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Mathematical Analysis of Root Growth

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It grows externally in its own kidney-shaped hard shell at the end ofB. Fractal Analysis of Mangosteen and Cashew Root Growth Patterns Figures 5-7

    Nature and Science, 3(1), 2005, Puzon, Mathematical Analysis of Root Growth

    Mathematical Analysis of Root Growth in Gamma-irradiated Cashew (Anacardium occidentale L.) and Mangosteen (Garcinia

    mangostana L.) Using Fractals

    Klarizze Anne M. Puzon

    Quezon City, Philippines, klarizze@gmail.com

    Abstract: Root growth is related to the acquisition, distribution, and consumption of water and nutrients of plants. As a vital organ, roots directly take the effect of environmental change and its behavior is closely related to the growth of the whole plant. With such, the importance of root systems has motivated botanists to seek a better understanding of root branching complexity. This complexity, which has been difficult to comprehend using simple Euclidean methods (i.e. lines and circles), is important to the survival of plants, especially when the distribution of resources in the environment is scarce. Mathematical models using fractals and computers can be applied to accurately understand the growth and form complexity of plant root systems. This study was conducted to analyze the root growth of gamma-irradiated cashew and mangosteen using fractals. [Nature and Science. 2005;3(1):59-64].

Key words: fractals; root growth; cashew; mangosteen; mathematical model

    As people's views on using modern means such as computers extend, it has been difficult to use traditional 1 Brief Summary methods like simple lines and circles to comprehend biological systems. Biological systems, like root Seeds of cashew (n=360) gamma-irradiated at 0 Gy, branching, display fragmentations that cannot be 150 Gy, 300 Gy, 450 Gy, 600 Gy and 750 Gy, and modeled and comprehended by simple shapes alone. mangosteen (n=75) gamma-irradiated at 0 Gy, 10 Gy, Mathematical models using fractals have recently been 20 Gy, 30 Gy, and 40 Gy were germinated in perlite applied to explore the relationship between plant growth plots. The plants’ primary root lengths were measured. and structure. Image analysis using Fractal Dimensions software was A. Problem conducted to determine the fractal dimensions, D, of the Complexity in root systems, which reflects nutrient plant roots. exploitation efficiency, is important for plant survival, Findings for mangosteen reveal that as the gamma-especially when the distribution of resources in the soil irradiation dose increases, the primary root length environment is scarce. However, root complexity is decreases and the root D increases. Roots irradiated at difficult for scientists and researchers to study. 40 Gy showed the highest average D at 1.657. This B. Objective implies greater root branching complexity which results This research study was conducted to analyze and to better plant nutrient exploitation efficiency. For compare the root growth branching patterns of gamma-cashew roots, D did not vary significantly with irradiated cashew and mangosteen using fractals. increasing gamma-irradiation dose. However, cashew C. Significance seeds irradiated at 150 Gy exhibited the highest ; The study addresses the real-world problem of germination rate, highest average primary root length, making accurate quantitative observations and an average D of 1.613. General trends also reveal regarding root growth. The fractal dimensions that cashew roots’ D increased with time. may reflect the plants’ root branching This study demonstrates that fractal dimension can complexity and reflect nutrient uptake be a useful tool in characterizing the complex branching efficiency. characteristics of root systems. This may pave the way ; Since radiation causes genetic mutations, for further applications of fractals in other areas of fractal analyses of root patterns in gamma- research. The findings from this study can also be used irradiated cashew and mangosteen provide to improve the production of cashew and mangosteen information on the growth mechanisms of which are the two of the world’s most economically these plants. Data from this study can be used valued fruits. to improve the agriculture and production of

