Making connections

In my last blog, I talked about layers and the importance of “digging deep” to find the past and future of a system. I didn’t really discuss how to dig or what one is digging for. An essential part of “digging” into a system is finding and following its internal connections.

Geodesic dome
Geodesic dome distributes force (red arrow) throughout structure with compression (red) and tension (blue).

Looking at systems through their interconnections helps us understand how changes in one part of the system affect other parts of the system. Natural biological systems are deeply complex and interconnected within each individual and in the social and ecological connections between individuals.  One principle I focus on for understanding such connections is known as tensegrity. This is a term coined by Buckminster Fuller combining the words tension + integrity, to refer to architectural structures which distribute force through balanced tensional and compressional elements. Geodesic tensegrity is illustrated in the geodesic dome, which Fuller was famous for popularizing.  These spherical structures are composed of smaller triangles which are themselves very strong and stable when force is applied. A geodesic shape interconnects smaller triangles so that they can mutually extend their ability to transfer force in a larger configuration, providing distributed resistance to force from multiple directions.

For hands-on experience, playing with toothpicks and balls of clay to form joints is instructive (or marshmallows for those with a sweet tooth). One quickly finds that triangular pyramids are not only easier to build and more stable than our usual cubes, but that you can support more weight if you test structures with a compressive force (like loading with a book).

Buckminsterfullerene molecule
Buckminsterfullerene. 60 carbon atoms each connected to three other atoms with two single and one double bond.

The efficiency of the geodesic structures appears in nature. Buckminsterfullerene is the name given to the spherical carbon molecule composed of 60 atoms with 20 hexagon faces and 12 pentagon faces in the shape of a soccer ball (more formally a truncated icosahedron). Not only is it naturally occurring on earth and in outer space (Zhang, Y., & Kwok, S. (2013). On the detections of C60 and derivatives in circumstellar environments. Earth, Planets
and Space, 65(10), 1069-1081.), but its structure makes it an extremely strong molecule that maintains shape with high temperatures and pressures. Doping with an alkali metal can give it superconducting properties, and films of “buckyballs” have been used to make organic solar panels (although they have a lower conversion efficiency than silicon based panels).

Carbon nanotube
Carbon nanotube also has each carbon connected to three other carbons with two single and one double bond, but forming a sheet of all hexagons. Image by Arnero [CC BY-SA 3.0], via Wikimedia Commons
The technology to analyze and create buckyballs is the basis of nanotechnology.  Carbon nanotubules are hexagonal lattices of carbon, shaped as tubes rather than spheres. Like buckminsterfullerene, these structures are not only strong (carbon nanotubules have the greatest tensile strength of any currently known substance), but they also possess unique thermal and electrical conduction properties.

Snelson's sculpture, Mozart I
Snelson’s sculpture, Mozart I, illustrates structure generated from tension distribution.

While use of geodesic shapes yields efficient and stable structures, we can find more dynamic and complex structures when the tensional elements are “prestressed” to create similar structural stability while allowing the structural nodes to float unanchored. Some beautiful examples of this include the sculptures of Kenneth Snelson which you can find on his website:

The use of tension in these structures to provide support can also be understood from examining the difference between the wagon wheel and the bicycle wheel. The hub of a wagon wheel rests on whichever one of its spokes is between it and the ground, compressing that spoke then the next in turn as the wheel rotates. It needs thicker and heavier spokes in order to support increases in weight. A bicycle wheel, on the other hand, uses evenly distributed tension from all the spokes to pull the rim into a circle that “floats” around the hub. Increased weight transmitted from the hub pulls down on the rim and tends to deform the whole structure, but this deformation is constrained by the spokes that all work together to hold the rim in a circle. Distribution of tensional force is more efficient as it uses all the spokes, allowing bicycle wheels to be lighter and use less materials, as well as being more efficient in load distribution.  Compression is distributed throughout the rim as the spokes pull it towards the hub.

tensegrity icosahedron
tensegrity icosahedron
Distorted tensegrity isosahedron
Application of force is transmitted to all points of the model

To get a better feel for prestressed tensegrity, you can create your own models with elastic bands providing tension with dowels providing compression.  Many models are described online, and one site I like is Levin’s, which has a series of videos on how to make them.  The tensegrity icosahedron is one of the most common with six compression struts and 24 tendons.

If you are having problems visualizing it from the photos (or just prefer manipulating virtual objects),  here is a 3D model  I made on which you should be able to rotate after clicking on the graph so the coordinates appear.

