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A Digital Method of Measurement and a Logistic Growth Model for the Blooming Angle of Miniature Roses.. Luo, Violet 2009

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Figure 1. Angle θ is the angle of bloom of rose petal P at the moment of photograph. One sepal is labelled S. The corolla is combination of all of the pedals, labelled C. A DIGITAL METHOD OF MEASUREMENT AND A LOGISTIC GROWTH MODEL FOR THE BLOOMING ANGLE OF MINIATURE ROSES AND THEIR APPLICATION ON CYCLAMEN AND HYACINTH-- A SCIENCE ONE SECOND-TERM PROJECT REPORTViolet LuoScience One 2008-09ABSTRACT  In this project, the angles of miniature rose petals throughout the course of their bloom were measured with a digital protractor constructed using a function plotting software. The angle was then modeled as a function of time, and an analogy was drawn from the blooming of rose petals to the logistic growth of populations. The digital method of measurement and the logistic growth model were also applied to cyclamen and hyacinth petals.INTRODUCTIONCurrently there are 258,650 species of flowering plants (angiosperms) known to exist in the world [1], contributing to more than one tenth of the total number of living species of all organisms. Within this vast variety in the angiospermae group, each species still blooms in its own fashion. In this project, three flowering plants were chosen for a measurement of the angular position of the petals as they bloom. The miniature rose, the cyclamen and the hyacinth used for this project were identified as cultivars of Rosa chinensis minima [2], Cyclamen persicum [3] and Hyacinthus orientalis [4], respectively. The rose shows many characteristics of a common flower, and thus it was at focus of this project. The corollas of roses are aligned with the axis of the stem, the petals unfold outward and down during its bloom, and the petals are well distinguished from the sepals (Figure 1). Not as typical a flower as the rose, the cyclamen is a nodding flower that unfolds its petals upward during its bloom, and the hyacinth flower is composed of six lily-like tepals instead of  differentiated petals and sepals (Figure 2). The angle of petals was measured as the factor to be modeled instead of the actual arc length traveled by the petals, so that the different petal structures could not affect the ability of the data to reflect the actual state of bloom; For example, the arc length data may reflect that a longer petal blooms faster than a shorter petal due to a greater diameter of arc traveled, even though the angular data may indicate an equal speed for both. For the purpose of this project, the angle of a petal at any particular moment of its bloom is defined as the angle subtended by the tip of the petal from the axis of the corolla, with a vertex at the base of the petals (labelled as θ in Figure 1).1Figure 2. The nodding flowers of a cyclamen (left) and the tepals of a hyacinth (right).METHODSSource of the Plants and Controlled variablesA miniature rose, a cyclamen and a hyacinth were purchased from a local greenhouse, Burnaby Lake Greenhouse Ltd., Vancouver, BC. Initially, the plants were approximately 10 cm in height and the buds were ready to bloom. The following controlled variables were imposed to allow possible discoveries of any common trends in the blooming angle of the petals: The plants were kept in plastic flowerpots (502mL in volume) and placed on a west-facing windowsill with room temperature kept within the range of 20 to 25˚C. Each plant received 50mL of water daily, and excessive water was allowed to drain. No fertilizer was applied to any of the plants throughout the course of the experiment.Collecting Images of the PetalsA Sony DSC-T30 digital camera was used to collect photographs of the petals. The camera was placed 20cm away from the base of the petals and photographs were taken at a f-stop of +3.0f and an image size of 3072 x 2304 pixels. Only petals with their profile aligned in view of the camera can be used for data collection. The time interval for taking the photos varied between the three species due to their different blooming speed: once every 12 hours for the miniature rose and once every hour for the hyacinth and the cyclamen. Replications of this procedure were done for more data from different petals on each plant.Construction of the digital protractorA digital protractor was constructed using the function plotting freeware Math GV™ Version 3.1 developed by Greg Van Mullem. The function, = 180tan)( pinxxy (1)where n is the angle in degrees, was input into the software repeatedly for all integer values of n  from 0 to 89, since function (1) has a discontinuity at n=90. The two-dimensional Cartesian plot generated by the software was saved as a bitmap file. A copy of this half of protractor image was rotated 90˚ to the left and imposed upon the original image using an image editor (included in Microsoft Word), the combined image is the complete digital protractor as shown in Figure 3. Because the protractor was digitally generated using mathematical functions, it is reasonable to assume that the increments are indefinitely precise, and thus the uncertainty in the measurement of the protractor is simply half of the smallest increment, ±0.5˚.Treatment of the PhotographsIn the Microsoft Word interface, the bitmap image of the digital protractor was imposed upon each of the photographs of the petals with one increment (defined as 0˚) aligned to the axis of the flower stem to read the angles of each chosen petal as defined in the introduction of this report. Thus, there is a systematic error of ±1˚ degree for every angle measured, since two alignments of the protractor were required to obtain the angle, and there was an uncertainty of ±0.5˚ in every reading of the protractor. The time data was taken from the record of the camera and rounded to the whole hour, resulting in an uncertainty of ±0.25 hour for each data. Both the angle and the time data were then analyzed graphically.2Figure 3. A minimized (5%) image of the digital protractor constructed using Math GV with a 30% view in the box. The blue and red lines show increments of 10˚ and 1˚, respectively. The actual size of the protractor is 2000 x 2000 pixels.Figure 4. Data collected for the blooming angles of the miniature rose (petals 1 to 4 with numbers of data n =17, 16, 16 and 14, respectively), the cyclamen (petals 5 and 6, n= 13 and 16) and the hyacinth (petals 7 and 8, n=18 and 18) was plotted against time.  RESULTSFour petals from the miniature rose, two petals from the cyclamen and two petals from the hyacinth were monitored during the course of their bloom (Figure 4), and the data from each petal was modeled as a logistic growth,jtoo eJJt− −+=θθθ 1)( (2)where J is the maximum (or critical) angle the rose petal would be able to unfold to, j is the relative rate of change of the angles of the petals and θo is the initial angle of the petal prior to the bloom. The averages of the parameters that yield the minimum standard deviation, R2, from each of the plants are shown in Table 1. Although the logistic growth model applied to the miniature rose petals well, it failed to apply to the cyclamen and hyacinth data, hence the significantly large standard deviation values shown in Table 1. Thus, a species-specific model for the blooming angle of miniature roses was determined to betrose et 0251.06.33182.8)(−+°=θ        (3)                                           with a standard deviation of R2=3.48±0.53. Further analysis yielded the following models, (4) and (5), to represent the data collected for cyclamen and hyacinth more accurately:with R2 = 4.60±3.20, andwith R2 = 0.974±0.013.DISCUSSIONJustification of the Logistic Growth Model for Rose PetalsIt is reasonable to assume that the analogy drawn between the blooming rose petals and the logistic growth of a population of organisms is valid, because the factors represented the parameters and variables in the 3Table 1: Logistic Growth Model for each of the PlantsMiniature Rose Cyclamen HyacinthJ 82.8˚±5.3˚ 175.3˚±5.9˚ 42.2˚±3.2˚j 0.0251±0.008 0.0498±0.007 0.0348±0.008θo 4.8˚±3.8˚ 7.9˚±5.8˚ 4.3˚±2.3˚R2 3.48±0.53 15.94±1.35 20.93±2.13Blooming Angles of Miniature Roses, Cyclamen and Hyacinth0.050.0100.0150.0200.00 50 100 150 200 250 300Time  (hour)Blooming Angle (degrees)Petal 1 Petal 2 Petal 3 Petal 4Petal 5 Petal 6 Petal 7 Petal 822727.2ln157)( −−= tttcyclamenθ7.1561.1)(hyacinth += ttθ(4)(5)model for either system can be related (Table 2).Table 2. A Comparison between the Factors Represented by the Logistic Growth Model in Two SystemsA Population of Organisms A Rose Petalθ(t) The size of the population as a function of time The angle of bloom as a function of time t The time The timeJ Carrying capacity, the maximum population that the environment is capable of sustaining with its limited resourcesBloom capacity, the maximum angle the petal is capable of reaching, constrained by the sepals, before it starts to fadej The relative growth rate The relative rate of change of the angleθo The initial population size at t = 0 The initial angle of the petal with respect to the stem axis at t = 0Not only does the values of θo (4.8˚±3.8˚) and J (82.8˚±5.3˚) obtained from the model agrees with the observations of the initial and final angles of the petals, the sigmoid curve