MATHEMATICAL MORPHOGENESIS
Christopher J. Marzec
Institute for Biomolecular Stereodynamics
cmarzec@csc.albany.edu
Introduction
Welcome.
This site is intended as a visual venue for presenting some research
results in the field of morphogenesis.
The text only adumbrates the ideas that generated the images.
These are developed more fully, and less polemically, in the
references .
We speak here informally, on the internet, in the interest of better
communication across disciplines; we will try to avoid using semicolons.
But all of the models are rigorously formal, in that they have been
generated by systems of equations.
So, except for Felix as Natural Philosopher at the bottom of
this page, the animations are not cartoons.
All animations are listed in an INDEX OF
ANIMATIONS.
The enthused reader may view
pre-prints of articles in
PDF format.
Morphogenesis is the creation of form, as it appears in both biological
and non-biological systems.
We are concerned with shapes that exist in the three spatial dimensions and
that evolve in time, and particularly with shapes that show high symmetry.
As embryology, morphogenesis has been contemplated since Aristotle
conjectured about the homunculus within a sperm.
Today, geneticists tinker with embryogenesis processes at the nuts and bolts
level of genes.
Although their efforts have developed quite specific information about
individual pieces of this gigantic puzzle, very little
is known about its outlines.
Genetics tends not to consider questions concerning time and space, which
are obviously crucial to the understanding of the development (over time)
of form (in space).
Modern microscopy has yielded an explosion of information about the
high-symmetry forms of the sub-micron world.
One finds helices realized in DNA, in the thread-like filamentous
bacteriophages, and in coiled muscle and hair fibers.
The symmetry of the icosahedron, a Platonic solid, is realized in the
capsids (outermost protein shells) of the spherical viruses.
All of the other Platonic solids are realized in diatoms, several orders
of magnitude larger than the sub-micron structures.
Any putative solution to this set of morphogenesis problems must be able to
accommodate the errors and anomalies that arise both in Nature and in the
laboratory.
For example, virus capsids can mis-assemble in Nature into sheets or
cylinders, and one capsid protein has been induced to assemble in vitro
into octahedra. (See Salunke et al. in the
references.)
None of this is deeply understood.
Appealing to the eye, elegant form is easy to appreciate.
Undone by reductionist thinking, it is difficult to understand.
Upon examining closely any of the many interacting pieces of a complex thing,
we find that its form is always elsewhere, hidden among the pieces.
What has usually been missing when these questions are pondered is any
application of mathematics to the abstract question of form.
(The study of phyllotaxis, the development of form in the plant world, is
an interesting exception.)
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The mathematical study of morphogenesis is in its infancy.
It is possible to measure data concerning form: the positions, lengths, and
angles that arise in a developing structure, be it an embryo or a virus capsid.
Many practical obstacles intervene, and
overcoming them is the patient work of an army of unsung heroes, the
microscropists,
spectroscopists, molecular biologists, morphologists, and embryologists.
But, supposing that these workers had all of the numbers that
they seek -- what then?
The overarching relations among the parts of a form, what we wish to
understand, are implicit in the numbers, but not measured by
the numbers.
Relationship per se is not the subject of measurement.
Take as an example the five-fold symmetry evident upon slicing an apple
equatorially.
Individual apples display this symmetry imperfectly, and any strictly
numerical assay of symmetry would yield a small residuum of assymmetry.
Yet, even in a world of imperfect apples, we still know that
five-fold symmetry is in the nature of apples, and that our understanding
of apples is incomplete if it does not explain this feature.
What we wish to understand is a relation that lies among the apples, not
strictly within any one of them, except as a morphogenetic tendency.
In this connection,
consider D'Arcy Thompson's classic observations of the homologies between
skulls, pelvises, and entire body profiles of different species.
(See the references.)
Morphology measurements depend upon species, even vary among individuals,
but the relationships underlying the homologies do not.
The word "homology" is telling, as it is a word that appears in
mathematics as well as in biology.
It is our
belief that an understanding of morphogenesis, concerned with abstract
spatial relationships that develop in time, must be given in the language
of mathematics.
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So some preliminary observations about the connection between mathematics
and morphogenesis are in order.
Students of morphogenesis self-assemble under three flags:
1) The historically dominant viewpoint posits an organismic "holism,"
some organizing
center or principle that determines the creation of form by fiat.
This view comes naturally to the biologist.
Whether he or she has studied whole organisms or their parts, biologists
can't help noting apparent purposefulness ("teleology"), structures
and systems poised to make something useful happen.
(An example with morphological relevance is the structural protein,
exquisitely designed, in this view,
to compel it to assemble with others into one, specific final form that
must somehow be encoded within its structure.)
And whatever benign agency is ultimately responsible for this arrangement,
that is the organizing principle.
DNA is the modern exemplar of such an organizing center, a view found
in sneaky phrases such as "genetic program," that likens the actions of
genes to the sequential, rigorously deterministic unfolding of a computer
program.
The genetic
determinists (to a first approximation, all geneticists) belong in this camp.
This group has two big problems.
