HELICAL TENSEGRITY AS A STRUCTURAL MECHANISM IN HUMAN ANATOMY
International Journal of Osteopathic Medicine 2011;14:24-32.
Graham Scarr
ABSTRACT
Tensegrity is a structural system popularly recognised for its
distinct compression elements that appear to float within a tensioned
network. It is an attractive proposition in living organisms because
such structures maintain their energy-efficient configuration even
during changes in shape. Previous research has detailed the cellular
cytoskeleton in terms of tensegrity, being a semi-autonomous system
amenable to such analysis because of its size. It has also been
described at higher levels in the extracellular/fascial matrix and
musculoskeletal system, but there are fewer syntheses of this.
At a fundamental level, the helix and tensegrity share common origins
in the geometries of the platonic solids, with inherent hierarchical
potential that is typical of biological structures. The helix provides
an energy-efficient solution to close-packing in molecular biology, a
common motif in protein construction, and a readily observable pattern
at many size levels throughout the body. The helix and tensegrity are
described in a variety of anatomical structures, suggesting their
importance to structural biology and manual therapy.
1. INTRODUCTION
The world of biology is full of weird and wonderful shapes, some with
no obvious purpose, and others that suggest some hidden meaning. Even
human anatomy has its fair share of the bizarre in the shapes of bones
and limbs. How and why does each one develop its characteristic form,
and how does that relate to function? Is there more to shape than
genetics and Wolffs’ Law?
Three
thousand years ago, the Greeks believed that just five archetypal forms
could describe everything in the universe, because they were pure and
perfect, and part of natural law. Recent research reinstates these
physical laws as a major determinant of biological complexity in the
sub-cellular realms, and significant to structures at higher scales.
1-4
Tensegrity (tension-integrity) is a structural mechanism that
potentially integrates anatomy from the molecular level to the entire
body, and is popularly recognised for its distinct compression elements
that appear to float within a tensioned network. It is a most attractive
proposition in living systems, because such structures automatically
assume a position of stable equilibrium, with a configuration that
minimizes their stored elastic energy. Tensegrity structures allow
movement, with the minimum of energy expenditure, without losing
stiffness or stability.
1,5-7
This contrasts with the orthodox view that explains the
musculo-skeletal system through classical Newtonian mechanics, using
pillars, arches and fixed-fulcrum levers to counteract the force of
gravity. In this approach, bones stack on top of one another like a pile
of bricks, restrained by soft tissues that permit movement in a local
piece-meal like way.
8 Comparisons of tensegrity and
biological structures show them both to have non-linear visco-elastic
properties, with fluid-like movements that result from integration of
all components in the system.
1,5,6,9
The molecular helix provides an energy-efficient solution to
close-packing in biology and also displays tensegrity properties. It is a
common motif in protein construction, and a readily observable pattern
at many size levels throughout the body. It is proposed that helical
tensegrity is a key mechanism in structural biology and consequently has
significance for manual therapies.
2. THE HELIX
The
helix is like a coiled spring, or put mathematically, “A spiral curve
lying on a cone or cylinder, and cutting the generators at a constant
angle” (Walker, 1991).
10,11 In biology, it can be appreciated as a regular stacking of discrete components, such as the nucleotides and bases in DNA, or the steps in a spiral staircase.
Globular proteins, often containing multiple helical domains, can themselves polymerize into helixes (fig. 1a,b).
12 Similar helixes can wind around each other to form coiled-coils (Fig. 1c),
13
and assemble into mechanically rigid rods or filaments, or further
combine into more complex structures with specialized functions (fig.
2).
In collagen type I, repeating sequences of amino acids spontaneously
form a left-handed helix of procollagen, with three of these helixes
combining to form a right-handed helix of tropocollagen. Five
tropocollagen molecules then coil in a staggered helical array,
14 which
lengthens longitudinally by the addition of more tropocollagen to form a
microfibril, with higher arrangements forming fibrils, fibres and
fascicles.
15 Collagen appears at several different hierarchical levels within bones, tendons, ligaments and fascia (fig. 2).
2.1 Structural hierarchies
Hierarchies
link structures at multiple levels and are widespread in living
organisms. They provide an efficient mechanism for packing in 3-D
16 by
using components that are made from smaller components, with each made
from smaller still, often repeating in a fractal-like manner (fig. 2).