    2 Brief introduction

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    Nature and Science, 3(1), 2005, Puzon, Mathematical Analysis of Root Growth

    cashew and mangosteen which are defined botanically as the fruit. It grows externally in its economically valued fruits. own kidney-shaped hard shell at the end of this pseudo-

     fruit, or peduncle. It is commonly found in Brazil and in

    other tropical countries. Besides being a popular food 3 Detailed introduction export, it is now being used as an alternative medicine against asthma, diabetes, fever, and the like. A. Background of the Study The emergence of forms in the growth process, like B. Mangosteen root branching, is one of the most exciting problems in The mangosteen fruit, usually found in tropical biology. Most biological systems, like root branching, are countries, is 2-3 inches in diameter and has a thick reddish-difficult to comprehend, displaying fragmentations which purple rind that covers the segmented pulp (Morton, 1987). cannot be easily modeled by simple shapes (Kaandorp, It is usually eaten fresh, but can be stored successfully for 1994). Mathematical models using fractals have recently short periods of time. It is also canned, frozen, or made been applied to explore relationship between growth and into juice, preserves, and syrup. Mangosteen is also used form (Kenkel & Walker, 1996). Cashew (Anacardium as a pharmaceutical. occidentale L.) and mangosteen (Garcinia mangostana L.) are one of the most recognized tropical fruits. Both have C. Fractals universal appeal and high economic values because of their Fractals are unusual geometric structures that can be quality in color, shape and flavor. used to analyze many biologic structures not amenable to conventional analysis (Richardson & Gillepsy, 2000). B. Statement of the Problem Mandelbrot introduced the term 'fractal', from the Latin The demand on cashew and mangosteen often fractus, meaning 'broken', to characterize spatial or exceeds supply. Further studies about the growth and temporal phenomena that are continuous but not agriculture of both are needed. Also, the qualitative differentiable (Kenkel & Walker, 1996). Fractals possess characteristics of cashew and mangosteen root systems are properties that include scale independence, self-similarity, already known, the problem is to make accurate complexity, and infinite length or detail. Fractals have been quantitative observations on their root growth. The major recently used to analyze the root architecture of some objective, therefore, of this study is to analyze and compare plants. Correlations between fractal dimension and the root growth patterns of irradiated cashew and topology of root systems of legume plants grown in root mangosteen using fractals. boxes were studied (Tatsumi & Takagai, 1996). It was suggested that when roots develop under favorable C. Significance conditions, D is a good indicator for estimating the This study would be a means of new knowledge system’s size and root branching. about root branching growth of cashew and mangosteen. The fractal analysis of the root patterns of irradiated

    5 Methodology cashew and mangosteen can help understand their growth The method used is summarized by the dynamics. The fractal dimensions would reflect their

    FLOWCHART (Figure 1). branching complexity and growth velocity. Moreover, a

    PLANT comparison of the fractal models and the actual growth

    5.1 Plant materials and seed germination forms can be used to detect the effects of slow changes in

    Cashew seeds were obtained from University of the the environment, like gamma radiation.

    Philippines Los Banos- Agriculture Department. The 360 seeds were randomly divided into 3 blocks with 6 groups D. Scope and Limitations of the Study containing 20 seeds each. All groups were soaked in water This study focuses on having quantitative for 48 hours and were randomly irradiated at 0 gy, 150 gy, observations on the root systems of the samples, not on 300 gy, 450 gy, 600 gy and 750 gy. The seeds were their already known qualitative characteristics. One germinated in plots with perlite. For four weeks, the length limitation of this study is that the root structure being three of the primary roots of the samples was obtained. The plant dimensional will be modeled using a two dimensional

    materials for mangosteen were obtained at the Philippine fractal analysis software due to the inavailability of a three

    Nuclear Research Institute. The same preparation were to dimensional fractals software to the author.

    mangosteen, but the radiation doses were 0 gy, 10gy, 20 gy, 30 gy, and 40 gy. 4 Review of related literature 5.2 Fractal analysis A. Cashew Roughly once a week, 9 cashew root samples per The cashew tree is a medium-sized tree with oval radiation dose (3 from each block) were digitally blunt alternate leaves (Grieve, 2004). The cashew nut is photographed. Then, 3 from 5 samples per radiation

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    Nature and Science, 3(1), 2005, Puzon, Mathematical Analysis of Root Growth

dose of mangosteen root pictures were randomly chosen. After such, the pictures were turned into monochrome 6 Data and results format. Then, fractal analyses using the Fractal Dimensions software’s box-counting method were done. A. Mangosteen and Cashew Root Growth Observ- The data from the fractal counting were tabulated and ations (Figures 2-4) plotted on a log-log plot graph. A linear regression was B. Fractal Analysis of Mangosteen and Cashew Root done to find the best fit line. The fractal dimension was Growth Patterns (Figures 5-7) calculated. It is equal to 1 minus the slope of the best fit

    line, relative to 1 or simply D=slope.