It is a symmetric and stable system with a mathematical analysis of its stability in the Wikipedia article on tensegrity.  This is the same model used by Ingber in his application of tensegrity principles to cellular structure (see below)  His model used an isosahedron as a model for a cell nucleus, which was placed inside a larger isosahedron for the cell itself.

The musculoskeletal system uses similar principles to maintain lightness and efficiency.  Muscles provide prestress with a constant level of muscle tone, while bones take compression.  The shoulder girdle and the pelvic girdle do not have to apply the full transmission of weight to their respective ball joints, but rather use all the supporting muscles, ligaments and tendons to create a floating system that distributes tension throughout the body. Although this is much more energy efficient, it also means that we have to look at more than local phenomena when examining injury and function. Other systems will come into play, providing counterbalancing forces to compensate for weaknesses or when pain creates a tendency to protect certain areas. Transmission of force from a foot strike while walking will not just affect directly supporting knee and hip, but the entire spine and skull and connected organs as they constantly stabilize themselves together in connected balanced tension. Conversely, energy for initiating action, such as swinging a club or sword, can come from more than just wrist and arm, but also from the hips, the connection of feet to the ground, and perhaps even the balance of tension and relaxation between .

More details about the shoulder or spine as tensegrity systems is available from Levin’s biotensegrity website:

Note that to model more complex systems, simple tensegrity models are often linked to other tensegrity models in hierarchical or modular fashion so that each component can optimize the balance of structure and tension in different ways.

These tensional interconnections exist not only between connected modular sections, but are also transmitted to systems at different scales. Cells also have skeletons, composed of 25 nanometer wide cylindrical microtubules, thinner, stretchable intermediate filaments, and very thin, 7 nm wide, actin microfilaments. External receptors on the cell surface can transmit motion and electrochemical information to the nucleus and various organelles through this network, which appears to use prestressed tension and tensegrity principles.

Donald Ingber has been a key researcher in the application of tensegrity to cellular systems.  Much of his work is summarized in his articles:

  • Ingber, D. E. (2003). Tensegrity I. Cell structure and hierarchical systems biology. Journal of cell science, 116(7), 1157-1173.
  • Ingber, D. E. (2003). Tensegrity II. How structural networks influence cellular information processing networks. Journal of cell science, 116(8), 1397-1408.

He also wrote an excellent scholarpedia article which reviews the history of the concept of tensegrity and its applications which I recommend reading.  If you are obsessive enough to want to delve into the debate on the applicability of tensegrity principles to cell systems, read Ingber, D. E., Heidemann, S. R., Lamoureux, P., & Buxbaum, R. E. (2000). Opposing views on tensegrity as a structural framework for understanding cell mechanics. Journal of Applied Physiology, 89(4), 1663-1678.  It is instructive in thinking about the importance of precise definitions (as well as how use of words affects communication).

compression on cytoskeleton of breast cancer cells
Changes in actin microfilaments and microtubules in breast cells when compressed increases likelihood that cancer cells (67NR) will migrate. From Janet, M. T., et. al. (2012). Mechanical compression drives cancer cells toward invasive phenotype. PNAS, 109(3), 911-916.

Balanced internal forces enable the cell to rapidly adapt to external information, shifting shape and producing proteins for growth or for rest and recuperation.  To reproduce and divide, cells have to change their internal tensional elements, move different internal organelles, and trigger biochemical cascades for duplicating genetic information.  Uncontrolled growth as seen in the cancerous transformation of cells, particularly metastatic cancer, appears to involve disruptive changes to cytocellular tensegrity networks, reducing the normal internal tension through loss of microtubules and actin microfilaments. The ability to grow in unusual shapes or to break their connection to other cells is paralleled by changes to the external integrin receptors that bind cells to the extracellular matrix. Cancer cells trigger coordinated shifts in nearby supporting stromal cells (which usually maintain tissue structure) as well as release of factors to increase the number of nearby blood vessels (angiogenesis) to support accelerating growth. Because of the interlocking systems with multiple layers of connections that normal tissues maintain, transformative change such as cancer requires concerted impact at multiple levels and timescales.

