Phuc Le

ratatouille17042007@gmail.com
TOPIC
Quantifying and Comparing the Information Transfer Efficiency Across Phenotypic Traits of
Common Beans (Phaseolus vulagaris) in Response to Different Light Wavelengths
Abstract. This study models plant-light interactions as a biological communication channel,
using the lens of Information Theory with common beans (Phaseolus vulagaris) as the study
subject. It aims to quantify how effectively information is transferred within 4 common beans’
phenotypic traits when interacting with light, by experimentally growing plants under different
wavelengths. By doing this, it seeks to reveal plant information processing strategies and provide
perspectives to fields such as plant neurology and computational biology.
Introduction.
Information theory has been widely applied to biological systems, ranging from genetic
sequences (Tkačik and Walczak, 2011) to cellular computation (Mehta and Schwab, 2012).
Moreover, plants have long been investigated as a system which receives, and processes
environmental information (Brenner et al., 2006). Thus, this study also models common beans as
a communication channel receiving light wavelengths as input variables(𝑋 ∆ ) and produce
phenotypic traits as output variables (𝑌𝑘∆ ), with 𝑘 = 1,2,3,4 corresponding respectively to Root
length, Leaf Width, Chlorophyll content and Plant height. This model is conceptualized in
Figure 1.
To ground this research, it is necessary to briefly introduce 3 following concepts:
Entropy of a continuous variable measures its average uncertainty (in bits). In this research, we
focus on the Entropy of each output variables 𝑌𝑘∆ (phenotypic traits), which is defined as:

1 1
𝐻(𝑌𝑘∆ ) = ∑𝑗𝑘 𝑝(𝑦𝑗𝑘 )∆𝑦𝑘 log 2 ( ) + ∆𝑦 (1)
𝑝(𝑦𝑗𝑘 ) 𝑘

where ∆𝑦𝑘 is the length of discrete intervals (or bin width) used for discretizing 𝑌𝑘∆ and 𝑝(𝑦𝑗𝑘 )is
the relative frequency of any random variable falling into the 𝑗𝑡ℎ bin of 𝑘 𝑡ℎ phenotypic trait. The
concept of “bins of data” and “bin width” is demonstrated in Figure 2.
Mutual Information is the reduced amount of uncertainty about output variable 𝑌𝑘∆ when
observing input variable 𝑋 ∆ :
𝑝(𝑥𝑖 ;𝑦𝑗𝑘 )
𝐼(𝑋 ∆ ; 𝑌𝑘∆ ) = ∑𝑖,𝑗𝑘 𝑝(𝑥𝑖 ; 𝑦𝑗𝑘 )∆𝑥∆𝑦𝑘 log 2 ( ) (2)
𝑝(𝑥𝑖 )𝑝(𝑦𝑗𝑘 )

where ∆𝑥 is the bin width of discretized 𝑋 ∆ , 𝑝(𝑥𝑖 ) is the probability of a random variable
assigned to the 𝑖 𝑡ℎ bin of light wavelengths, 𝑝(𝑥𝑖 ; 𝑦𝑗𝑘 ) is the joint probability of a random
variable in both 𝑖 𝑡ℎ bin of light wavelengths and 𝑗𝑡ℎ bin of 𝑘 𝑡ℎ phenotypic trait.
In the context of this research, we define the Information Transfer Efficiency as follow:
𝐼(𝑋 ∆ ;𝑌𝑘∆ )
Information Transfer Efficiency = (3)
𝐻(𝑌𝑘∆ )