1,5,17
Hierarchies enable mechanical forces to be transferred down to a
smaller scale with the dissipation of potentially damaging stresses.
18-21At
atomic and molecular levels, the basic forces of attraction and
repulsion automatically balance those stresses in the most energetically
efficient configuration.
12,22-24
2.2 Helical tubes
The tubular nature of the helix scales up into blood vessels,
25 the urinary system and intestinal tract.
26,27
Carey (1920) observed left and right-handed helical patterns in the
epithelium during formation of the oesophagus and trachea, respectively,
in the early embryo.
28 In the walls of elastic arteries,
such as the aorta, helical collagen reinforcement resists high loads
from the pressure of blood. The middle layer organizes into lamellar
units, with the orientation of collagen fibres and smooth muscle cells
forming a continuous helix. Collagen is more dispersed in the outer
adventitia, but still forms two helical groups of fibres.
25
Within the spine, the intervertebral disc contains collagen arranged
in concentric lamellae, with opposing orientations in alternate helical
layers of 65
o (axial).
29 The inner lamellae of the
annulus fibrosus consist of collagen type II fibres, cross-linked to
type IX on the fibre surface, within a highly hydrated proteoglycan
matrix; gradually changing to collagen type I fibres in the outer
lamellae.
30,31 The higher proteoglycan/water content in the
inner lamellae acts as a thick-walled pressure vessel containing the
nucleus pulposus, while the higher concentration of collagen type I in
the outer lamellae provides tensile reinforcement during bending and
torsion.
29,32
Pressurized
tubes cause circumferential and longitudinal stresses in the tube wall
that are typically contained by collagen under tension within a helix.
Clarke and Cowey (1958) showed that an optimum fibre angle of ~55
o
(axial) balances both these stresses, with a reduced angle resisting
tube elongation, and a higher angle resisting circumferential and volume
increases.
33,34 Such helical fibre arrays allow pressurized tubes to bend smoothly without kinking, and resist torsional deformation;
32 collagen has itself been described as a tube.
35
Cardiac muscle fibre orientation varies linearly between inner and outer walls, from 55
o (axial) in one direction to 55
o in the opposite, with tangential spiralling in a transverse plane.
36
The entire heart has also been described as a helical coil of muscle
with contractions that cause clockwise and anti-clockwise twisting
motions.
37 This typically produces a left ventricular ejection fraction of 60%, for a muscular contraction of just 15%,
38 confirming the mechanical efficiency of a helix.
2.3 Tubes within tubes
Traditionally considered as mere packing tissue, fascia has been shown
to exert considerable influence over muscle generated force
transmission.
39-42 It naturally develops into compartments,
or ‘tubes within tubes’, particularly noticeable in cross-sections of
the limbs. Within muscle, a delicate network of endomysium surrounds
individual muscle fibres and is continuous with the perimysium
ensheathing groups of fibres in parallel bundles, or fasciculi.
Perimysial septa are themselves inward extensions of the epimysium that
covers the muscle and is continuous with the fascia investing whole
muscle groups. All these sheaths (tubes) coalesce and transmit the force
generated within muscle fibres through tendons and inter/extra-muscular
fascial attachments.
39,42 These fascial tissues are all reinforced by two helical crossed-ply sets of collagen,
36 with the ‘ideal’ resting fibre orientation of 55
o (axial)
33 that varies with changing muscle length.
Tubular
organs that maintain constant volume throughout changes in shape, due
to crossed-helical arrangements of muscle and fascial tissue, have been
described in the tongues of mammals and lizards, the arms and tentacles
of cephalopods, and the trunks of elephants.
43 Helical winding and its functional significance have also been described in the body walls of worms;
33 squid;
44 amphibians;
45 eels;
46 fish and dolphins;
47
suggesting that a similar helical arrangement is likely to occur
throughout the human. However, although the thoraco-lumbar and abdominal
muscle/fasciae appear to be partial spirals, information on the fibre
orientation of other fascial compartments is incomplete.
Stecco (2004) described helical fascial sheaths that transfer
tensional forces within and between themselves, and control movement in a
way that the nervous system is incapable of.
