The computational power of plants

Martin Howard and Alison Smith are research scientists at the John Innes Centre (JIC) in Norwich, England. JIC was founded as a horticultural institution by the philanthropist John Innes in 1910, and is today one of several independent institutes affiliated to the University of East Anglia. Howard and Smith are the senior scientists on a paper that claims to show that plants can perform arithmetic division. Their co-authors are Antonio Scialdone, Sam Mugford, Doreen Feike, Alastair Skeffington, Philippa Borrill, and Alexander Graf.

Today I thought we might look at their claim and see if theory can shed any light on it.

Their paper is titled “Arabidopsis plants perform arithmetic division to prevent starvation at night,” and appeared last June in the on-line journal eLife. It attracted quite a bit of general press coverage, and also created some not very nice comments by the paper’s readers. The claim is definitely a bit wild—many of us think of plants as, well, just plants. They have no neurons, so how could they do arithmetic operations? This was the cause of the negative comments, which I think are unfair.

Let’s take a high level look at the reason they believe that plants, in particular Arabidopsis plants, can divide. Note, I know nothing about plants, but apparently Arabidopsis is an important “model” plant that is often used in research.

## From The Plant’s Point of View

A plant creates starch during daylight and then consumes it at night, when it is dark. Its survival is dependent on having enough starch to make it through the night. Otherwise, it will stop growing, and could even die. If the plant has ${S}$ units of starch and there are ${T}$ hours left till dawn, it wants to consume ${S/T}$ roughly units of starch. This is clearly optimal: a nice uniform rate of consumption of the available food.

Howard and Smith, with my apologizes to the rest of their team, varied the amount of light and found that the plants were very good at appearing to do the above calculation. The plants almost always consumed about ${95\%}$ of the total starch that was available at the start of the night. Thus, they seemed to be able to do the calculation and divide ${S}$ by ${T}$.

## From the Scientists’ Point of View

One model that Howard and Smith propose is that the plants literally compute ${S/T}$ and use that to decide what amount of starch to consume. They are right that doing this at any time scale works from the mathematical point of view. That is, if every hour the plant consumes ${S/T}$ and updates ${S}$ and ${T}$ according to

$\displaystyle S' \longleftarrow S-S/T \text{ and } T' \longleftarrow T-1,$

then the right portion will be consumed. Note this is unit-invariant, in the sense that if we update every minute, the plant will still consume the same about per unit time. This is a very nice little fact. At first I thought this might have been an issue, but they are right that the plant needs a “clock” of some kind, but does not need to tell the exact time in hours and minutes till dawn.

The conclusion Howard and Smith make is:

The plants must have some biological mechanism that performs division.

This is the claim that I am not convinced follows.

They suggest a mechanism which at the high level is this—quoting them in the above-linked newspaper article:

We propose there is a molecule called S which tracks the amount of starch in the plant, and a molecule called T which tracks the time until dawn. The closer to dawn you get, the less of T and S you have. If the “S” molecule prompts the use of starch and the “T” molecule prevents it, then dividing the number of “S” molecules by “T” molecules would enable the plant to use up its reserves at a steady rate, [so that the reserves] expire at dawn.

This is correct, but must the plant do the arithmetic operation of division to create this effect? That is our question.

## From a Theorist’s Point of View

Okay, let’s agree that there are quantities ${S}$ and ${T}$, and that we want the ${S}$ to be consumed at the rate of ${S/T}$ per unit time. One obvious way is to have the plant do a division. Is there some way to avoid doing a division? This is the question that I find interesting.

From a theory point of view, imagine that we have a black box ${\cal B}$ that inside has ${S}$ and ${T}$. Every unit time it “ouputs” ${S/T}$. Actually it only needs to output approximately ${S/T}$. My question is, can we simulate such a black box ${\cal B}$ by another box that avoids division and yet outputs about the same quantity per unit time as ${\cal B}$?

This seems to be a nice, basic, and interesting theory problem. Certainly not one that the biologists would perhaps consider. So can we simulate division without doing division?

