Symmetry and symmetry breaking in cancer: a foundational approach to the cancer problem

Abstract

Symmetry and symmetry breaking concepts from physics and biology are applied to the problem of cancer. Three categories of symmetry breaking in cancer are examined: combinatorial, geometric, and functional. Within these categories, symmetry breaking is examined for relevant cancer features, including epithelial-mesenchymal transition (EMT); tumor heterogeneity; tensegrity; fractal geometric and information structure; functional interaction networks; and network stabilizability and attack tolerance. The new cancer symmetry concepts are relevant to homeostasis loss in cancer and to its origin, spread, treatment and resistance. Symmetry and symmetry breaking could provide a new way of thinking and a pathway to a solution of the cancer problem.

Keywords: cancer; complexity; scale-free; symmetry; symmetry-breaking.

Figure 1. Symmetry breaking in mutually inhibitory feedback loops (from Jolly et al)

Different levels of mutual inhibition and self-activation between two molecules, A and B, results in symmetry breaking from the configuration of equal numbers of A and B to bistability and tristability. In the tristable configuration, the intermediate state could represent a hybrid cell state. Red and black curves describe nullclines for A and B, and their intersections are the steady states. Green filled circles represent stable steady states, and green hollow circles show unstable steady states. For more information, see Jolly et al, 2015.

Figure 2. The Koch curve

The Koch curve is created by dividing each line segment into thirds and replacing the middle segment with an equilateral triangle. Each iteration of the Koch curve produces a curve that is self-similar to the previous ones. It can be readily shown that the total length scales the power law relationship (4/3)n for n iterations and thus the length approaches infinity in the limit. The fractal dimension of a Koch curve is defined as log4/log3 = 1.2619. The Koch curve is continuous, but not differentiable, i.e., it has no tangent at any point.

Figure 3. Graphs of differing symmetry

Graph 1 is a complete graph with all nodes connected; it has 6 rotation and 12 reflection symmetries. Graph 2 is the smallest asymmetric graph. Graph 3 is the smallest asymmetric graph with each node possessing 3 edges (degree 3); it is known as the Frucht graph. Graph 4 is a complex graph, such as might exist for interacting proteins, for which any imbedded symmetries are difficult to discern by inspection. Software programs, such as nauty and SAUCY2, can compute the graph automorphisms.

Figure 4. Visual illustration of the difference between an exponential and a scale-free network (from Albert et al)

a, The exponential network is homogeneous: most nodes have approximately the same number of links. b, The scale-free network is inhomogeneous: the majority of the nodes have one or two links but a few nodes have a large number of links, guaranteeing that the system is fully connected. Red, the five nodes with the highest number of links; green, their first neighbours. Although in the exponential network only 27% of the nodes are reached by the five most connected nodes, in the scale-free network more than 60% are reached, demonstrating the importance of the connected nodes in the scale-free network Both networks contain 130 nodes and 215 links ((k)=3.3). For additional information see Albert et al, Nature 2000.