First, it is unable actually to explain how any given shape is made, even
less why the perturbations experienced by all living things during development
are not inevitably fatal.
Second, it has to avoid the question of infinite regress: whence came the
UR-principle that organized the
organizing principle that organized the organizing principle that organized...
The organismic holism viewpoint resists mathematization, and this is not
held against it because one is not
intended to fathom the workings of an organizing principle.
Like an axiom, an organizing principle is a monad that can only be posited,
but not dissected.
But, unlike an axiom, it is posited to operate at such a high level of
organization that it cannot be used to generate theorems.
As undoubtedly useful as post double-helix genetics is, it has not
engendered any theorems, nor has anybody expected it to do so -- unlike,
say, Newtonian mechanics.
2) A second position attends the self-organizing properties of the
constituents of whatever system is being studied, considering
observed forms to be "emergent phenomena" that arise spontaneously from
the mutual interactions of the parts of the system.
This viewpoint lends itself to mathematical treatment in terms of dynamic
systems theory.
It comes naturally to the physical scientist with a
mathematical viewpoint, and it has built-in the sort of teleology and
resistance to perturbation found
in dissipative dynamic systems that evolve to some attractor.
However, mathematical treatments of biological systems draw the criticism
that they deracinate their subject,
typically abstracting the sub-system under study from its biological
environment and treating the rest of the organism as only a static boundary
condition.
In the worst misapplications of this approach, the biology is brutally
deformed to fit into a mathematical straightjacket, and the resulting
product does not look very biological.
Alternately, the entire organism and its environment (sometimes including
even the earth itself) are supposed to form
a single gigantic dynamic system containing multiple levels in a vast
organizational hierarchy.
This is a compelling big thought, but
it is too big to be of practical use -- certainly not for the biologist,
and perhaps not even for Nature.
Paradoxically, this approach denatures its subject -- Nature -- because it
also leads to a biology without organisms.
Where does one begin to understand something, even in principle, that
has no truly distinct pieces?
And, when Nature re-deploys a solution to an old problems in a new context,
doesn't this imply not only that discrete sub-systems exist, but that they
can be lifted from here and moved to there?
3) A third position recognizes that adherents of the other two
can point to situations where their analysis is useful and powerful.
(For example, geneticists can change a fruit fly leg into an antenna -- maybe
more powerful than useful --
and students of phyllotaxis, the arrangements of leaves on the stalk of
a plant, have long employed an elegant mathematical analysis that leads
dynamically to the golden mean.)
This position proposes a straightforward resolution of the dichotomy between
organizing principle (eg., genes) and deracinated or denatured
dynamic system.
In this view, a morphogenesis process can be treated as a dynamic system
(some "Rule of Change") that is subject to a set of geometrical
boundary conditions ("Informing Geometry")
that constrain its development, and that contains some sort of
dissipation (means of "Forgetting" previous forms).
The constraints act somewhat as a local
organizing principle, but
the informing geometry constraints do not in themselves encode a form.
The form develops spontaneously via the rule of change, and the same
rule of change can yield different forms under different informing
geometry constraints.
Genes play a central role in setting up an informing geometry, via
their gene products, so this viewpoint does not discard genetics.
The need for dissipation (or forgetting) in biological morphogenesis is
obvious.
Mathematically speaking,
only a dissipative system can evolve to an attractor that is stable to
perturbations incurred during development.
Physically speaking, it is necessary to shed, or overgrow, or subsume --
somehow undo and surpass-- earlier forms before later ones can
be realized.
This three-fold view is expounded in the vertebrate lens article listed
in the
references ,
and examples are given in the applications.
(The lens morphogenesis model
has an interesting, typically biological, way of forgetting its early forms.)
We believe that Nature tends to dissolve oppositions, even as it continues
to create them.
In this, Nature takes the third position, and we try to side with Nature.
The merit of this viewpoint can be seen in the models that it generates.
Although they are presented in the language of mathematics, they allow
for the action of genes in establishing the informing geometries.
Also, to the eye, they look biological.
Common Themes
The models presented here and in the references
span an unusual range (length scales alone vary
from several hundred angstroms in the Capsids, to millimeters for the
vertebrate lens, to centimeters for the lattices of phyllotaxis).
But we are concerned here with their similarities, not their differences,
because the similarities speak to the nature of morphogenesis processes, as
they can be treated abstractly, via mathematics.
A discussion of the insights to be had from the appearance of common themes
will be found in the article "Phyllotaxis: a Bridge to Other Disciplines"
(see the references ).
Each model is a three-part dynamic system, as mentioned above.
Such systems can evolve in ways that look like biological morphogenesis, and
dynamic metaphors for embryogenesis go back to Waddington
(see the references ).
He spoke of an "epigenetic landscape," a kind of morphological space
through which a developing embryo moves.
A position in the morphological space represents a developmental
configuration of the embryo.
The embryo moves through this space like a marble rolling
through a hilly terrain, avoiding the hills and heading for the
valleys.
The operative idea is that some configurations seem to "attract" the
embryo, so that even if it is perturbed from its developmental path by
some accident (a nutritional lack, chemical insult, or mechanical trauma),
it tends to return to its proper path.
