Can a formal model unify biology? What inductive informatics offers to evolutionary biology,...

Can a formal model unify biology?Can a formal model unify biology?

What inductive informatics offers to evolutionary biology, taxonomy, molecular biology, and

developmental biology

Lev Goldfarb ETS group

Faculty of Computer ScienceUNB

Fredericton, Canada

http://www.cs.unb.ca/profs/goldfarb/Cornell.ppt

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Outline

1. About the talk (4 slide)

2. The language of biology vs. that of physics: the metaphors scientists live by (6 sl)

3. On the reduction of biology to physics and chemistry (2 slides)

4. If not the “reduction” to physics and chemistry, then to what? (3 slides)

5. What is a numeric representation? (2 slides)

6. What is representation? (1 slide)

7. Inadequacy of the present day formalisms (5 slides)

8. What is a structural representation? The ETS model (12 slides)

9. Inadequacy of the string representation (5 slides)

10.The paradigm change (2 slides)

11.The ETS model and the above four main areas of biology (6 slides)

12.Conclusion: the future of biology (3 slides)

Lev Goldfarb, Cornell, March 2003 3

About the talk

Q: What put me in this “generous” (and professionally dangerous) state of mind?

A: An even more “dangerous” thought precipitated by what appears to me as the situation absolutely unprecedented in the history of science: emergence of a radically new, structural, mathematical paradigm, or language, as opposed to the numeric mathematical paradigm. The latter has dominated the science from the very beginning. The new formalism suggests a unified framework for understanding the information processing in nature (“representation”), as opposed to the conventional, “computational”, treatment of informational processing (“encoding”).


About the talk

In science, there is a conspicuous lack of the formal language that would suggest us how to properly think about various (evolving) classes of related objects. And, of course, all objects/events in the Universe both appear and evolve only as members of the corresponding classes of objects. Any kind of “evolution” in the Universe, including biological evolution, cannot be properly understood without insight into the nature of the relationship between an object and the class of related objects, including the organic connection between an object representation and the class representation.


About the talk

Over the last fourteen years, we have gradually discerned the outlines of a new and very general formalism for modeling the evolving relationship between an object and the relevant class of objects. The emerging structural object representation was developed, of course, as a far-reaching generalization of the classical numeric representation. However, the differences between them are so great that one should properly speak of emergence of qualitatively different mathematical language: the generalization is so substantial that its mastery will require quite significant efforts.


About the talk

Since the concept of an evolving class of objects is, quite likely, the very central one in life sciences (and biology in particular), I believe that the talk should be of quite general interest.

Given our UNB group’s relatively tiny resources, it is not surprising that the substantial applications of the new model I briefly outline here are still ahead. However, I thought that the formalism itself, as has always been the case in science, should be useful to some scientists in guiding their research efforts (and experiments in particular).

As for the term inductive informatics in the title of the talk, we associate it with the new science emerging around the development and various applications of this new formalism (recall the ideas of the true prophet of modern science, Francis Bacon).


The language of biology vs. that of physics

Scientists need metaphors: we first shape our metaphors . . . then they shape us! So, what are the metaphors that guide these two basic sciences?

• The development of physics suggests that, eventually, science involves the construction and development of the appropriate mathematical “languages” (as “metaphors”):

“I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them . . . which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics.”

(Albert Einstein, “On the method of theoretical physics”, 1933)


The language of biology vs. that of physics (Kuhn’s paradigms)

In particular, the nature of scientific revolutions, as proposed by Thomas Kuhn, is mainly explained by the (radical) shifts in the formalisms associated with the corresponding scientific paradigms: no formalism shift, no paradigm shift in Kuhn’s sense.

We will see shortly (slide 12) why, quite understandably, a prominent participant in the “evolutionary synthesis” Ernst Mayr objected to the above view.

Based on the history of biology, one can suggest that there are other kinds of “revolutions”, e.g. the one associated with the name of Darwin, but these, quite naturally, are, as Mayr himself suggests in the above slide, are more gradual and less intense.