Cancer is an important model for thinking about the balance of growth and death, and what is necessary for making balanced systems move out of balance.  A description of the importance of balancing physical forces in cellular tissues and the implications in cancer is found in Butcher, D. T., Alliston, T., & Weaver, V. M. (2009). A tense situation: forcing tumour progression. Nature Reviews Cancer, 9(2), 108-122.  This article does get quite detailed into the molecular biochemistry, but it you are really interested in this area, there is an excellent detailed review of cancer factors in Gupta, G. P., & Massagué, J. (2006). Cancer metastasis: building a framework. Cell, 127(4), 679-695.  For a more general view (with fewer details) that offers a broad perspective on all the ideas in this blog, read Saetzler, K., Sonnenschein, C., & Soto, A. M. (2011, June). Systems biology beyond networks: generating order from disorder through self-organization. In Seminars in cancer biology (Vol. 21, No. 3, pp. 165-174). Academic Press.

The growth of neurons is also guided by principles of balanced force. Physically, distance between neurons is important as greater separation increases the time and energy required for communication. Although high-speed myelinated fibers can have 50 times higher transmission speeds than unmyelinated ones, the extra layer of insulation to achieve this has costs in terms of physical space and metabolic support. By folding in three dimensions, the growing brain can bring distant neurons closer together, creating easier cross connections across separate processing areas as well as the physical gyri for which it is famous.

An excellent review of tensegrity, networks and the brain can be found in the book Buzsaki, G. (2006). Rhythms of the Brain. Oxford University Press. Cycle 2.  The article most associated with tension and folding of the brain is Van Essen, D. C. (1997). A tension-based theory of morphogenesis and compact wiring in the central nervous system. Nature, 385(6614), 313.

Space and energy requirements dictate limitations on biological systems, and by constraining the topology of neural networks, create unique features and vulnerabilities. Network or graph theory gives us measures how effectively any network is connected. The measure of path-length tells us the largest number of steps between any two particular nodes. If you look at nodes being individuals on the planet and connections as being whether they know each other, the path length is how many steps you would have to take to get to anyone else in the system, no matter how far away.   So for a particular world leader or artist, you might need six steps:  friend of a friend of a friend of a friend times six to get to that person. A path length of six gives us the famous “Six degrees of separation” that connects everyone. Of course, by the time you get to a path length of four (the friend of a friend of a friend of a friend), you may really feel you are talking about a stranger. Clustering gives a more familiar measure of how likely it is that two people you know will know each other, expressed as a percentage probability. It says how strongly interconnected a network is in a local scope.

I mention these two measures, because optimizing one can occur at the expense of the other, yet both are important for communication. If you can only keep up a thousand connections, you could focus on local connections which are more likely to cluster and overlap. If on the other hand, you wanted to quickly reach distant points, so you can influence that particular world leader or artist, then you need to have add a lot of connections that reach across groups. In what is known as a small-world network, we see a balance of these two measures, and many natural networks have been shown to demonstrate small-world characteristics. In the brain, the presence of highly connected hubs helps reduce the path-length between what might have otherwise have been distant neurons, while maintaining a high clustering.

For a popular introduction to small world networks, I recommend reading the book  Six degrees: The science of a connected age, by Duncan Watts, published by WW Norton & Company in 2004.  Watts describes his process for coming up with the ideas for these networks, and some of the ways they have been found to apply.  He also touches on many important ideas like Granovetter’s Strength of Weak Ties or Hardin’s Tragedy of the Commons, which illustrate the social consequences network researchers are dealing with.

Brain network changes from aging
Brain network changes from aging Brain network changes from aging. from Achard, S. & Bullmore, E. (2007). Efficiency and cost of economical brain functional networks. PLoS Comput Biol, 3(2), e17.

Network analysis techniques have been increasing applied to brain studies and the new area of connectomics.  For example, when measuring path length and clustering in a functional network of different brain areas defined using EEG beta activity, Stam et al. showed that normal brains demonstrated small world values for these measures, while people with Alzheimer’s lost small world characteristics due to an increase in path-length.  Similar results have been shown using fMRI measures of cortical thickness and metabolism, not only in Alzheimer’s, but with aging and dopamine blockade as well.

If you are interested in graph theory and its applications to understanding the brain,  look at the review article Bullmore, E., & Sporns, O. (2009). Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10(3), 186-198.  Even more information is available from Sporns’ book Networks of the Brain. MIT press, 2010.

We are all connected. Our technologies are making that more apparent, decreasing the path-length between people in virtual spaces and increasing the clustering of information in ways that can make it difficult to find ideas that close enough and also different enough to trigger change. Systems that dynamically learn and grow must maintain an equilibrium between flexible learning of new and stable memory of old. Truly listening to a social system, or to any of the constituent individuals in their layers of physical, electrical and cellular forces and tensions, involves digging into the intricacies of that balance: when and why and how long it tilts in one direction or the other.  How the human brain seeks to achieve such an understanding is what I will blog about next time!