be equal to the force-deflection ratio. Because of the rate of deformation and visco-elastic behaviour of the plant material, a discrepancy of about 10% was found between the dynamic and static stem bending resistance. The presence of the tip inflorescence caused vibrating vertical stems to behave as compressed columns in which the effective spring constant was strongly biased by the apical load due to the weight of the inflorescence. At the late milk stage, in the freely swaying stems of wheat and Triticale, the resistance to dynamic lateral loads was reduced by about 30% as a result of compression exerted by the inflorescence. So the prominent effect of the tip inflorescence on the dynamic behaviour (the effective spring constant and the natural frequency) of the stem is attributed to the non-negligible magnitude of the inflorescence weight relative to the critical load producing elastic buckling in slender vertical structures. Stem softening as a consequence of increasing inflorescence weight is assumed to be one of the essential factors reducing the lodging resistance in cereal crops at the late milk stage. The feasibility of the compressed-column approach for predicting the dynamic bending performance of slender vertical plant organs is discussed. Received: 4 March 1998 / Accepted: 20 July 1998     

An experimental study on in vivo dynamic mechanical properties of the knees of cats in valgus loading

In these experiments an attempt was made to investigate in vivo dynamic mechanical properties of the knee in valgus loading. Fifteen adult cats were used. Sinusoidal bending oscillation of 0.3, 1.0, 1.6, 3.3, 5.0, 8.6 and 10.0 Hz was transmitted to the knee-leg complex under each pre-bending strain in the valgus direction, and the bending moment was measured. Dynamic mechanical properties of the knee-leg complex were given as a complex bending elastic coefficient. Bending elastic and bending damping coefficient of the knee could be calculated by the least square method. As a first approximation, the knee was represented as a non-linear Kelvin model. Both bending elastic and bending damping coefficient of the knee increased as pre-bending moment was increased, and decreased as amplitude of bending oscillation was increased. Mechanical loss tangent of the knee-leg complex decreased within the range of pre-bending moment larger than 1.0 X 10(6) dyn cm as bending moment was increased, and decreased as amplitude of bending oscillation was increased.     

Polarotaxis,gravitaxis and vertical phototaxis in the green flagellate,Euglena gracilis

A fully automatic computer-controlled video analysis system has been used to study the movement of the green unicellular flagellate, Euglena gracilis in a horizontal or vertical cuvette. In darkness, in the absence of gaseous gradients, most cells swim straight upwards. While in a horizontal cuvette the transition between positive and negative phototaxis is found at about 1.5 W m^-2, an excess of 30 W m^-2 is required to reverse the upward swimming (due to the combined stimulus of negative gravitaxis and positive phototaxis) in a vertical cuvette. By studying the swimming direction in horizontal and vertical cuvettes in polarized light irradiated from above or from the side, respectively, the dichroic orientation of the photoreceptor molecules can be determined in three dimensions with respect to the axes of the cell: In a horizontal cuvette, in a linearly polarized beam from above, the cells orient predominantly at an angle of about 30° clockwise off the electric dipole transition moment as seen from above. The behavior in a vertical cuvette with polarized light entering from above indicates that the photoreceptor pigments are dichroically oriented 60° counterclockwise from the flagellar plane (seen from the front end of the cell). Experiments with horizontal polarized light indicate that the photoreceptor transition moment deviates 25° clockwise off the long axis of the cell.Abbreviation PFB paraflagellar body Dedicated to Prof. Dr. W. Nultsch on the occasion of his 60th birthday     

Microtubule sliding in swimming sperm flagella: direct and indirect measurements on sea urchin and tunicate spermatozoa [published erratum appears in J Cell Biol 1991 Nov;115(4):1204]

Direct measurements of microtubule sliding in the flagella of actively swimming, demembranated, spermatozoa have been made using submicron diameter gold beads as markers on the exposed outer doublet microtubules. With spermatozoa of the tunicate, Ciona, these measurements confirm values of sliding calculated indirectly by measuring angles relative to the axis of the sperm head. Both methods of measurement show a nonuniform amplitude of oscillatory sliding along the length of the flagellum, providing direct evidence that "oscillatory synchronous sliding" can be occurring in the flagellum, in addition to the metachronous sliding that is necessary to propagate a bending wave. Propagation of constant amplitude bends is not accomplished by propagation of a wave of oscillatory sliding of constant amplitude, and therefore appears to require a mechanism for monitoring and controlling the bend angle as bends propagate. With sea urchin spermatozoa, the direct measurements of sliding do not agree with the values calculated by measuring angles relative to the head axis. The oscillation in angular orientation of the sea urchin sperm head as it swims appears to be accommodated by flexure at the head-flagellum junction and does not correspond to oscillation in orientation of the basal end of the flagellum. Consequently, indirect calculations of sliding based on angles measured relative to the longitudinal axis of the sperm head can be seriously inaccurate in this species.     