The language of biology

• The development of biology, on the other hand, have relied often on the old physical, “mechanistic”, sociological, or linguistic metaphors:

“In October 1838, that is fifteen month after I had begun my systematic inquiry, I happened to read . . . Malthus on Population, and being well prepared to appreciate the struggle for existence which everywhere goes on from long-continued observation of the habits of animals and plants, it at once struck me that under these circumstances favourable variations would tend to be preserved, and unfavourable ones would be destroyed. The result of this would be the formation of a new species. Here, then, I had at last got a theory by which to work.”

Charles Darwin, Autobiography, 1876



Here is a relevant testament from an interesting recent book:

“Of all the inanimate objects in the universe, few have so captivated the imagination of biologists as our own machines and automata. Nowadays it is the computer that is held up as the most instructive analog of living organisms, with cellular architecture as hardware and the DNA tape as software.”

Franklin M. Harold, The way of the cell: Molecules, organisms and the order of life, OUP, 2001



I am convinced that, in contrast to physicists, biologists rely on such informal metaphors simply out of necessity: as all scientists, they need guiding metaphors, and they don’t have any guiding formalisms.

The absence of such formal language results in reliance on inappropriate metaphors. This situation has profound consequences that carry a very high price tag: quite obviously, in the long term, the wrong metaphors do render our efforts much less productive. I will come back to this point throughout the talk and will also suggest what I consider to be “productive metaphors” in biology.


Mayr’s objections to Kuhn’s view of scientific revolutions

“Where in the history of biology were the cataclysmic revolutions and where were the long periods of normal science postulated by Kuhn’s theory? From what I knew of the history of biology, they did not exist. No doubt Darwin’s On the Origin of Species, published in 1859, was revolutionary, but ideas about evolution had been in the air for a century. Moreover, Darwin’s theory . . . was not fully accepted until almost a century after its publication. Throughout this time there were minor revolutions but never any period of “normal” science. Whether or not Kuhn’s thesis was valid for the physical sciences, it did not fit biology.”

Ernst Mayr, This is biology, 1997

Unfortunately, Mayr does not realize that biology simply has not reached sufficiently formal stage of development, at which time Kuhn’s observations begin to apply, as they must (see slide 8).


The reduction to physics and chemistry as the “salvation road”

“Many more [biochemists and molecular biologists] would agree with Francis Crick (1966) that ‘the ultimate aim of the modern movement in biology is in fact to explain all of biology in terms of physics and chemistry’.”

F. M. Harold, The way of the cell


The reduction to physics and chemistry as the “salvation road”

Ironically, a number of leading physicists of the last century, e.g. Niels Bohr, Erwin Schrödinger, Eugene Wigner, as well as less known ones, e.g. Walter Elsasser, disagreed with the above view (and, I believe, they where in a better position to compare the overall nature, and estimate the complexity, of physical and biological systems):

“Niels Bohr was apparently the first to suggest that special laws not found in inanimate nature might operate in organisms. . . . Erwin Schrödinger and other physicists supported similar ideas. Francis Crick (1966) devoted a whole book to refuting the vitalistic ideas of the physicists Walter Elsasser and Eugene Wigner.”

Ernst Mayr, This is biology, HUP,1997


If not the “reduction” to physics and chemistry, then, to what?

Not to keep you in suspense about my answer to the above question, here is a short one. The “reduction” I am advocating is the “reduction” to the radically new formalism which we called the evolving transformations system (ETS) model, which forms the core of inductive informatics (recall slide 6).

Again, inductive informatics is an emerging science dealing with the structural object and class representation across all sciences. The important point to keep in mind:

inductive informatics is the science of (structural) representations.



I anticipate that the new formalism, because it is the “representational” formalism, should change the conventional pyramid of natural sciences

biology

chemistry

physics

mathematics

(which reflects the historical order in the development of natural sciences, as well as the dependence on the numeric models) to the following simple scheme:

inductive informatics

biologychemistry physics



In connection with the last slide, I can’t help quoting British biologist John Henry Woodger (1894-1981):

“writing more than 70 years ago [Biological Principles, 1929]. . . he regarded it as ‘somewhat of a scandal’ that biologists have no adequate concept of organization: ‘The failure to take organization seriously is perhaps but another consequence of the rapid development of physics and chemistry as compared to other sciences, and the consequent dazzling effect this had on biological vision’. ”

Franklin M. Harold, The way of the cell

Let’s move on now to a more formal “story”, and we need to start it from the very beginning.