Too large a perturbation from the proper path may result in the embryo's
cresting a hill and heading toward the wrong valley, with unfortunate results.
Waddington offered this picture based on his knowledge of embryology,
reasoning that this is the way it must be, without any mathematics.
The vertebrate lens model presents a mathematical
rendering of such a morphological space, complete with an attracting valley
that we call the "morphogenetic groove."
In the lens models, each position on the morphogenetic groove represents one
member in a family of special lenticular shapes called "scaling solutions."
The scaling solutions have the property that, in an appropriate limit, their
time evolution entails exponential growth without change in shape.
In addition to corroborating the Waddingtonian viewpoint,
we can develop this picture further.
Through studying the lens model,
it has become clear that the timescale for movement to the
attracting grove is much shorter than that for movement along the groove.
This means that the system accesses the groove quickly, and that perturbations
from the groove are damped quickly, compared to the stately movement along
the groove, which represents the normal development process.
If we can generalize this example to the developing embryo, we find:
It is the developmental trajectory that is the stable
attractor,
not any particular configuration of the developing system.
The bulk of our work concerns the morphogenesis of high symmetry, particularly
the icosahedral symmetry observed in the capsids of spherical viruses.
(The capsid is the protein or mostly-protein shell that encapsulates the
viral genome. See the capsid morphogenesis pages).
Helical symmetry is widely observed in the molecular biology realm: DNA,
microtubules, the alpha-helix, 310
helix, muscle fibers, etc.
It is also observed macroscopically, in the positions of the leaves of some
plants, a phenomenon treated in the classical botanical subject of phyllotaxis.
We have studied the morphogenesis of high symmetry structures abstractly
by considering systems composed of generalized morphological units (MU's)
that interact pairwise.
(These can be leaves, the capsomeres that cover a viruse capsid, the
amino acids that make up a protein, etc.)
These studies are not intended to be models of actual systems, but of
the processes that engender their symmetry.
When more detailed models of specific systems are made, such as those that
model leaf placement in phyllotaxis studies
(see the references),
they implicitly (will) include such processes.
If the MU's are confined to a closed surface, such as a sphere, then they
can be made to repel one another.
If the surface is open, or if they can move freely in space, then the MU's
are made to repel at small distances and to attract at large distances,
having a roughly Van der Walls shaped force curve.
By summing up all of the pairwise interactions, a global interaction
energy is calculated.
A model is generated by applying the simplest possible dynamic rule of
change, gradient descent:
vary each coordinate in an amount proportional to the gradient of the energy
with respect to that coordinate.
Heuristically, this means: go downhill.
Gradient dynamics arise when frictional damping dominates the equations
of motion, so the mechanism of forgetting is common frictional dissipation.
This dynamic rule naturally causes the energy to decrease, and the system
evolves until it reaches a local energy minimum.
We can make three general observations about the behavior of such systems:
1) A high symmetry configuration is a local minimum of the interaction
energy.
So perturbations away from a high symmetry local minimum decay, and the
system returns to the high symmetry minimum.
2) The high symmetry configuration itself is an attractor.
We are not merely asserting in this that a local minimum is stable.
Consider an icosahedrally symmetrical capsid structure in a local
minimum configuration.
Perturb it away from the local minimum, while
retaining its icosahedral symmetry, and then further perturb it away from
its icosahedral symmetry.
When allowed to follow the gradient descent dynamic,
this system will evolve quickly to restore its
icosahedral symmetry, then more slowly return to the original local minimum.
We meet again with the two timescales mentioned above, in the context of
the vertebrate lens model.
3) Let us loosely define a "warping" as any set of changes in
the coordinates and
orientations of the morphological units that causes only small changes
in the relations among neighboring morphological units.
This definition is intended to encompass common situations in which
the small local changes accrue as one passes over a structure, so that
they amount to large absolute changes between distant parts of the structure.
If the interaction between MUs falls off quickly with distance, then
a warping causes only a small change in the net interaction energy,
because only neighboring MUs contribute strongly to the interaction
energy, and their interactions are changed only slightly by the warping.
As a result of this, a structure that is a warping of a local minimum
returns slowly to the local minimum when allowed to evolve.
Warping a high symmetry structure may significantly alter its
appearance, but dynamically the warping acts like only a small
perturbation.
By comparing the evolution of a set of abstract MUs to the evolution
of the lens model, we find differences as well as surprising similarities.
For both sorts of systems, the development process entails finding (rapidly)
and then moving (slowly) along a developmental trajectory that contains
high symmetry structures.
For the lens, the high symmetry
structures are the scaling solutions, whereas for the systems of
MUs (eg., viruses), they are structures bearing the conventional
symmetries, such as the Platonic symmetries, helical symmetries, and others.
The systems are different in that the lens model does not sustain any version
of warping, because warping entails interactions that fall off quickly with
distance, whereas the lens dynamic involves the shape of the lens, which
is a global property.
| Morphogenesis is deep.
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There is much to ponder.
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We are very pleased to acknowledge the unfailing hospitality of Rama and
Mukti Sarma.