Biomechanical features of eccentric cambial growth and reaction wood formation in broadleaf tree branches

Tree branches and stems have different physiological functions that work collaboratively to maximize light interception. Light penetration in tree crowns is controlled by the orientation of the branches. However, mechanisms of branch bending have not received the attention they deserve. This study approached the problem by investigating the growth strain distribution in the upper and lower sides of branches of broadleaf trees, estimating the bending tendency of branches, and observing the branch eccentricity and the distribution of gelatinous fibers. The strain distribution was compared between the branches of 11 species (including 8 examined species and 3 referenced species) and tilted stems of 37 species from both our data and previous reports. Compressive strain was generally observed on the lower side of branches, but little was measured in tilted stems. The pith eccentricity of branches was in a reverse pattern to the corresponding strain distribution of stems. The radial growth of branches was hypotropic in contrast to the epitropic eccentric growth in inclined trunks. Furthermore, on the upper side of branches, G-fibers within the fiber arcs formed in an intermittent manner rather than in the continual manner found in artificially inclined stems. The resultant upward bending moment might not suffice to counteract the branch’s own weight; therefore, most of the measured branches, differing from tilted stems, tended to bend downward. In conclusion, by comparing the biological and mechanical aspects of the strain distribution, bending tendency, and eccentricity, our experiments could discriminate the bending dynamics and role of G-fibers in tree branches from that of main stems.     

Orientation selectivity and visual acuity in schoolchildren and adults

We continued our study of the mechanisms of visual acuity (VA) in ontogenesis. We measured the VA and sensitivity to lines orientation and determined the minimal length of lines for discrimination of the vertical and the horizontal ones in subjects aged 7–25 years. It was found that the thresholds of discrimination of vertical and horizontal lines in subjects with normal VA decreased with age up to 9–10 years and then remained constant, while the orientation selectivity was improving up to the age of 14–16 years. The average VA almost did not depend on age. Individual thresholds of line lengths and orientation discrimination correlated with the VA of subjects.     

Effect of shoot tip and leaf removal on gravitropism in pea

Intact, light-grown pea (Pisum sativum L. cv. Alaska) seedlings were subjected to continuous horizontal gravistimulation and their growth and bending response compared with seedlings whose shoot tip and youngest leaf had been excised and with seedlings to which a counterweight to replace the mass of the decapitated tissue was added. While all seedlings achieved vertical orientation in 2 to 3 h, seedlings that were counterweighted bent upward at a significantly slower rate than the non-counterweighted, decapitated plants. In addition to this effect of mass on the rate of bending, decapitation also removed a major supply of auxin to cells in the bending zone which resulted in the slower bending of treated plants. Thus when using decapitation both the loss of mass and the time course of the response must be considered to understand its effect on gravitropism.     

Horizontal-vertical filters in early vision predict anomalous line-orientation identification frequencies.

A characteristic of early visual processing is a reduction in the effective number of filter mechanisms acting in parallel over the visual field. In the detection of a line target differing in orientation from a background of lines, performance with brief displays appears to be determined by just two classes of orientation-sensitive filter, with preferred orientations close to the vertical and horizontal. An orientation signal represented as a linear combination of responses from such filters is shown to provide a quantitative prediction of the probability density function for identifying the perceived orientation of a target line. This prediction was confirmed in an orientation-matching experiment, which showed that the precision of orientation estimates was worst near the vertical and horizontal and best at about 30 degrees each side of the vertical, a result that contrasts with the classical oblique effect in vision, when scrutiny of the image is allowed. A comparison of predicted and observed frequency distributions showed that the hypothesized orientation signal was formed as an opponent combination and horizontal and vertical filter responses.     

Ultimate strength of the lumbar spine in flexion--an in vitro study

The ultimate strength in flexion of 16 lumbar functional spinal units (FSU) was determined. The specimens were exposed to a combined static load of bending and shearing in the sagittal plane until overt rupture occurred (simulated flexion-distraction injuries). The biomechanical response of the FSU was measured with a force and moment platform. Mechanical displacement gauges were used to measure vertical displacements (flexion angulation) of the specimens. Photographs were taken after each loading step for determination of horizontal displacements and the centre of rotation. The lumbar FSU could resist a combination of bending moment and shear force of 156 Nm and 620 N respectively, before complete disruption occurred. The tension force acting on the posterior structures was 2.8 kN. The flexion angulation just before failure was 20 degrees and the anterior horizontal displacement between the upper and lower vertebrae was 9 mm. The centre of rotation was located in the posterior part of the lower vertebral body. The bone mineral content in the vertebrae appeared to be a good predictor of ultimate strength of the lumbar FSU. Knowledge of the biomechanical response of the lumbar spine under different static traumatic loads is a first step to better understand the injury mechanisms of the spine in traffic accidents.     

