Monkeys ape molecules

An online exhibition by N. Michael Green, Division of Mathematical Biology

Receptor specificity determines cell tropism of influenza viruses in differentiated cultures of human airway epithelium.'Pick up Monkeys' was originally produced as a children's game (1965) and they have proved very versatile. I discovered in 1968 that they were ideal models for protein subunits, being asymmetric, having multiple interaction sites and available in several colours. This exhibition illustrates their use in modelling the geometry of multi–subunit protein structures.

Every living organism produces a variety of complex structures by self assembly of identical building blocks of one or more types. These structural units are usually protein molecules, which have evolved to assemble spontaneously, using multiple weak bonds. The principles of thermodynamics ensure that the most stable links are used. The number of such links is maximised in symmetrical structures, in which all the units are identically bonded. Natural selection ensures that assemblies with useful biological properties are perpetuated (e.g. virus coats, muscle fibres, mitotic spindles, intercellular junctions and many enzymes). The geometric principles involved also apply in architecture, tiling patterns and fabric design. They may be explored ‘hands on’ using any asymmetric building unit.

Plastic monkeys (available from Hasbro USA, as ‘Barrel of Monkeys’) are particularly convenient since there are over eighty different ways of linking a pair of monkeys (Nature, 210, 413, 1968). If one or two of these links are chosen and repeated systematically, large structures are generated as shown here. Any pair can be used as a nucleus for further growth.

If the first link is asymmetricassymetric
asymmetric
(A) it joins a site X on the first monkey and site Y on the second (1). The link can be repeated, using the Y site of the first monkey and the X site of the second monkey to generate an extended helixhelix
helix
(3). If the two sites are joined by a symmetric link (S) then a pair with twofold rotational symmetrysymmetry
symmetry
(2) is produced and this cannot grow further without introducing a new link, which in turn maybe either symmetric or asymmetric. (These comments are best understood by manipulating a few monkeys.)

If the pitch of the helix is zero, the helix will short circuit after one turn to form a ringsymmetry
ring
(4), the size of which is determined by the angle between successive monkeys. A simple way to generate such rings is based on similarity between the back of the monkey and its front – the outlines are essentially mirror images. It follows that a pair made with hand contacts running from back to frontassymetric_A
back to front (A)
(A) will be a mirror image of a similar pair on which the hand goes from front to backassymetric_A
front to back (A')
(A’) and that alteration of A and A’ pairs will generate a flat ring. The top and bottom surfaces of such a ring will differ but their outlines will be the same.

A symmetric pair cannot grow further without using a new link to join them. If this link is also symmetric an antiparallel double helixantiparallel double helix
antiparallel double helix
is formed. This will again form a ring if the pitch is zero, but its two surfaces will now be the sametetramer
Tetramer
(5), additional exampleshexamer
Hexamer
(6). If the new link is asymmetrical, a parallel double helixparallel double helix
parallel double helix
will be formed (9).

More complex structures can be assembled from rings by joining them together with new symmetric links, giving a linear rod-likerod-like
linear rod-like
(7), additional linear examplelinear
additional linear example
(8), planarplanar_10
planar
(10) or polyhedral structuresdodecahedral
dodecahedral
(11).

protein molecule Any asymmetric unit such as a protein molecule (an example is shown in the image), with an affinity for others of is kind can generate structures such as these, limited only by the principles of thermodynamics (stability). The forces between globular protein molecules are weak and the stability of complex assemblies depends on achieving the maximum number of stable links. This enforces a symmetrical structure. Many examples of this can be seen in natural protein structures, such as multi-subunit enzymes, muscle fibres, bacterial flagella, virus coats mitotic spindles (microtubules) and specialised cell membranes. The monkey provides a large scale unit for simulating the geometrical relationships within such structures as well as illuminating some general principles of symmetry.

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