48 Anecdotally, palpatory phenomena with a helical component are observed within the soft tissues of the extremities.
49
A normal pattern exhibits right-handed helical motion in the limbs on
the left side, and left-handed helical motion on the right, although
current anatomical knowledge is unable to explain this.
The helix has long been recognized in joint motion,
8 and
its widespread appearance at multiple size-scales throughout the body
suggests that it has some special significance. At a fundamental level,
the helix and tensegrity are linked through a common origin in the
geometries of the platonic solids.
1,4,50
SIMPLE GEOMETRY
3.1 The platonic solids, geodesic geometry and close-packing
The
platonic solids are regular polyhedra distinguished by having faces
that are all the same shape, and naturally form through the efficiencies
of geodesic geometry (the connection of points over the shortest path)
and principles of symmetry.
1,4,50 In two-dimensions, objects
of similar size close-pack and form stable triangular configurations
(fig. 3a). Adding another sphere to each triangle creates a tetrahedron,
and the addition of more spheres allows the octahedron and cube to
emerge (fig. 3b-c), because of the same packing arrangement. These
platonic shapes are generally only found as fixed inorganic crystals,
but there are many consequences of close-packing.
The icosahedron differs from the other platonic shapes by packing spheres around a nuclear
space to form the geodesic dome (Fig. 3d).
50
It is also triangulated and has multiple symmetries which allow it to
stack in a column or helix and form more complex patterns and shapes.
1,2 Some naturally occurring structures based on the icosahedron are carbon fullerenes; pollen grains and ‘spherical’ viruses.
22-24
Both the tetrahedron and icosahedron spontaneously form through the
interactions of natural physical forces, and are the basis for
appreciating complex shapes in human anatomy.
2,4,51
3.2 Chirality and Equivalence
The
property of chirality is intrinsic to the helix, and the platonic
solids demonstrate this as they polymerize into left and right-handed
helixes (fig. 4).
51-54 At a basic level, four spheres
close-pack to form a tetrahedron, the shape that occupies the smallest
proportion of unit space; minimum volume within maximum surface area.
50
The addition of more spheres as in the lattice packing of figures 3b
& 3c, alters that proportion because of the squares within the
octahedron, but a tetrahelix comes closer to the optimum, making it a
more suitable model for molecular packing because of this margin of
energy-efficiency (fig. 4a).
51,53,54 A tetrahelix also displays inherent hierarchy within its sub-helixes of different pitch (fig. 5).
Mapping
a tetrahelix onto a plane surface, by ‘unzipping’ one of its long
helical edges, displays the packing efficiency of a triangular pattern
(fig. 3a). Rolling that map into a cylinder demonstrates equivalence,
where each component is in the same relative position to all the others.
53,54
Equi-valence implies that components are arranged symmetrically, and
the only shapes that can accommodate it have surfaces based on the
platonic solids and cylinders.
22-24,55,56 Because molecules
in a peptide sequence are unlikely to match the points on a geometric
lattice precisely, evolution has evaded this constraint through the
device of ‘quasi-equivalence’, where component proteins contort slightly
but still relate to the geometric template.
1,23,24,53
Tropocollagen (fig. 2) has been described as three stretched quasi-tetrahelixes surrounding a central core.
53,54
Each glycine residue, from the three procollagen peptides, contributes a
hydrogen atom that forms the corner of a regular tetrahedron, and
together they form the right-handed tetrahelical core of the
tropocollagen molecule. The left-handed procollagens are the sub-helixes
shown in figure 5b; and this configuration also gives rise to a stack
of slightly contorted icosahedra.
53,54 Most (if not all) molecular helixes are geometrically related to the tetrahelix and icosahedron,
12,22,53,54,56 including the alpha-helix of DNA, which has been described as a [triple stranded] tetrahelix with one strand missing.
53
Molecules automatically assume a state of minimal-energy as they
balance the attraction and repulsion of their constituent atoms. As the
helix is a more efficient close-packing configuration it is
understandable that it should be such a common structural shape.
At
a larger scale, the bacterial cell wall contains actin homologues
arranged as a structural helix determining cell shape and elongation.
57,58 Plants display similar configurations in their cell walls
59 and geometric patterns at a higher level.
