# Why is chlorophyll green?

The paper that I want to discuss today is Quieting a Noisy Antenna Reproduces Photosynthetic Light Harvesting Spectra, by Arp, Aji, Gabor et al. I first read about this in a Quanta article. The scientific question that was answered in this paper was amazing- why are plants green?

So, why indeed are plants green? One might say it is because of chlorophyll, which is green in color. But why is chlorophyll green? Of course, one answer to this question is “because it is green”. But a more satisfying answer is that chlorophyll being green ensures that plants receive a steady, not-wildly-fluctuating supply of energy.

## Introduction

A plant, or at least the light harvesting parts of a plant, may be thought of as an antenna. The extremal points of a plant absorb energy from sunlight, and transfer this energy via various pathways to the parts where this energy is needed in order to make food and sustain other life-giving activities. Sunlight contains a spectrum of wavelengths, and plants probably want to absorb all wavelengths in order to maximize their intake. However, absorbing the green frequency would lead to a lot of variance in the amount of energy absorbed. Hence, to reduce this variance, plants just reflect this green part of the solar light, and absorb the red and blue parts.

And that is why plants appear green.

## Antenna Network and Noise

Ensuring that the energy input into a network or grid is equal to the energy output is a fundamental requirement of networks. If excess energy is absorbed, it may destroy the system, and if not enough energy is absorbed, an underpowered system will soon shut down. However, the environment of a plant can vary rapidly with time. The sun can be covered by clouds, the plants above a light absorbing leaf may sway with the wind and hence block access to sunlight at intervals, etc. How can a plant ensure that it receives a steady supply of energy? Clearly, much like we need a constant amount of food everyday, a plant’s energy output to its food making parts needs to be constant in order to survive.

If the energy absorbed by a plant at a fixed point in time can be plotted on a graph, with different probabilities given to different amounts of energy absorbed, the greater the variance in this graph, the more the variance in energy absorbed by a plant. This variance, which is called noise, should be reduced. Reducing noise is going to be our main motive as we do the analysis below. Methods of reducing noise like adaptive noise filtering require external intervention, and hence are not available to plants.

## One node network

Imagine a network with a single input node $A$, that absorbs light at wavelength $\lambda_A$ with (maximum) power $P_A$. Note that the average power absorbed does not have to equal $P_A$ due to changing external conditions like swaying plants blocking sunlight, etc. Hence, the average power absorbed by a plant is $p_A P_A$, where $p_A<1$, and can be thought of as the probability of the plant absorbing $P_A$.

Let the energy output be $\Omega$. If $P_A=\Omega$, then the average energy input, which is $p_A P_A, would always be less than output. Hence, in this model of only one input node $A$, $P_A>\Omega$ so that the average energy input is equal to output. In other words, $p_A P_A=\Omega$.

Let us now calculate the variance of the energy received. If $p_A$ is the probability that the plant is able to receive $P_A$ energy, and $(1-p_A)$ is the probability of the plant receiving $0$ energy, then the variance is $\sigma^2=p_A(P_A-\Omega)^2+(1-p_A)(\Omega)^2$. This can be simplified as

$\frac{\sigma^2}{\Omega}=\frac{P_A}{\Omega}-1$

We should look for ways to reduce this variance. We can do this by having two nodes instead of one.

## Two node network

Let us now have a network with two input nodes, and see if we can reduce variance. Let the input nodes $A$ and $B$ absorb light at frequencies $\lambda_A,\lambda_B$. Let the power absorbed be $P_A,P_B$ with probabilities $p_A,p_B$. We will assume that $P_A<\Omega. Also, we want $p_A P_A+p_BP_B=\Omega$, as the average power input should be equal to the output. One constraint of the system is that the plant shouldn’t absorb $P_A+P_B$ power, because $P_A+P_B>>\Omega$. Hence the possibilities of power absorption are that the plant absorbs $P_A$ power, $P_B$ power, or $0$ power. The variance of the model is now $\sigma^2=p_A(P_A-\Omega)^2+p_B(P_B-\Omega)^2+(1-P_A-P_B)(\Omega)^2$. This can be simplified as

$\frac{\sigma^2}{\Omega^2}=(\frac{P_A}{\Omega}-1)-p_B\frac{P_B}{\Omega}(\frac{P_A-P_B}{\Omega})$

Clearly, this is smaller than the variance we get from the network with just one node. The good news is that plants also have two input nodes- chlorophyll a and chlorophyll b. The presence of two input nodes for a given wavelength probably served as an evolutionary advantage in order to minimize noise in energy absorption.

## Optimization

We want plants to absorb energy at a steady rate, to ensure that energy input=energy output. We want to maximize $P_A-P_B$, where $P_B<\Omega, so that the noise, or variance in energy absorption, is minimized. Our constraint is that $p_A P_A + p_B P_B=\Omega$. And we want to do this for all the wavelenghts of light that we can.

Now the maximum power available $P_A$ depends upon the sunlight available, and is given below as the black curve in the graph. Ignore the two peaks for now.

Hence, we can ideally select two nodes each for the blue, green and red regions of the wavelength spectrum, and absorb energy from each of them. In order to reduce noise however, we need to maiximize $P_A-P_B$. This can be done if we place two nodes each in each of the three regions, and let the two nodes have very similar wavelengths, but different $P_i$‘s. This can be done easily where the slope of the irradiance is high. We can see in the graph above that the slope of the irradiance graph is high in the blue and red regions. However, in the green region, the slope is close to $0$. Hence, if we place two nodes there with similar wavelengths, $P_A-P_B$ will be almost $0$, and hence there will be a lot of noise in the energy input.

This is the reason why plants have two nodes each only in the red and blue regions of the light spectrum, and not the green region. The green light is reflected, and this is why plants are green.

Purple bacteria and green sulphur bacteria can be modeled using the same constraint of reducing noise in energy absorption. Hence, the scientific model developed by the authors is robust, and can explain the color of much of the flora and fauna found on the planet.

## References

1. Quieting a Noisy Antenna Reproduces Photosynthetic Light Harvesting Spectra