Ruliobiology and the Theoretical Foundations of Biological Form

Biology is (as Charles Darwin put it a century and a half ago) full of “endless forms most beautiful and most wonderful”. But just where do such forms fundamentally come from? And is there any kind of general theory one can make of them? Darwin seemed to imagine that natural selection alone would (like “the fixed law of gravity”) inevitably lead to such forms. But it’s never been clear just why that would be so. Or how one might in fact set about constructing a theory of such things. But from work I’ve done in the past couple of years, I think we begin to see some possibilities. And that’s what I’m going to be talking about here.

The starting point is what we can call “ruliobiology”. We can think of ruliobiology as a metamodel of biology: an abstract framework not intended to provide a specific model of any particular biological system, but rather intended to capture the essence of certain core features of what’s fundamentally going on in biology. Ruliobiology is ultimately about what happens in systems based on simple rules subject to adaptive evolution—and we can explore it using the general methods and concepts of ruliology.

And what we’ll discover here is that a remarkable range of what we might consider to be “quintessentially biological” phenomena already show up in ruliobiology. And indeed ruliobiology seems to give us a way to see that, yes, in some sense Darwin’s supposition was correct: there is a certain theoretically-derivable inevitability to the generation of “endless forms most beautiful and most wonderful”.

One of the ingredients that’s needed is indeed Darwin’s idea of natural selection—and the process of adaptive evolution associated with it. But another crucial ingredient is what we can think of as the central phenomenon of ruliology: computational irreducibility. Computational irreducibility is what makes even simple underlying rules very often lead to behavior that is in effect irreducibly complex. And in effect it supplies the “raw material” for ruliobiology—on which adaptive evolution can act. It is then the interplay of adaptive evolution with computational irreducibility that leads to the core phenomena of ruliobiology.

Darwin seems to have imagined that there might be “laws of biology” analogous to laws of physics (like “the fixed law of gravity”). And here indeed we’ll see that there are “laws of ruliobiology”—though they are different in character from laws of physics. For traditional laws of physics tend to concern themselves with aspects of physical behavior that are computationally reducible, and that can be described, say, with mathematical formulas. Ruliobiology—and, I believe, biology—have computational irreducibility at their core. So, for example, whereas in physics one can discuss a phenomenon just as something happening “here and now”, in ruliobiology—and biology—there is an inevitable dependence on history, and on the computationally irreducible process that led us to here and now.

In the past one might have imagined that this would mean there could be nothing general one could say about ruliobiology—or biology—and that everything would have to be attributed just to “accidents of history”. But by thinking about the ruliad and about spaces of all possible rules we discover that there are in fact general things that can be said. One approach is to make use of the concept of the rulial ensemble: the set of rules that are reached by adaptive evolution with some computationally simple fitness function. And the key point is that while the details change with different fitness functions (and different classes of underlying rules), certain emergent features do not—and can therefore be seen as general to ruliobiology.

As is often the case, the introduction of the new methodology of ruliobiology lets us start addressing new kinds of questions—that in many cases one had never even thought to ask before. Ruliobiology is ultimately a rich and vast domain—and here I’ll only begin to scratch its surface. But from what we see here I think it’s already clear that ruliobiology can finally provide the basis for broad theoretical thinking in biology—with the potential to transform how one approaches many important issues in biology.

The basic setup in ruliobiology involves looking at “rulioorganisms” that “grow” according to simple fixed rules—with the rules being determined by an overarching process of adaptive evolution. Here I’ll be using rules based on symmetric 5-color cellular automata, grown from “seeds” consisting of a single cell:

The rules (shown on the right) can be thought of as corresponding to the genotype of the rulioorganism; the pattern (shown on the left) obtained by following these rules can be thought of as the phenotype of the rulioorganism. The rules are progressively updated by a process of adaptive evolution—with successive steps of this evolution involving random mutations that change the outcomes for particular rule cases:

The mutations are preserved if they increase (or at least leave constant) a fitness function; otherwise—in an analog of natural selection—they’re rejected. The particular fitness function we’ll mostly use here is “lifetime”, defined as the number of steps for which a phenotype pattern survives before dying out. (We take infinite patterns to have zero fitness.) With this setup we can plot (as red dots) the fitness values obtained by a sequence of mutations:

We see that many mutations decrease fitness—which means they’re not preserved. But every so often there’s a “breakthrough” that increases fitness. And looking at the rulioorganisms corresponding to these breakthroughs we see—a bit like in the fossil record—that they involve a sequence of “discoveries” that build on one another to progressively increase the fitness:

It’s immediately striking how “biological” these patterns look—with lots of complexity, but also identifiable patches of regularity reminiscent of the structures like organs that we see in biology.