    A. ACQUISITION OF PLANT MATERIALS

    B. SEED IRRADIATION AND GERMINATION

    C. OBSERVATION OF

    ROOT GROWTH

    D. ROOT LENGTH MEASUREMENT

    E. FRACTAL ANALYSES

    C. STATISTICAL ANALYSES (Mann-

    Whitney & Kruskal-Wallis Tests)

    Figure 1. Flowchart

    146

    12

    1048

    6

    24

    2Average Primary Root0Average Primary Root Length0Length (cm)0102030400150300450600750(cm)

    Gamma-Irradiation Dosage (Gy)Gamma-Irradiation Dosage (Gy) Figure 2. Average primary root length of Figure 3. Average primary root length of cashew at

    mangosteen at increasing gamma-irradiation doses increasing gamma-irradiation doses

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    Nature and Science, 3(1), 2005, Puzon, Mathematical Analysis of Root Growth

     907 Analysis and discussion summary 80 7060; For mangosteen roots, fractal dimension, D, 5040decreased as the primary root length increased. The 30highest gamma-irradiation dose for mangosteen, 40 2010Gy, resulted in the highest D, 1.657. This high 0Over-all Germination 0150300450600750value implies greater root complexity, which in Rate (%)Gamma-Irradiation Dosage (Gy)turn could result to enhanced efficiency for soil nutrient exploitation. Figure 4. Changes in cashew (Anacardium

    ; For cashew roots, D did not vary significantly with occidentale L.) seed germination rate at increasing

    gamma-irradiation doses. increasing radiation dose. However, primary root

     length measurements (5.40 cm) and germination

    rates (81.6%) revealed that cashew grows best at 1.8150 Gy. Such results might have happened maybe 1.5because cashew is less radiosensitive compared to 1.2mangostee. 0.9; From a cellular perspective, gamma-irradiation 0.6might have altered chromosomal structure (e.g. 0.3Average Fractalintroduction of transitions, deletions, and 0Dimension (D)010203040frameshifts in the genetic material) of mangosteen Gamma-Irradiation Dosage (Gy)and cashew root cells. The radiation might have also affected transmission of the genetic material through inhibition of cell mitosis. It is Figure 5. Changes in average fractal dimensions of

    hypothesized that these alterations in genetic mangosteen roots at increasing gamma-irradiation

    doses makeup might have led to the changes in root cell

     growth, which in turn affected the root systems’

    complexity. 1.7

    8 Conclusions and recommendations 1.65

    ; Fractals are useful in analyzing complex biological 1.6systems accurately. The fractal dimension (D) Average Fractalserved as the summary statistic of the branching 1.55Dimension (D)0150300450600750characteristics of cashew and mangosteen roots. Gamma-Irradiation Dosage (Gy); The best gamma-irradiation dose for mangosteen was 40 Gy, which showed the highest root fractal Figure 6. Changes in average fractal dimension of

    dimension. While the best dose for cashew was cashew roots at increasing gamma-irradiation doses

    150 Gy.

    ; In this study, the process of determining fractal 1.70 Gydimensions of gamma-irradiated roots and

    150 Gy1.65correlating it to primary root lengths showed that

    300 Gyvariations in D exist due to plant differences 1.6450 Gybrought about by genetic makeup (e.g. species) 1.55600 Gyand/or environmental factors (e.g. radiation dose). Average Fractal1.5750 GyDimension (D); The fractal dimension could be of interest to 1st wk2nd wk3rd wk4th wk

    botanists because it is directly correlated with the Time (weeks) efficiency at which the roots exploit soil resources.

    The use of other types of plant species and the Figure 7. Changes through time in the fractal application of other forms of environmental stress, dimensions of cashew roots

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    Nature and Science, 3(1), 2005, Puzon, Mathematical Analysis of Root Growth

    like drought and mineral deficiency, is ; The author is a graduate of Philippine Science recommended. High School- Main Campus and a BS Mathematics

    ; This study may pave the way for further freshman at the University of the Philippines- Diliman. applications of fractals in other areas of research,

    especially in agricultural engineering, computer References

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