What is a numeric representation?

Our natural numbers have gradually emerged over a hundred millennia as the most popular/convenient form of representing information about objects or events, and all our scientific paradigms are built on the foundation of this, numeric, representation. Measurement is the corresponding process for (numeric) representation of objects or events, i.e. it is a procedure or device that realizes the mapping from the set of objects to the set of numbers.

It is very important to realize that, in mathematics, there has existed only one underlying/basic representation model (specified by the classical Peano axioms). This model captures the structure of natural numbers and implicitly includes the standard distance measure on them.


What is a numeric representation?

Classical measurement is a systematic method for representing objects/events by numbers.

Peano representation

Fixed property

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What is a representation?

Representation structure

In general, representation can be defined as a systematic process of assigning to “concrete objects” from classes some entities, let’s call them structs, which also belong to the appropriate classes. Moreover, we postulate that all classes have “generative” structure.

It is important to keep in mind that “laws” in biology are about classes of entities, e.g. about taxonomic categories. In particular, predictive generalizations about the entities are taxonomic statements.


Inadequacy of the present day mathematics

Over the last 15 years, it has gradually become clear to me that none of the current mathematical and computational formalisms (including Chomsky model) is adequate for studying various, but particularly inductive, relationships between objects/events and their classes.

To appreciate a complete lack of the “class” perspective in mathematics, it is interesting to note an attempt by the originator of set theory, George Cantor, to describe the concept of a set (as opposed to class), the very foundational concept of modern mathematics:

a set is any collection of definite, distinguishable objects of our intuition or of our intellect to be conceived as a whole (“shades” of a class concept).

What are the difficulties involved?


Inadequacy of the present day mathematics

• The form of class representation in statistical/numeric models is not related to the form of object representation: any family of decision surfaces can be chosen, but none of them can be justified on the basis of the vector space axioms (where the latter have to be taken more seriously as specifying the object operations). This is an inherent limitation of the classical, numeric, mathematical models: in them, the class elements cannot be directly expressed via the object operations.

• As a result, we cannot construct/predict any new class elements.


Inadequacy of the present day formalisms

Chomsky’s concept of generative grammar is closely related to the concept of Post’s production system, a classical computational formalism. The problem with Chomsky grammar (as with all computational “models”) is that it is not a “representational” model, which should be a basic requirement for an (applied) scientific formalism:

the string is not an adequate representation of the original object; in particular, a string over a finite alphabet does not carry within

itself enough information to link it reliably with the corresponding grammar, i.e. to identify the class to which it belongs.

Note that the classical numeric representations also do not carry such information. This is the main reason why elaborate and quite “artificial” classification models, including various statistical models, have been proposed.



Inadequacy of the present day formalisms

Thus, in spite of their central and strategic importance for all sciences, the concept of “object/event representation” and the closely related issues concerning the relationships between objects/events and their classes have not been addressed so far, until the ETS model was recently developed.


ETS model basics: primitive transformations

Primitives with the “interface” sites labeled. Labels have structural types.

Some examples of structural types in biochemistry:

• for atoms — various bonds

• for nitrogenous bases — hydrogen bonds in A – T, C – G

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ETS model basics: segments (segments of history)

Formal notation:


ETS model basics: struct as the final form of representation

The final form of object representation (for the simplest case of a single level representation) is derived by merging all segments of the same structure into a single entity, struct.

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ETS model basics: transformations

transformation

segment the same segment with marked context

after application of the transformation

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ETS model basics: transformations as new primitives, levels


ETS model basics: class representation

In the model, a class is specified by a progenitor plus a finite set of weighted transformations:

progenitor + transformations (with their weights) .

Such class representation can be directly expressed via the class generating process.