The particular rulioorganisms in the picture above are the result of a particular sequence of random mutations. As we will soon see, other sequences of random mutations will lead to quite different rulioorganisms. But despite their detailed differences, what we’ll find is that there are features of these rulioorganisms—and their evolution—that are widely shared, and in effect represent the core phenomena of ruliobiology.

What can rulioorganisms be like? Here are a few examples:

The resonance with Darwin’s “endless forms most beautiful and most wonderful” is certainly striking. Each of these rulioorganisms was ultimately evolved with the same simple objective of “living as long as possible”. But what we see is that the rulioorganisms achieve this objective in a remarkable diversity of different ways. Notably, not a single one of these ways is “simple”, in the sense that one can readily give a simple “mechanical” description of how it works. And at some level all of them “just happen to work”, in the sense that they represent frozen pieces of computational irreducibility that turn out to achieve the objective—even though one can’t see that this will be the case except in effect by explicitly running them and observing what happens.

So how did these rulioorganisms evolve? Here are some examples of the sequences of breakthroughs that were involved, in all cases starting from the trivial “null” rule:

The details are different in each case. But there’s an important common theme. In the course of the adaptive evolution, some “innovation” is invented (like “use triangular regions of purple”), and then, as the adaptive evolution proceeds, this innovation is progressively (and often “creatively”) elaborated.

Look at a biological organism at different scales and you’ll see different things. Indeed, it seems as if in biology there’s—quite ubiquitously—“novelty” at every scale. When biological organisms were first examined under microscopes it was a big surprise how much unseen structure was discovered. And with essentially every new level of microscopy that’s introduced, yet more—often very different—new structures seem to be revealed.

Nesting and the idea of fractals captures the notion of having smaller copies of the same structure at different scales. But what we’re talking about here is something very different: we’re talking in effect about having a whole hierarchy of novelty, where fundamentally different things are happening at each scale. It’s something that’s been hinted at—particularly in biological contexts—for centuries. But so far as I know, it’s not something that’s ever explicitly been given a name. Here I’ll refer to the phenomenon as polyplexy. (As we’ll discuss below, it’s a different phenomenon from what one sees in something like the Mandelbrot set.)

So do we see polyplexy in ruliobiology? The answer is yes, quite ubiquitously so. Consider for example this rulioorganism:

Provide a zoom and pan version of this ...

What we probably first notice is its complicated overall shape:

Next we might notice its areas of purple—and the random-looking “connective tissue” between them. But let’s zoom in a bit, say near the lowest part of the pattern:

There are areas of purple alright. But they’re laced with (irregularly placed) vertical stripes. And we can now see that the connective tissue isn’t really random: there’s actually all sorts of structure in it. Zooming in a little more (and showing only the left half of the pattern) we start to see a whole variety of specific highly ordered structures dotted around:

At a still smaller scale we begin to see how those structures are embedded in the larger pattern:

And zooming all the way down to “ruliological atoms” we can finally identify some of the “lowest level structures” in our rulioorganism:

We can think of such structures as corresponding to patches of computational reducibility. But sometimes—even at the lowest level—there’s just too much immediate computational irreducibility for us to be able to discern any kind of “describable structure”:

Looking at our rulioorganism as a whole, we see that—like a typical biological organism—it contains many small structures that are basically repeated in different places

though often in slightly different forms:

Biological organisms are typically viewed as having a hierarchy of types of parts (from organ systems to organs to tissues to cells to organelles, etc.) And one might have imagined that cellular automata of the kind we’re using here—with their simple, uniform underlying discrete structure—would never be able to exhibit any such hierarchy. But in fact we find that our rulioorganisms are full of hierarchy—usually with visibly different kinds of structures on different scales:

Another feature of rulioorganisms is that their different parts are often differentiated from each other. In the end, the parts must all “work together” to achieve the fitness objective that’s been defined. But in general, many of them will in some sense “do different things”—much like different parts of biological organisms.

(It’s useful to contrast this with what happens in something like the Mandelbrot set, where there’s plenty of visual diversity, but where at every coordinate position and every zoom level one’s just seeing geometric distortions of a single object, with the distortions defined essentially by the digit sequences of the coordinates.)

We saw above various examples of rulioorganisms. But what is the space of all possible rulioorganisms like? I’ve argued elsewhere that any “rulial ensemble” of what we now call rulioorganisms selected according to any computationally simple fitness function will have certain overall features in common. But to be specific here, let’s start by considering (symmetric) rules selected purely on the basis of lifetime, and let’s look at the rulioorganisms we get after 1000 steps of adaptive evolution (here with the lifetime in each case indicated):

“discuss later”