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What Makes the Beauty of a Butterfly? Organized Chaos?

Take a look at a butterfly’s wing, and you can learn a lesson about life.

Not that it’s beautiful, or fragile, or too easily appreciated only when it’s fading—though all that is true, and evident in a wing.

Look very close, at the edge of a pattern, where one color turns to another. The demarcation isn’t so abrupt as it seems at arm’s length. It’s not a line, but rather a gradient.

RANDON KEIM posted this JUN. 24, 2013

Organized Chaos Makes the Beauty of a Butterfly

Butterfly wing

Butterfly wing via squinza

This is a lesson about uncertainty.

A butterfly’s colors come from its scales, each a single cell, pigmented a single hue.

At pattern boundaries, scales of different colors intermingle. Transitions and shading are achieved by varying the proportions of the mix. It’s beautiful. It is also, in the language of molecular biology, a model for a stochastic mechanism of gene expression.

Each scale’s fate is not preordained. For example, cells on the surface of a swallowtail’s wings were not originally specialized to be yellow or blue or black. Instead, they contain genes potentially capable of producing each of those pigments.

What determines the color of each butterfly scale is, in a word, chance.

A molecule hits a piece of cellular machinery at just the right moment, in just the right place, and a gene produces a certain pigment. There’s no guarantee it will happen. It’s a matter of probability and moment-to-moment randomness. (The same probabilistic mechanism underlies the portions of the wings with solid colors, too. In those parts, molecules that trigger genes for one color are present at such high concentrations that the final color outcome is assured.)

In biology, there’s a tendency to conceive of randomness as noise, an accidental factor, a product of error—random genetic mutations, For example, random genetic mutations are mistakes in a chromosome-duplicating system that’s supposed to make perfect copies. Random mutations might be harmful, or insignificant, or beneficial, but they’re fundamentally mistakes, a disorderly deviation from an orderly system.

What makes a butterfly’s wings so remarkable isn’t just that unpredictabilities underlie their colors, but that they’ve harnessed the probabilities 1.

Randomness and uncertainty are translated into the ordered, functional patterns of a monarch or checkerspot. And in this, the butterfly’s wing is not unique, but a manifestation of principles ubiquitous in biology.

Let’s put on our Powers of Ten goggles and increase our magnification to where cell activity occurs, the level of so-called cellular machinery.

We’ll have to abandon that metaphor, though: Cells indeed contain complicated, task-performing structures, but the word “machine” is a product of the macroscopic world. We think of machines as rigidly assembled, with predefined purposes. At the cellular level, the analogy falls apart.

Out at the leading edges of theoretical and computational and experimental biology, where known and unknown meet, cellular machines have been redefined 2. The proteins of which they’re made don’t fold and unfold and operate according to some stepwise blueprints. Shape and function are exquisitely sensitive to infinitesimal energetic shifts, to the motion of atoms and the forces they exert.

Rather than a cellular factory, imagine a restaurant with a kitchen where blenders turn into convection ovens and whisks into knives when someone walks by, raising ambient temperatures by a fractional degree.

Imagine that the whole kitchen is like this, that cooks and prep staff, though they move with intent, can’t help but wander around—and still the 7-course meals come rolling through the doors.

Undoubtedly this metaphor has its own problems, but it gets the point across: The cellular world is an ever-fluctuating place. It’s full of randomness and, when things aren’t narrowly random, with uncertainty. Atoms and molecules and gradients change from moment to moment, and proteins with them.

Butterfly wing 2
Butterfly image via rougenair

Here one might ask where uncertainty comes from: Is it truly uncertain?

Might every molecular fate be predicted, if only we knew the motions and properties of every particle in a cell?

Or does quantum physics enter the equation at some level, with all its spooky uncertainties and strange probabilities shaping biology in some fundamental way?

That we don’t know, and might never. It’s a question too hard to study. Whatever the case, certain molecular activities are, best as we can describe them, random or probabilistic.

And we know, up at the cellular level, that a cell’s fate—whether an embryonic stem cell becomes specialized for service in kidney or liver, whether a blood stem cell grows up to carry oxygen or identify pathogens—is to some extent stochastic, determined by a signal that may or may not appear.

What’s extraordinary is that from all this uncertainty, form arises. Two identical twins, after trillions upon trillions of cell divisions, actually look the same to our eyes.

Out of disorder/order, though having identical genomes,is no guarantee of identical outcomes. Indeed, stochastic role in cells became evident when genetically matched yeast colonies, raised in the exact same environments, developed in very different ways.

What seems to explain that is variation in so-called epigenetic responses—processes that alter gene activity according to environment and circumstance, allowing organisms to change their biology in response to life’s unpredictable demands.

The different yeast colonies had different epigenetics and they responded to uncertainty differently. That might itself have been the product of chance, some “inherited stochastic variation,” though the benefits are obvious.

It’s evolutionary bet-hedging, a way of increasing the adaptive possibilities for one’s descendants, despite their genetic similarities.

