The correlation of structure and function is a central theme in biology so this is a good question to consider. To make sense of biological function, one needs to look through the lens of evolution, which is seldom clear. In contrast to function, mechanism is often examined by going from existing structural elements to test how something works to achieve an activity. Function in biology exists via a process of selection – the result of a chain of causes unlike an engineered machine. Relationships of structure and function exist on molecular, biochemical, cellular level, physiological, and organism levels and each has to be seen through the lens of evolution. In practice, trying to define function from structural data depends upon having additional information. At the molecular level, perhaps the most useful type of prior information is being able to recognize previously understood functional domains and motifs and then apply tests of activity and ultimately genetic mutation and complementation. The biological function problem is hard because evolution does not design for a predetermined purpose as in engineering, but represents the outcome of a selective process.

So solving a macromolecular structure does not reveal its function in a straightforward fashion. However the structure often provides a detailed framework to integrate other types of information and with which other types of data must be reconciled. Biology becomes more predictable at the molecular level where function is informed by catalysis, conformations, and interactions. My first crystal structures were for Cu,Zn superoxide dismutase (SOD) and the structures explained the activity, specificity and faster than diffusion rate via electrostatic guidance of the anion substrate to a positively charged active site (see review Biochim Biophys Acta 2010 Feb;1804(2):245-62. doi: 10.1016/j.bbapap.2009.11.004). Yet despite the structural evidence supporting its biological function, the discovery of SOD by biochemists Irwin Fridovich and Joe McCord is viewed as “the most important discovery in modern biology never to win the Nobel Prize ” (www.nejm.org/doi/full/10.1056/NEJM200309113491125). ‎ This was because although SOD was the first evidence for the great importance of reactive oxygen species and its regulation in cells which certainly merited a Noble, the biological function was challenged for many years by chemists who argued it was a metal storage protein or had some as yet unknown function, despite the strong structural evidence.

I am uncomfortable with the “physical science of teleology” wording because evolution is not analogous to human purpose in terms of working toward specific ends. But I do think we can “differentiate between living (propagative) and dead (non-propagative) configurations of the organism” empirically and by the types of molecular machines necessary for life. Life (as we know it) has to eat and breathe and reproduce itself. In chemical terms this means that there must be an electron donor and electron acceptor to sustain life and a stored form of the instructions for organism development and functioning that can be accurately copied and maintained. So I agree with your idea and axiom, as far as I understand it.

Thanks for your thoughtful response, John! You write:

"The biological function problem is hard because evolution does not design for a predetermined purpose as in engineering, but represents the outcome of a selective process....I am uncomfortable with the “physical science of teleology” wording because evolution is not analogous to human purpose in terms of working toward specific ends."

I agree. The word 'teleology' is so contaminated by theistic/Aristotelian connotations that similar considerations promoted the coinage of 'teleonomy' by Pittendrigh.

The problem is, when we're talking about the 'perpetuation of the living state' (the structural biologist's usual proclivity, as you mention in the last paragraph) then a structure-function relationship IS the same thing as a structure-purpose relationship (with the purpose being 'staying alive').

And are we really so far from telos if we're talking about 'purpose'?

Perhaps the thought experiment in the first few pages of my Thesis (determining the structure-function relationships of an inclined-plane) will illustrate where I'm at with this. I've attached that.

The problem with the thought experiment is that, as you know, the only known constraint on structure-function relationships is that these are *evolved* structures producing *adaptive* functions.

So in the context of SOD the discussion was (I'm supposing) centered on its adaptive role in extant mammals. But it's difficult to show that, given some radical (haha) change in the gene product's environment, it would not improve an organism's fitness through metal storage. I couldn't see that happening in the course of primate evolution, but let's say it gets horizontally transferred to a bacterium that lives in anoxic conditions....then SOD may well act as a metal storage protein.

This problem -- the structural ambiguity of biological adaptiveness -- is what I'm trying to work out. I'm not sure if the problems solvable though.

Consider the requirement for life that it have a stored form of instructions for organism development and functioning that can be accurately copied and maintained. In this light, the structures of molecular machines that repair damaged DNA (the instructions for organism development) have non-ambiguous relationships to function. In particular we gain knowledge about the molecular basis of normal biological functions in genome integrity and how they go awry in human diseases when we examine the structures of DNA repair complexes with mutations associated with predispositions to cancer and aging.