3.3 Fibonacci and the Golden Mean
The
number of elements within each opposing spiral is nearly always two
consecutive numbers of the Fibonacci sequence, where each new term is
the sum of the two preceding ones (1,1,2,3,5,8,13,21,34…). The ratio of
any two consecutive numbers approximates to the Golden Mean (1.61804),
and becomes closer as the sequence gets higher. The helical pattern on
the side of a pineapple, arrangement of branches on a plant stem
61 and position of coronary artery lesions
62 relate to the same sequence. The Golden Mean often appears in the proportions of biological structures and platonic solids,
63 including the icosahedron, which is the model that takes us into the tensegrity of macro-anatomy.
50
4 TENSEGRITY
Descriptions of tensegrity in biology have appeared in the literature since the early 1980’s,
64,65 and include the cellular cytoskeleton;
5 developing neurites
66 and cerebral cortex;
67 spider silk
6,68 and wasp arcus;
69 mammalian
70-72 and avian lung;
73 fascial matrix;
74-76 shoulder;
75 spine;
51 pelvis
77 and cranium.
78
Fuller (1975) described a tensegrity structure as a set of struts
under compression, and an arrangement of cables under isometric tension,
that always balances in the most energetically efficient configuration.
50
It is geodesic by its very nature, because tension always acts in
straight lines, and automatically reduces itself to a minimum.
Tensegrity structures make possible an infinite variety of stable shapes
through changes in the lengths of their compression members, and
changes in those shapes that require very little control energy. As each
component influences all the others, stresses distribute throughout the
system, creating a structure that can react to external forces from any
direction without collapsing.
6,7,51 An organism utilizing
such a system would be able to move with the minimum of energy
expenditure without losing stiffness or stability.
6,7,51
Because tension and compressional forces are separated, the material
properties of components can be optimized, and in biological systems
this typically occurs through hierarchies. Tensegrity hierarchies
achieve a significant reduction in mass,
6,7 and provide a
functional connection at every level, from the simplest to the most
complex, with the entire system acting as a unit.
5,51,76
4.1 The tensegrity helix
The
icosahedron is a fundamental geometric shape because it encloses a
greater volume, within minimum surface area, than any regular structure
apart from a sphere (fig. 6a). It is developed into a tensegrity
structure by using six compression struts to traverse the inside (fig.
6b). These connect and hold opposite vertices apart with the outer edges
of the icosahedron now replaced by cables under tension. The resultant
pull of the cables is balanced by the struts, which remain distinct from
each other and do not touch. They provide structural integrity so that
the compression elements float within the tension network.
50,79
Considering the six struts in different groups of three, joined on the
surface by ‘tension triangles’ (fig. 6c), shows that each strut within
the group is oriented at 90
o to the others, and together they
create a chiral twist. On the other side of the structure is a similar
group with a twist in the opposite direction, which means that a
tensegrity icosahedron already contains helical precursors of both
chiralities.
When three struts are modelled on their own (Fig.7), they form a shape called a tensegrity prism.
6,7
Increasing the number of struts causes their centres to position more
towards the outside of the structure, enlarging the central space and
eventually forming a cylindrical ‘wall’ due to the changing orientation
(fig. 7b-d). The struts are equivalent, and all form part of an infinite
series of left or right-handed helixes; the model in figure 8
demonstrates their tubular nature. Each strut could be made from a
smaller helix, or the whole structure become part of a strut within a
larger helix ie it has hierarchical capability. Helical molecules are at
the ‘lower’ end of structural hierarchies that fill the entire body,
but have physical properties that continue into those higher levels.
Helical tensegrity is a structural mechanism with many properties useful
to organic life.
5 THE HELICAL-TENSEGRITY BODY
Helical
molecules behave as tensegrity structures in their own right, as they
stabilize through a balance between the forces of attraction (tension)
and repulsion (compression).
79,80They readily combine into more complex structures that retain some of the same properties.
2,12
The cellular cytoskeleton is described as a multi-functional
tensegrity structure that influences cell shape, and activates multiple
intra-cellular signalling pathways.
5 Helical microfilaments
of actin and microtubules of tubulin are the tension and compression
elements, respectively (fig. 1a,b); while spectrin fibres and actin
bundles may have similar roles within the cell cortex (Figs. 1c).