An evaluation of the uniform stress hypothesis based on stem geometry in selected North American conifers

The uniform stress hypothesis of stem formation was evaluated by comparing stem taper of Abies balsamea, Abies lasiocarpa, Picea rubens, Pinus contorta, Pinus elliottii, Pinus palustris, Pinus ponderosa, Pinus taeda, and Pseudotsuga menziesii to the taper expected if stems develop to uniformly distribute bending stress. The comparison was conducted by regressing stem diameter at height h (D_h) against bending moment at h (M_h) using the model D_h=J (M_h)^' where J and ' are fitted coefficients, and testing for '=0.333, the hypothesized value. Twelve curves were fitted with the model. Seven of the fitted values of ' were significantly different from 0.333, but eight of the values were within ᆞ% of 0.333 and eleven values were within ᆣ% of 0.333. Where the fitted value of ' was >15% of 0.333, residuals were biased with height. Fit by relative height, values of ' were within ᆞ% of 0.333 for large portions of these stems. While most of the fitted values of ' support the uniform-stress hypothesis, the values of ' for Pseudotsuga menziesii trees clearly did not. Many of the fitted values of J were inversely related to the modulus of elasticity (E) of green wood reported for these species. With the exception of Pseudotsuga menziesii, growing conditions appeared to account for extraordinary values of J. Increases in J with stem height corresponded with reported decreases in E with height. The covariance between J and E suggests some regulation of bending curvature by adjustments in cross-sectional area. These results suggest that stems taper to maintain a uniform bending curvature and that when E is relatively constant within and among stems, diameter along the stem or across stems can be predicted from bending moment using a simple power function.     

Safety and streamlining of woody shoots in wind: an empirical study across 39 species in tropical Australia

? Wind is a key mechanical stress for woody plants, so how do shoot traits affect performance in wind? ? We used a vehicle mounted apparatus to measure drag, streamlining and mechanical safety in 127 vertical lead-shoots, 1.2 m long, across 39 species in tropical Australia. ? Shoot dimensions and stem tissue properties were closely coupled so that shoots with low stem specific gravity or larger projected area had thicker stems. Thicker stems provide larger second moment of area (I), which increased shoot safety and bending stiffness but impeded shoot reconfiguration in strong winds, including frontal area reduction. Nonetheless, increasing I also improved streamlining. Streamlining was unrelated to traits except I. Stem tissue material properties only had small effects. Higher modulus of rupture increased shoot safety and higher Young's modulus impeded shoot reconfiguration. ? We found no conflict between bending stiffness and streamlining for woody shoots. Stiffness might help streamlining by increasing damping and stability, thereby reducing flagging in wind. Tissue-level traits did influence shoot-level mechanical safety and behaviour, but shoot geometry was much more important. Variable shoot and stem traits, which all influenced shoot biomechanics, were integrated in shoots to yield a relatively narrow range of outcomes in wind.     

Interaction with gravitropism,reversibility and lateral movements of phototropically stimulated potato shoots

Phototropic (PT) and gravitropic (GT) bending are the two major tropic movements that determine the spatial position of potato shoots. We studied PT bending of potato plantlets grown under long-day photoperiods in several prearranged position setups providing different interactions with the GT response. Starting with the standard PT stimulation setup composed of unilateral irradiation of vertically positioned shoots, experiments were also done in antagonistic and synergistic setups and in treatments with horizontal displacement of the light source. In the standard setup, PT bending suppressed the GT bending, which could occur only if the PT stimulation was cancelled. The antagonistic position, with phototropism and gravitropism attempting to bend shoots in opposite directions, showed phototropism and gravitropism as independent bending events with the outcome varying throughout the day reflecting diurnal changes in the competence of individual tropic components. Whilst gravitropism was constant, phototropism had a marked daily fluctuation of its magnitude with a prominent morning maximum starting an hour after the dawn in the growth room and lasting for the next 6 h. When phototropism and gravitropism were aligned in a synergistic position, stimulating shoot bending in the same direction, there was little quantitative addition of their individual effects. The long period of morning PT bending maximum enabled multiple PT bending events to be conducted in succession, each one preceded by a separate lag phase. Studies of secondary PT events showed that potato plantlets can follow and adjust their shoot position in response to both vertical and horizontal movements of a light source. PT bending was reversible, since the 180° horizontal change of a blue light (BL) source position resulted in reversal of bending direction after a 20-min-long lag phase.