The complementary statement to 'life requires high-fidelity DNA replication' is that 'evolution requires imperfect DNA replication.' I don't understand how any protein participating in a DNA repair mechanism could evolve if its ancestor also had a completely unambiguous structure-activity repertoire. For example, at some point, wasn't a nominal polymerase 'recruited' to specialize in editing/repair?

You raise important and fundamental questions. In fact, life is impossible without DNA repair; this was the importance of the DNA double helix that Watson-Crick missed because they were taken by the ideas in physicist Erwin Schrödinger’s book “ What Is Life” published in 1944. In this book, the idea from physics was that the genomic material had to be unusually stable. DNA has to be stable enough of course but the double helix reveled that biology took the much more feasible and efficient solution of having two copies so that the damaged genetic information could be repaired based upon a second copy. The double helix is about repair as well as replication. Replicative polymerases and repair polymerases have undergone gene duplication and variation to become the lesion bypass polymerases that keep replication from getting blocked by DNA damage, which is a constant result of endogenous and exogenous damage.

Perfection in genome fidelity, like perfection in anything, becomes infinitely expensive in an imperfect world. So high fidelity DNA replication and repair needed for organisms does not mean perfect – there will always be variation upon which evolutionary selection can act as completely avoiding changes is unsustainable in terms of biological cost. Organisms can selectively gain variation by reducing or eliminating some (but not all) DNA repair functions, such as DNA mismatch repair, which is lost in many emerging pathogens, so that they can take up and incorporate rather then degrade foreign DNA with mismatches. Similarly defects in all the various DNA repair pathways are associated with cancer predispositions, as the loss of accurate damage responses creates a mutator phenotype that allows cancer cells to escape normal cellular controls on growth and metastasis.

I guess I was expecting that some would argue that the Goedel Incompleteness Theorems set bounds on the whether a biological function was computable, but I didn't have a clear sense (it's easy to abuse those concepts and I lack the formal training.) It seemed to me that a set of axioms that are flexible enough to anticipate novel biological functions would not be self-consistent.

I talked to a friend (Paul-Jean Letourneau) (http://www.rule146.com/about/) that works with cellular automata and on the problem of computability, which took me down an interesting rabbit hole. In particular, the work of Gergory Chaitin -- previously unknown to me -- is a wellspring of concrete elaborations. I attached a reference of Chaitin that I'm chewing on.

I'm also blown away by this other paper; it almost completely preempts and encapsulates my thinking thus far (showing, as usual, that I should be reading about ten times more than thinking!):

http://arxiv.org/pdf/1210.5908v2.pdf

The point I make in the intro of my Thesis is nicely subsumed by "...living is information processing, in which memory is maintained by both molecular states and ecological states as well as the more obvious nucleic acid coding..." The elaborations I attempted also seem to be well-explained by that paper.

Also, the blog of Hector Zenil is proving to be extremely enriching:

http://www.mathrix.org/liquid/archives/living-organism-algorithmic-complexity-logical-depth-genetic-toolkit#comments

Just a quick note here: the term "function" might well be itself flawed. I'd propose to just speak of effects, as this does not imply a preexistent criterium of utility (which happens with function). Everything there is has one or more effects and these effects are dependent on the context. In line with this, I consider life the ability and spontaneous activity of a system to take parts of its environment and rearrange them so they become parts of an instance of its own class, be it the original system itself (growth, manteinance and repair) or another instance (reproduction).

There are many implications to this kind of definition, but addressing the original question, the only proper biological function that could be defined would be the main process of life itself, that is, the very definition used. What is generally referred to as "biological functions" would be "effects" and "mechanisms", secondary to the main process and dependent on the context for their contribution or hindering of the biological function, so not objectively qualifying as functions themselves.

Now, that just came out of my head, so I'll better read the attached documents and links and see for myself the multiple mistakes that you'll be pointing in these thoughts.