81,82Tensioned intermediate filaments link everything together, from the nucleus to the cell membrane.
83
Tension is generated through the action of actomyosin motors and
polymerization of microtubules, and any change in force at one part of
the structure causes the cytoskeleton to alter overall cell shape.
5
Many enzymes and substrates are situated on the cytoskeletal lattice,
and changes in its configuration alter their activity, leading to a
switch between different functional states such as growth,
differentiation or apoptosis.
5
The cytoskeleton connects to the matrix and other cells through
transmembrane proteins, such as integrins and cadherins, respectively.
These create a mechanical coupling that transfers tension, generated
within the cytoskeleton, to the matrix and adjacent cells. A prestressed
state of isometric tension thus exists between them, so that a change
in matrix tension causes a realignment of structures within the
cytoplasm, and a change in cell function. This reciprocal transfer of
mechanical forces is likely to orchestrate cellular growth and
expansion, allowing the emergence of complex multi-cellular tissue
patterns, based on the same principles.
5,84,85
5.1 Helical tubes
The
formation of capillaries results from tension-dependent interactions
between endothelial cells and an extra-cellular scaffold of their own
construction, and is described through tensegrity.
86 The growing matrix causes changes in the configuration of cytoskeletal components,
5 and initiates chemical signalling cascades that influence further development of the capillary network.
87
The capacity for fluid flow through a tube depends, in part, on the
porosity of the tube wall. The helical tensegrity ‘wall’ in figure 8 has
many gaps, but if the struts were expanded into plates that just
touched each other, they could be made to ‘seal’ the internal space.
This compares with the selective barrier of endothelial cells that
allows vascular contents to pass out between capillary walls. The
internal cellular cytoskeleton determines cell shape and orientation,
through tensegrity;
5 is affected by signalling mechanisms and
variations in fluid flow; and alters the tension between cells through
adherens junctions,
88 ultimately affecting tube permeability.
89,90
In tensegrity terms, there is no specific need for a compressional
element within the tube wall if this is provided by outward pushing
radial pressure, although arterial walls are pre-stressed even when load
free. It is likely that wall components under tension are linked to
other structures under compression at different hierarchical levels;
Fuller (1975) emphasized that tension and compression must always
coexist.
50 Collagen type I fibrils are the predominant tensors, and are virtually inextensible under tension (<5%);
30
but the mechanical properties of more than twenty other types are
poorly understood. Proteoglycans and glycosaminoglycans tend to increase
in tissues under compression. Combining these and other components into
tissue specific matrices contributes to huge histological variation.
Confirmation that they are tensegrity configurations, however, will
depend on analysis of their physical interactions.
A
fundamental principle of tensegrity is that the forces of tension and
compression are separated into different components, and always act in
straight lines; which means that there are no shear stresses or bending
moments. The model in figure 8 shows curved struts that seem contrary to
this, but they can be understood in terms of hierarchies.
Curved
struts only remain stable if their crystal/molecular structures are
strong enough to resist the potentially damaging shear stresses that
lead to buckling; or they are part of a tensegrity hierarchy that
eliminates those stresses by its very nature. Curves may appear at one
level within a tensegrity hierarchy, but when looked at in more
detail, have structural components that handle tension and compression
in straight lines.
Undoubtedly, the fibre angle within any particular tissue depends on
the functional context. The model in figure 8 shows struts arranged in a
self-similar array and tension cables with differing orientations.
Previous descriptions of “random” collagen orientations
may have misinterpreted what were actually functionally ordered tensegrity alignments,
91 and the sensitivity of newer imaging techniques and their analysis may resolve this.
92,93
5.2 Helixes within helixes
Axial
compression of a tensegrity helix initiates rotation in a direction
dependent on the helical angle and strut orientation (chirality), with a
corresponding decrease in the central diameter. Axial extension causes
it to expand demonstrating a negative Poisson ratio; most man-made
materials reduce their width when stretched,
9,16 but this unusual response is common in biological structures.
1,51
Surrounding a helix with another one of opposite chirality increases
resistance to axial compression, as each helical layer counteracts the
rotation of the other; crossed helixes have been shown to alter tubular
properties.
33,34
The intervertebral disc contains collagen arranged in concentric
lamellae, with opposing orientations in alternate helical layers that
provide tensile reinforcement.