Again, we see biology using uncertainty, building on it, making it integral to life. And it’s evident not only in epigenomes, but in genomes: When we look at our own, which for each of us contain some number of apparently random errors produced by copy-making glitches, we find that the errors are not randomly distributed. Mutations occur at different rates in different parts of the genome.

This isn’t the same thing as saying that certain sequences tend to stick around over evolutionary time, because errors there are more likely to cause problems. Instead, the potential for a random error to occur in the first place fluctuates across the genome. At every cellular level, randomness is harnessed.

Life is a study in contrasts between randomness and determinism,” wrote biochemists Arjun Raj and Alexander van Oudenaarden in an article entitled, “Nature, Nurture or Chance,” in the journal Cell. “From the chaos of biomolecular interactions to the precise coordination of development, living organisms are able to resolve these two seemingly contradictory aspects of their internal workings.”

Do these resolutions occur at even higher levels? Pattern from chance in populations, species, communities, ecology?

And in our own lives?

We can’t look at societies or lives the way we do cells, but certainly we feel it, at some intuitive level. “I think back on the trajectory of my life, and I think: I happened to bump into this person on the train, and it led to this or that,” Raj told me. “So many things are unpredictable on a long time scale, though it feels like they are predictable in the moment.”

I think on my own life: My parents met on a train. My closest friends came from chance encounters in a subway station, a class, a hockey team, a writing assignment. I can’t imagine my life without them, yet each of those meetings was profoundly, unsettlingly improbable.

And why stop with friendships? Why not scale up to the level of the universe itself, where order and disorder interpolate in random patterns?

There, perhaps, from the perspective of God or some alien cosmologist or deep time or whatever you use to imagine inconceivable vastness, we might find order yet again. Who knows for certain; we likely never will. But we can look at a butterfly’s wing and wonder.

(1) The pigment-synthesizing genes that guide coloration in a butterfly’s wing are an ideal model system because they’re relatively simple and straightforward. That’s the exception, not the rule.

Most traits involve multi-layered cellular and genetic relationships: networks of interactions nestled inside networks of interactions, often behaving in nonlinear ways, a whole biological nesting doll of complication. Which makes their harnessing of randomness and uncertainty all the more extraordinary.

(2) There’s a certain amount of uncertainty to these descriptions. It’s fairer to say, some scientists who study these matters think this is happening, and the fragments of biochemical data we’re able to retrieve at molecular and atomic scales inside cells match our computational models, but it’s slow going.

Take, for example, the modeling of protein folding and unfolding described in this PNAS article, produced by a custom-designed, massively parallel piece of dedicated hardware that’s roughly 100 times more powerful than any other machine used for this purpose. Running at full power over the course of a day, it can model a scant 10 microseconds of molecular cell dynamics. Run it for 2,737 years, and you’d describe one second.

Brandon Keim (@9brandon) is a freelance journalist specializing in science, environment, and culture. Based in Brooklyn and Bangor, Maine, he often carries sidewalk caterpillars to safety.

One thing we know is that life reinforces the hypothesis that the world is infinitely complex and most of its phenomena will remain incomprehensible, meaning unexplained.  For example, no theory of life evolution was able to predict the next phase in evolution and the route taken to the next phase.  We don’t know if laws in biology will exist in the same meaning of laws of physics or natural phenomena.

For example, is the universe simple or complex, finite or infinite? The mathematician Chaitin answered: “This question will remain without any resolution, simply because we need an external observer outside our system of reference, preferably non-human, to corroborate our theoretical perception.”  (A few of my readers will say: “This smack of philosophy” and they hate philosophy or the rational logic deducted from reduced propositions that cannot rationally be proven)

So many scholars wanted to believe that “God does not play dice” (Einstein) or that chaos is within the predictive laws of God and nature (Leibniz), or that the universe can be explained by simple, restricted set of axioms, non-redundant rules (Stephen Hawking).

Modern mathematical theories and physical observations are demonstrating that most phenomena are basically behaving haphazardly.  For example, quantum physics reveals that hazard is the fundamental principle in the universe of the very tiny particles:  Individual behaviors of small particles in the atomic nucleus are unpredictable; thus, there is no way of measuring accurately speed, location, and direction of a particle simultaneously; all that physics can do is assigning probability numbers.

Apparently, hazard plays a role even in mathematics.  For example, many mathematical “true” statesmans cannot be demonstrated, they are logically irreducible and incomprehensible.  Mathematicians know that there exists an infinity of “twin” prime numbers (odd number followed by even number) but this knowledge cannot be proven mathematically. Thus, many mathematicians would suggest to add these true “propositions” but non demonstrable theories to the basic set of axioms.  Axioms are a set of the bare minimum of “given propositions” that we think we know to be true, but the reason is unable to approach them adequately, using the logical processes.

Einstein said: “What is amazing is that the eternally incomprehensible in nature is comprehensible”; meaning that we always think that we can extend an explanation to a phenomenon without being able to proving its working behaviors.  Einstein wrote that to comprehend means to rationally explain by compressing the basic axioms so that our mind can understand the facts; even if we are never sure how the phenomenon behaves.