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ETS model basics: class generating process

class progenitor

weighted transformation


ETS model basics: class typicality measure

For each class C, based on the corresponding sequence of transformations, the typicality measure typC (defined on structs) can be introduced. This turns the class into a fuzzy set, but the fuzzy membership function is not defined by an expert but is induced naturally by the generating process.

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Example: class representation for the ALC class

ProgenitorOH

Transformation set

O

O

1

2

3

4

5

6

7

8

9

10

11

12

H

OH H

Type w(1 F

118.7

2 F 0

3 F 37

4 F 16.7

5 F 17.3

6 CN -53

7 CN -56.3

8 CN -55.3

9 CN -54.3

10 CN -28.3

11 CN -8.3

12 CN -7.7


Example: part of the generating process for the above ALC class

H

OH

OH

O

OHOH

OH

OHOH

O

OH

OH

OH

O

OH

O

H

HH

OH

OOH

O

H

OH

O

HO

OH

OH

OH

OO

OH

OH

O OH

O

OH

O

H

OH

O

H

H

OH

OH

OH

OH

OH

O

O

OH

H

H

m3 m2 m4 m5 m1

OH

OH

OHH

O O O

H

OH

H

H

H

H

The hierarchy of object & class representations in the ETS model

Ci is a class at level m whose description is Ti : Ci = <Ti > .

I.e., the set of transformations of class Ci consist of the

transformations from level m.

structs transformation sets

Ti

level m class descriptions

Ci = <Ti >

level m object representations

progenitor


ETS model basics: the evolution of a class

Since any class is specified by a progenitor plus a finite set of weighted transformations

progenitor + transformations (with their weights) ,

the class evolution is readily understood via modification of the set of transformations (structural change) and/or their weights (quantitative change). Hence the typicality of an object w. r. t. the class also evolves.

Inadequacy of the conventional string representation

Three different structs corresponding to the same string abc

Primitives for a a three letter alphabet

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contexts

Below are the two corresponding structs:

Two possible formative histories for string abaca

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Three relevant transformations (see the last slide) and the corresponding next level primitive transformations (shown in gray):

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Several of the many possible (2nd level) structs corresponding to the string below.




Thus, the conventional string is not an adequate/reliable form of representation: there are just too many object histories that are “hidden” behind this representation.

The related observation applies to various numeric representations.


The formal paradigm change

The ETS model was motivated by the desire to “bring together” the forms of object and class representations. The impossibility of achieving this goal within the confines of the known formalisms became gradually clear to me after analysis of the substantial attempts (undertaken in the areas of pattern recognition and artificial intelligence during the last 40 years) to unify two fundamentally incompatible formal models:

• the basic applied mathematical model of the measurement space, the vector space model, and

• the Chomsky’s version of classical computational model, the generative grammar model.


The paradigm change

In the ETS model, a fundamental gap between an object and the class that “gave rise” to that object is removed. Thus, the model offers us the tools (our guiding “metaphors”) for investigating a large variety of natural phenomena that satisfy the main “generative” hypothesis.

This hypothesis suggests, in particular, that classes are, in an “informational” sense, more “primary” than the objects: whenever an object appears, its class (description) must “precede” it as a blueprint for its (as well as other class elements) formation.


The present situation in biology

“So are we all waiting . . . for new techniques of apprehending the utterly remote past. Without such a breakthrough, we can continue to reason, speculate and argue, but we cannot know. Unless we acquire novel and powerful methods of historical inquiry, science will effectively have reached a limit.”

Franklin M. Harold, The way of the cell

My belief in the ETS model offering “powerful methods of historical inquiry” will probably not surprise you.


Evolutionary biology & the ETS model

What does our model for structural representation offer to evolutionary biology? For the first time, we have a precise language for capturing and studying the relevant “historical” processes (last slide) at various levels.

• We can now view the evolution process not just as a slogan but as a result of the interaction of many generating processes, some of which are more “passive” (those associated with the “environment”) while others are more active (“biological” generating processes). Obviously, the more active processes evolve faster than the more passive ones. Each generating process is specified by a hierarchically structured weighted set of transformations.