29 Whether this is a tensegrity
configuration is yet to be assessed; but the widespread view that discs
provide resistance to spinal compression as a prime function is
probably too simplistic, and the whole spine has been looked at from a
tensegrity perspective.
51 Although disc failure usually occurs in tension,
94 this is usually due to abnormal loading.
The negative Poisson ratio may also have relevance to the helical
dynamics of the heart and has been described with the tensegrity
‘jitterbug’ mechanism. When any two tension triangles of a tensegrity
‘icosahedron’ are pushed together or pulled apart (Fig. 6c), the entire
structure contracts and expands, respectively.
1,50,51,95
5.3 PUTTING THIS ALL TOGETHER
51,74,76 Helical ‘tubes within tubes’ mean that fascial compartments
of the trunk and limbs can be considered in the same way. Objections
that fascia is too flexible to contain compression struts can be
overcome by considering the diameter of muscle, and its increase during
contraction, as such struts. This would undoubtedly alter the tension
pattern of surrounding fascia, which has itself been shown to influence
the force appearing at tendons.
39,40 In a tensegrity sense,
fascia is the bodies main component of tension suspended between bones
under compression, with smaller compartments taking origin from larger
ones. Muscle fibres can then be considered as mere motors.
Helical and tensegrity structural systems complement each other, and
are based on the fundamental properties of the tetrahedron and
icosahedron. A chain of tensegrity icosahedra simply contains the
crossed-helical fibres of a tube. Putting all this together from a
helical-tensegrity perspective necessitates a reappraisal of structural
biology and manual therapeutic techniques in terms of fundamental
geometry.
6 CONCLUSION
The
observation of a geometric pattern doesn’t necessarily imply anything
meaningful, as Johannes Kepler (1571-1630) found out with his early
description of a platonic solar system. However, the simple tetrahedron,
octahedron, cube and hexagon are recognised in the structures of
inorganic crystals, a result of atomic close-packing and principles of
symmetry. (fig. 3b,c).
1,4,50 
Carbon fullerenes and viruses appear as icosahedra and are related to the geodesic geometry of a sphere(fig. 3d).
1,2,23 The hexagonal packing of muscle fibrils and cells occurs because of the same physical laws.
4,5,65
There are many possible consequences of close-packing, and the
tetrahelix as one of them provides a more energy-efficient solution in
molecular biology (fig. 5).
53,54
Molecules assemble spontaneously and automatically balance the
attraction and repulsion of their constituent atoms in a state of
minimal-energy.
24,79 The helix forms because of the same
‘platonic’ rules, those of organic chemistry and the dynamic nature of
biological systems. The tetrahelix and its geometry then describe the
helical hierarchies of protein structures and DNA.
Concurrent with the molecular helix is the principle of tensegrity.
Tension and compression (attraction and repulsion); geodesic geometry
and minimal-energy; and the inherent ability to form hierarchies are
characteristics of both these structures. At the cellular level, the
tensegrity principle describes the mechanical behaviour of the
cytoskeleton, being a semi-autonomous system amenable to such analysis
because of its size.
5 As a structural mechanism, tensegrity
depends on the integration of every part, and it has been proposed that
this includes the whole body from molecules, cells, extra-cellular and
fascial matrix to the entire musculo-skeletal system.
1,4,5,74-76
Although it has been described at higher levels of anatomy, detailed
multi-scale syntheses of its components are few. The helix, however, is a
readily observable pattern at many different levels and may be
inseparable from tensegrity, but there is a caveat.
If the structure of the human body is considered as a vast hierarchy
of interacting sub-tensegrities, structurally and functionally, the
examination of any part in isolation can be misleading, as it is
inevitably incomplete.
39-41 The possibilities for enquiry become virtually endless and make it unlikely that ‘bio-tensegrity’
51
could ever be proved. However, if it describes biological systems more
thoroughly, it is only a matter of time before this becomes the standard
approach to biomechanics.
Human anatomy and physiology have been described in terms of
tensegrity, and the volume of supporting evidence is steadily
increasing. The helix is a well-known structural motif in biology. The
fundamental links between tensegrity, the helix and platonic solids
support a comprehensive view of human anatomy that is best appreciated
as a complex interaction of natural physical forces.