For example, Platon said that the universe is comprehensible simply because it looks structured by the beauty of geometric constructs, the regularity of the tonality in string instruments, and steady movement of planets… Steven Weinberg admits that “If we manage to explain the universal phenomenon of nature it will not be feasible by just simple laws.”

Many facts can be comprehended when they are explained by a restricted set of theoretical affirmations:  This is called the Occam Razor theory which says: “The best theory or explanation is the simplest.”  The mathematician Herman Weyl explained: “We first need to confirm that nature is regulated by simple mathematical laws.  Then, the fundamental relationships become simpler the further we fine-tune the elements, and the better the explication of facts is more exact.”

So what is theory?  Informatics extended another perspective for defining theory: “a theory is a computer program designed to taking account of observed facts by computation.  Thus, the program is designed to predict observations.  If we say that we comprehend a phenomenon then, we should be able to program its behavior.  The smaller the program (more elegant) the better the theory is comprehended.”

When we say “I can explain” we mean that “I compressed a complex phenomenon into simple programs that “I can comprehend”, that human mind can comprehend.  Basically, explaining and comprehending is of an anthropic nature, within the dimension of human mental capabilities.

The father of information theory, John von Neumann wrote: “Theoretical physics mainly categorizes phenomena and tries to find links among the categories; it does not explain phenomena.”

In 1931, mathematician Kurt Godel adopted a mental operation consisting of indexing lists of all kinds of assertions.  His formal mathematical method demonstrated that there are true propositions that cannot be demonstrated, called “logically incomplete problems”  The significance of Godel’s theory is that it is impossible to account for elemental arithmetic operations (addition or multiplication) by reducing its results from a few basic axioms.  With any given set of logical rules, except for the most simple, there will always be statements that are undecidable, meaning that they cannot be proven or disproven due to the inevitable self-reference nature of any logical systems.

The theorem indicates that there is no grand mathematical system capable of proving or disproving all statements.  An undecidable statement can be thought of as a mathematical form of a statement like “What I just said is a lie”:  The statement makes reference to the language being used to describe it, it cannot be known whether the statement is true or not. However, an undecidable statement does not need to be explicitly self-reference to be undecidable. The main conclusion of Gödel’s incompleteness theorems is that all logical systems will have statements that cannot be proven or disproven; therefore, all logical systems must be “incomplete.”

The philosophical implications of these theorems are widespread. The set suggests that in physics, a “theory of everything” may be impossible, as no set of rules can explain every possible event or outcome. It also indicates that logically, “proof” is a weaker concept than “true”.  Such a concept is unsettling for scientists because it means there will always be things that, despite being true, cannot be proven to be true. Since this set of theorems also applies to computers, it also means that our own minds are incomplete and that there are some ideas we can never know, including whether our own minds are consistent (i.e. our reasoning contains no incorrect contradictions).

The second of Gödel’s incompleteness theorems states that no consistent system can prove its own consistency, meaning that no sane mind can prove its own sanity. Also, since that same law states that any system able to prove its consistency to itself must be inconsistent, any mind that believes it can prove its own sanity is, therefore, insane.

Alan Turing used a deeper twist to Godel’s results.  In 1936, Turing indexed lists of programs designed to compute real numbers from zero to 1 (think probability real numbers).  Turing demonstrated mathematically that no infallible computational procedures (algorithms) exist that permit to deciding whether a mathematical theorem is true or false.  In a sense, there can be no algorithm able to know if a computer program will even stop.  Consequently, no computer program can predict that another program will ever stop computing.  All that can be done is allocating a probability number that the program might stop.  Thus, you can play around with all kinds of axioms, but no sets can deduce that a program will end.  Turing proved the existence of non computable numbers.

Note 1: Chaitin considered the set of all possible programs; he played dice for each bit in the program (0 or 1, true or false) and allocated a probability number for each program that it might end.  The probability that a program will end in a finite number of steps is called Omega.  The succession of numbers comprising Omega are haphazard and thus, no simple set of axioms can deduce the exact number.  Thus, while Omega is defined mathematically, the succession of the numbers in Omega has absolutely no structure.  For example we can write algorithm to computing Pi but never for Omega.

Note 2:  Bertrand Russell (1872-1970) tried to rediscover the founding blocks of mathematics “the royal highway to truth”  He was disappointed and wrote: “Mathematics is infected of non proven postulates and infested with cyclic definitions.  The beauty and the terror of mathematics is that a proof must be found; even if it proves that a theory cannot e be proven”

Note 3:  The French mathematician Poincaré got a price for supposedly having discovered chaos.  The article was officially published when Poincaré realized that he made a serious error that disproved his original contention.  Poincaré had to pay for all the published articles and for destroying them.  A single copy was saved and found at the Mittag-Leffler Institute in Stockholm.




March 2023

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