Andrés, after reading and re-reading your comment, I agree that the word 'function' has flawed connotations. A deeply related question to 'can we make a physical science of teleology' is 'can we only speak of causes and effects in biological systems.' So I like that, although I can't figure out what to do with it.

Parenthetically, I disagree that organisms are well-described as instances of a class (that class presumably being a species) because that (apparently) precludes the evolution of new species from an ancestral species. The organisms within a species must vary, and a class definition cannot (I think) also identify all the ways something is allowed to vary while still belonging in the class, effectively re-writing its own definition. To paraphrase Ghiselin (who originated this idea), organisms are parts of a whole, and that whole is the species. Species are individuals and organisms are their parts. (Probably material for a different Q&A thread.)

As the discussion below shows, I'm hung up on the consideration of effects without causes, and I'm very hung up on the assertion that "...effects are dependent on the context." I think 'context' may be a similarly problematic word as 'function.'

Say we're talking about gravitation; the movement of tides is an effect of the Moon orbiting the Earth. Say the movement of the tides sustains a biological niche (tide pools) and therefore the evolution of particular adaptations (strong anchoring points.)

When a biologist picks up a tide pool critter and exclaims "look how this structure facilitates adherence to a substrate" and later a structural biologist describes the molecular interactions that mediate this particular adaptation "look how these sidechains make strong ionic bonds to silicates" (I'm making that up) then I think (and I think you'd agree) that gravitation will not meaningfully direct the train of thought.

Although there is a causal chain that connects the Moon's orbit to the evolution of sticky tide pool critters, it doesn't seem to be informative -- indeed, substitute any other source of turbulence, and the same adaptive pressures would result, and similar adaptations would evolve.

This does appear to suggest a solution; the context to consider is shearing hydrodynamic forces at particular length scales, not the lunar orbit. A well-defined context is helping us to meaningfully interpret the evolutionary history! So the evolved sticky-structures have effects that are significant in terms of shear forces. But if that seems clear then what are the causes of these effects? Causality seemed simple and appropriate when describing the lunar orbit giving rise to tides...what am I missing here when describing this adaptation of sticky tide pool critters?

UPDATE:

A friend mentioned the work of Robert Rosen and the role of circular causation in living systems. Here's a summary (haven't read the source material so I don't know how accurate this is): https://lpei2.wordpress.com/the-waters-of-life/2-causation-complexity-and-life/. From that article:

"Rosen’s most often-quoted insight is that “a material system is an organism if, and only if, it is closed to efficient causation.” What the organism is open to from the outside world is material cause: its necessary physical environment, nutrients that it needs to stay alive. But it is closed to efficient causation in that its fundamental aliveness is not being caused from without. The organism causes itself to continue to exist; this is the whole point of the self-replicating repair system. No outside agency is required to intervene in order to keep the organism in being; and when a certain threshold of aging, disease, or injury is crossed, no outside agency can keep life going.

This is how an organism is fundamentally different from a machine."

Thanks, Keith, for the mention of this fascinating website. The page:

http://www.whatlifeis.info/pages/Themes/Philosophy/Function.html

directly addressed my current contentions. I seem to be leaning towards the 'dispositional' definition of a biological function (as described by Sandra Mitchell) rather than the etiological one.

I found this discussion very interesting and thoughtful for me, being a General Pathologist at Verona University. I can't define (but even speak to students) what is "pathologic" outside a teleonomic framework (in favor or against life of the individual and/or of the species). Fever, pain, cell growth, cell death, even infection can be useful or damaging life according the final outcome of those phenomena for the whole organism (or for the whole world?). But now I know that this view is scientifically insufficient. Thanks.

It seems to me that only operational definitions of a "function" are possible. The reason for it is a teleology-generating assumption that living things are kind of mechanisms like a car or a grammar of a language. Verbs in a grammar are functionally equivalent and so do spark plugs in different cars. The "function" then could be defined as a set of functionally equivalent parts of a well-understood machine / mechanism.

Relational biologists would strongly object to  entire concept of function not to mention its definition(s). They would probably be right if we believe that living things are realizations of complex systems and thereby cannot be adequately modeled as machines in all their aspects.

I like your proposal of an axiom but I don't think it can be useful in forming methodology of life sciences. (Too much of a belief system involved in it.)