I am not interested in hearing that we can do division by multiplication. Yes that is possible, but then we would still be left with “Plants can do arithmetic multiplication.” My conjecture is that there should be a random process that we can construct that would behave roughly the same as the black box ${\cal B}$, and yet does not do any division. It would seem to me that the assumption that the plants’ chemistry could be random is quite plausible, and would yield a much simpler and less strange conclusion.

Consider the following model:

There are ${X}$ bins and ${X}$ balls—we have rescaled things so that the denoted quantities of starch and time are the same. Suppose at each time interval we shake things up, and some ball lands in some bin. That causes the bin and ball to be consumed. We continue doing this. Then at roughly each step, one bin and one ball are taken out of play.

That is, we seem to divide, but just use a simple random action. This seems like a simpler model, and one that we might be able to make into a viable one that a plant could do. Note that division has been replaced by a simple random process.

## Open Problems

Do plants do division, or can we show they achieve the same effect with a much simpler type of operations? Does the above random model make any sense? Can we create an even better one?

Solving is believing

 Cropped from source.

Boris Konev and Alexei Lisitsa are both researchers at the University of Liverpool, who work in Logic and Computation. They have recently made important progress on the Erdős Discrepancy Problem using computer tools from logic. Unlike the approach of a PolyMath project on the problem, their proof comes from a single program that ran for just over 6 hours, a program administered only by them, but mostly not written by them.

Today Ken and I wish to talk about their paper and two consequences of it. Read more…

Do our computers live in a simulation?

René Descartes is famous for countless things in mathematics—Cartesian products, Cartesian coordinates, Descartes’ rule of signs, the folium of Descartes. He is also famous for his work in philosophy and the notion of an evil genius. The evil genius presents a full illusion of a reality, and “fools” Descartes into believing there is a reality, while actually there is none.

Today Ken and I want to talk about the evil genius, and its relationship to the simulation hypothesis.

Comments on three papers from the Conference on Computational Complexity

Michael Saks is Chair of the Program Committee of this year’s Conference on Computational Complexity (CCC). He was helped by Paul Beame, Lance Fortnow, Elena Grigorescu, Yuval Ishai, Shachar Lovett, Alexander Sherstov, Srikanth Srinivasan, Madhur Tulsiani, Ryan Williams, and Ronald de Wolf. I have no doubt that they were faced with many difficult decisions—no doubt some worthy papers could not be included. The program committee’s work does not completely end after the list of accepted papers is posted, but it is not too early to thank them all for their hard work in putting together a terrific program.

Today I wish to highlight three papers from the list of accepted ones. Read more…

How can we possibly see atoms?

John Sidles is a medical researcher and a quantum systems engineer. His major focus is on quantum spin microscopy for regenerative medicine. He is both Professor of Orthopedics and Sports Medicine in the University of Washington School of Medicine, and co-director of the UW Quantum Systems Engineering Lab. Watching various injury troubles at the Sochi Winter Olympics makes us wonder whether quantum sports medicine is an idea whose time has come. Well beyond some media’s overheated references to our athletes as “warriors” is a nice reality: John’s main project is for healing those injured in the armed services.

Today Ken and I wish to talk about John and his work in general. We especially like his title of quantum systems engineer. Read more…

What is better than how (?)

 History of filters source.

Wilhelm Cauer was a German mathematician and engineer who worked in Göttingen and the US between the two world wars. He is associated with the term “black box,” although he apparently did not use it in his published papers, and others are said to have used it before. What Cauer did do was conceive a computing device based on electrical principles. According to this essay by Hartmut Petzold, Cauer’s device was markedly more advanced and mathematically general than other ‘analog devices’ of the same decades. He returned to Germany in the early 1930′s, stayed despite attention being drawn to some Jewish ancestry, and was killed in the last days of Berlin despite being on the Red Army’s list of scientists whose safety they’d wished to assure.

Today Ken and I wish to talk about black boxes and white boxes, no matter who invented them, and their relation to computing.