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A Digital Method of Measurement and a Logistic Growth Model for the Blooming Angle of Miniature Roses.. 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 REPORT Violet Luo Science One 2008-09 ABSTRACT 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. INTRODUCTION Currently  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). 1 Figure 2. The nodding flowers of a cyclamen (left) and the tepals of a hyacinth (right). METHODS Source of the Plants and Controlled variables A 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 Petals A 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 protractor A digital protractor was constructed using the function plotting freeware Math GV™ Version 3.1 developed by Greg Van Mullem. The function,    = 180 tan)( 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 Photographs In 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. 2 Figure 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. RESULTS Four  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, jt o o e J Jt −    − + = θ θ θ 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 be trose e t 0251.06.331 82.8)( −+ ° =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, and with R2 = 0.974±0.013. DISCUSSION Justification of the Logistic Growth Model for Rose Petals It 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 3 Table 1: Logistic Growth Model for each of the Plants Miniature Rose Cyclamen Hyacinth J 82.8˚±5.3˚ 175.3˚±5.9˚ 42.2˚±3.2˚ j 0.0251±0.008 0.0498±0.007 0.0348±0.008 θo 4.8˚±3.8˚ 7.9˚±5.8˚ 4.3˚±2.3˚ R2 3.48±0.53 15.94±1.35 20.93±2.13 Blooming Angles of Miniature Roses, Cyclamen and Hyacinth 0.0 50.0 100.0 150.0 200.0 0 50 100 150 200 250 300 Time  (hour) B lo om in g A ng le  (d eg re es ) Petal 1 Petal 2 Petal 3 Petal 4 Petal 5 Petal 6 Petal 7 Petal 8 22727.2ln157)( −−= tttcyclamenθ 7.1561.1)(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 Systems A Population of Organisms A Rose Petal θ(t) The size of the population as a function of time The angle of bloom as a function of time t The time The time J Carrying capacity, the maximum population that the environment is capable of sustaining with its limited resources Bloom capacity, the maximum angle the petal is capable of reaching, constrained by the sepals, before it starts to fade j The relative growth rate The relative rate of change of the angle θo The initial population size at t = 0 The initial angle of the petal with respect to the stem axis at t = 0 Not 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 Hyacinth Although 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 Uncertainties One 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 4 parameters. 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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