• We now have a uniform language for describing the evolutionary processes at all levels: molecular, cellular, tissue, organ, organisms, etc.

• Moreover, the same formal language, allows us to describe and to study the evolutionary transition from one level to the next one.


Biological taxonomy & the ETS model

Taxonomy is a key area of biology, but the lack of formal models that incorporate naturally the evolutionary reality into a classification model split the area into several opposing groups.

The ETS model was developed as a framework for classification. In part, it was motivated by a natural unification (in a single model) of the ideas that were responsible for the splitting of taxonomists into at least three schools: phenetic, cladistic, and evolutionary schools.

I believe that upon closer examination, the ETS model should satisfy all taxonomists.


Molecular biology & the ETS model

With the advent of molecular biology after WWII, biology entered into a new, less intuitive, stage. As far as I am concerned, this stage more explicitly exposes the need for “formal metaphors”, or, as Franklin Harold mentioned above, for “novel and powerful methods of historical inquiry”.

Moreover, I believe that the concept of random mutation is neither the answer to this challenge, nor to the nature of evolution.

In studying actual biological processes, the main difficulties are related to the separation of the “software” (information driven) from “hardware” (energy driven) processes, including the separation of the two associated optimization processes, which are tightly integrated in nature. Current scientific paradigms cannot help us with this separation.


Molecular biology & the ETS model

The ETS class representation, together with the associated generating process, suggests a direct way of thinking about biomolecules and their evolutionary classes as inseparable concepts.

Moreover, one can immediately make some predictions, based on the model, that are currently considered quite controversial: the relevance of temporal order in the construction of a protein’s amino acid sequence. Note, however, that non-controversial predictions are much less interesting.

At the same time, the continuum of “near molecular” life forms—episomes, transposons, insertion elements, retrotransposons, retroviruses, hepadnaviruses, lysogenic phages, plasmid-viruses—gives us a glimpse into how ETS transformations might have been implemented biologically (“hardware” + “software”) during the early evolution of life on Earth.


Developmental biology & the ETS model

The model suggests that we should think of any biological taxon as a result of a long process whose “external” features are immediately visible (“the tip of the iceberg”), while formative/developmental features are not transparent.

The evidence discovered by evolutionary developmental biologists, including homeotic gene complexes, support this view of a taxon. It is interesting to note that the following expression, used by developmental biologists, is completely consistent with the ETS view: “during the embryonic development, a cell is influenced by its developmental history”.

Moreover, a taxon appears and evolves when the developmental process (ETS generating process) is accordingly modified. Thus, for example, Lewis Wolpert’s statement “evolution can be seen to be the modification of development” now has a much more precise expression in the ETS model: evolution can be seen as a modification of the generating process, i.e. as a modification of the class description.


Conclusion

Biology has developed historically as a descriptive science studying the diverse forms and categories in the plant and animal kingdoms. This is why the central place in biology is occupied by the analysis of, systematics of, and classification of an enormous amount of empirical material. This material has been collected by naturalists and, more recently, by molecular biologists and geneticists.

However, having been faced with more “abstract” data, i.e. molecular data, biology will need powerful (formal) “guiding metaphors” to fully mature as a natural science. The wrong metaphors/models lead away from that goal, rather than towards it.


Conclusion

I very briefly outlined a radically new formalism that may be able to guide us through this paradigm change and, in the process, mature as a formalism for structural representation. In particular, it should offer tools for delineating and clarifying the following issues involving the relationships between an individual and a corresponding class:

• the concept of a formative/generative history of a biological object

• the concept of a class, including the class representation (progenitor + weighted set of transformations)

• the class generating process and its relation to the developmental process

• the hierarchy of multilevel class descriptions

• the nature of the formation of a new biological level


Conclusion

Undoubtedly, the golden age of biology is still ahead of us, though its arrival depends on the choice of the “right” representational model.

I hope that some of you will participate in this most exciting development.

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Can a formal model unify biology? What inductive informatics offers to evolutionary biology,...

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