observed in each of the four sets of rose data points (blue in Figure 4) also suggest a good fit of the logistic growth model. It was observed that the sepals unfolded down to form an angle of approximately 90˚ prior to the bloom of the petals, and they supported the corolla afterward during the bloom. At the initial stage of the bloom, the petals accelerated due to the freedom of movement unleashed from the sepals, and the maximum speed was reached eventually. Since the lever-like mechanism of the blooming petals had pivot points at the point of connection between the petals and the sepals, the presence of the sepals also constrained the petals from unfolding to any more than the angle of the sepals. The greater the angle of the petals, the harder the petals had to push against the sepals to reach further, thus the petals decelerated as it reached its angle capacity. The acceleration in the first half of bloom and the deceleration in the second half of bloom resulted in the sigmoid curve observed in each of the four sets of data points (Figure 4). Since the sigmoid curve is a characteristic of the logistic growth curve, it is reasonable to assume that the logistic growth model is a proper fit for the blooming of rose petals.  Failure of Application on Cyclamen and HyacinthAlthough the logistic model is sufficient to satisfy the data obtained from the miniature roses, it did not apply well to the data from the cyclamen or the hyacinth. This discrepancy could be explained by the different structures of the petals. Whereas the rose petals are rounder and bloom in an upright position, the nodding flowers of the cyclamen have longer petals and are inverted so that they have to overcome the force of gravity to unfold upward. Comparing to the well distinguished petals and sepals of miniature roses, the undifferentiated tepals of hyacinth do not have the typical characteristics such as the flexibility of the thinner petals and the blooming capacity controlled by the position of the sepals, and the lack of such characterisitics caused the tepals of hyacinth to bloom differently. Treatment of the UncertaintiesOne of the uncertainties in this measurement was potentially inherent in the mechanism of the lenses of the camera to produce an image. The amount of any spherical aberration was investigated by taking a photograph of the calibrated digital protractor on the laptop screen from 20cm away. When the protractor was imposed on the photograph of itself, as in a measurement, the two images completely overlapped, thus there was no spherical aberration caused by the camera.Because each petal is a unique individual system that fits to a particular set of parameters, to combine the results together would create a substantial amount of uncertainty in the average of the four sets of 4parameters. This uncertainty mainly depends on the deviation between the four sets of results as shown in Table 1, since the uncertainty in the measurement of the angles and time (∆θ=±1˚ and ∆t = ± 0.25 hour) are rather small compared to the deviation between the replications.CONCLUSION Modeling data from organic systems could hardly be comprehensive because there are a great number of known and unknown factors at all scales that could affect even the simplest living system. The initial angle (θo), the bloom capacity (J) and the relative angular speed (j) considered in this project are only a few of the factor that affect the angle of rose petals during their bloom, and thus this model is still far from being inclusive for all miniature rose plants. More replications of the experiment on petals from several rose plants could improve the reliability of the result to represent a broader range of rose plants. Further investigation could also involve other factors such as the amount of water and fertilizer fed to the plants. Nevertheless, the digital method of measurement of angles developed in this project can be applied to a broad range of situation whenever a precise measurement of angles is required.REFERENCES[1] Thorne, R. F. The classification and geography of the flowering plants: dicotyledons of the class Angiospermae (subclasses Magnoliidae, Ranunculidae, Caryophyllidae, Dilleniidae, Rosidae, Asteridae, and Lamiidae). Bot. Rev.66: 441–647 (2000).[2] Gault, S. M., Synge, P. M. The dictionary of roses in colour. Ebury Press and Michael Joseph. London (1971).[3] Grey-Wilson, C. Cyclamen, a guide for gardeners, horticulturists and botanists. B T Batsford. London. (2002).[4] Pfosser, M., Speta, F. Phylogenetics of Hyacinthaceae Based on Plastid DNA Sequences. Ann. Missouri Bot. Gard. 86: 852-875 (1999). 5


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