since it quantifies what proportion of average uncertainty in 𝑌𝑘∆ is reduced through the mutual
relationship between 𝑌𝑘∆ and 𝑋 ∆ . Its value ranges from 0 to 1.
Literature review.
Previous studies in photobiology have primarily focused on how specific light wavelengths
positively or negatively affect phenotypic traits or stages of plant development, addressing
questions such as: “When plants receive information from wavelength A, how is trait B
influenced? Which light wavelengths optimize the growth of which trait?” For instance, red light
has been shown to enhance ramie growth, whereas blue and orange lights reduced it (Rehman et
al., 2020), comparisons have been made between blue, red, green, and natural light on the growth
of physiological traits of common beans (Ziegler, 2020), or the conclusion that yellow light is the
optimal light for the growth of E. pseudowushanense (Yang et al. 111550).
However, no study has addressed the reverse question: “When receiving diverse information
from different wavelengths, which phenotypic traits’ growth do plants optimize for information
processing?”. Thus, this study connects photobiology and Information theory to investigate
which phenotypic traits do plants maximize Information Transfer Efficiency and to explore
broader questions: If differences in efficiency are found across traits under varying wavelengths,
why do they occur? Have plants “evolved” to optimize specific functional traits, and for what
reasons?
Methodology.
Before the experiment, we discretized the light wavelengths into 7 bins in equal width of 44 nm
(in nm), or 7 colors of light: Violet (395 – 439), Blue (439 – 483), Cyan (483 – 527), Green (527
– 571), Yellow (571 – 615), Orange (615 – 659), and Red (659 – 703). Common beans are
grown inside 7 large blocks constructed from black-painted acrylic sheets to minimize outside
light interference. The top of each block has LED lights emitting 1 of the 7 wavelengths. Each
block houses 25 plant pots. Each pot contains 1 common bean seed. The seeds are sown 3-5 cm
deep in the fertile soil with a stable pH of 6.5. To prevent plants being damaged for receiving
excess light, the LEDs operate on a cycle: 12 hours of light and 12 hours of darkness. Plants are
cultivated for data collection after 1 month. This experimental set-up is illustrated in Figure 3.
On data collection, Root length (mm) is the length of the primary root, measured after gently
removing the plant from the soil. Leaf width (mm) is the widest point perpendicular to the
midrib of the largest chosen leaf. Chlorophyll content (µmol/𝑚2 ) is assessed using a chlorophyll
concentration meter on a chosen leaf. Plant height (mm) is measured from the base to the apex
of the main stem. Traits’ data, once collected, are grouped into bins of data according to
Freedman-Diaconis rule to reduce bias in choosing bin width. An ANOVA test will be
performed on each trait among the 7 groups of 7 wavelengths. If the ANOVA shows a
statistically significant difference among the 7 groups, Bonferroni adjusted t-tests will be done to
identify statistical differences between any 2 groups.
Using MATLAB or Python, we graph the data on 3D surface plots to visualize and calculate the
joint probability𝑝(𝑥𝑖 ; 𝑦𝑗𝑘 ). Since there are 4 phenotypic traits, a total of four 3D plots are
created, each graphed 𝑋 ∆ against respective 𝑌𝑘∆ . Using equations (1), (2), (3) and the data
collected, we calculate Information Transfer Efficiency for each phenotypic trait. Finally, we
compare them to identify patterns: Are they uniformly low or high, or varied across traits?
This framework can be extended to traits such as Seed germination rate or Flowering time to
gain further insights into how plants optimize information processing as time progresses.
Bibliography:

Brenner, Eric D., et al. “Plant Neurobiology: An Integrated View of Plant Signaling.” Trends in
Plant Science, vol. 11, no. 8, Aug. 2006, pp. 413–19,
https://doi.org/10.1016/j.tplants.2006.06.009.

Mehta, P., and D. J. Schwab. “Energetic Costs of Cellular Computation.” Proceedings of the
National Academy of Sciences, vol. 109, no. 44, Oct. 2012, pp. 17978–82,
https://doi.org/10.1073/pnas.1207814109.

Rehman, M., et al. Red Light Optimized Physiological Traits and Enhanced the Growth of
Ramie (Boehmeria Nivea L.). 23 June 2020,
www.researchgate.net/publication/342379971_Red_light_optimized_physiological_traits_
and_enhanced_the_growth_of_ramie_Boehmeria_nivea_L

Tkačik, Gašper, and Aleksandra M. Walczak. “Information Transmission in Genetic Regulatory

Yang, Qianru, et al. “Yellow Light Promotes the Growth and Accumulation of Bioactive
Flavonoids in Epimedium Pseudowushanense.” Journal of Photochemistry and
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https://doi.org/10.1016/j.jphotobiol.2019.111550.

Ziegler, Aaron. “Comparison of Violet, Red, and Green Light on Early-Stage Common Bean
(Phaseolus Vulgaris L.) Development.” The Ohio Journal of Science, vol. 120, no. 2, Aug.
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Appendix:
Figure 1. A Communication Channel model with light wavelengths as input and the effects on
phenotypic traits as outputs.

Figure 2. Demonstration of “bins of data” (denoted 𝑗1 , 𝑗2 , 𝑗3 , 𝑗4 , 𝑗5 , 𝑗6 , 𝑗7 ) and “bin width”

Figure 3. Illustrated diagram of the experimental set-up of common beans: The black blocks are
acrylic blocks housing the plants with their colored frames indicating their corresponding color
of LED lights.