Cancer and Chaos and the Complex Network Model of a Multicellular Organism

Abstract

In the search of theoretical models describing cancer, one of promising directions is chaos. It is connected to ideas of "genome chaos" and "life on the edge of chaos", but they profoundly differ in the meaning of the term "chaos". To build any coherent models, notions used by both ideas should be firstly brought closer. The hypothesis "life on the edge of chaos" using deterministic chaos has been radically deepened developed in recent years by the discovery of half-chaos. This new view requires a deeper interpretation within the range of the cell and the organism. It has impacts on understanding "chaos" in the term "genome chaos". This study intends to present such an interpretation on the basis of which such searches will be easier and closer to intuition. We interpret genome chaos as deterministic chaos in a large module of half-chaotic network modeling the cell. We observed such chaotic modules in simulations of evolution controlled by weaker variant of natural selection. We also discuss differences between free and somatic cells in modeling their disturbance using half-chaotic networks.

Keywords: Kauffman network; biology; cancer; chaos; complex network; deterministic chaos; genome chaos; half-chaos; multicellular organism.

Figure 1

Network elements: node, its function, signals and links. Network consists of nodes and links between them. We consider directed networks, and links transmit signals in particular direction. We use s ≥ 2 equally probable signal variants. For s > 2, such networks cease to be Boolean; however, (we propose) they may stay as Kauffman networks. Node has K input links—we keep K fixed for all nodes of a given network (for simplicity of program), but the number of output links k may be different. Node function calculates output signal (node state) using its input signals (input state of node). Here, K = 2 and k = 3, but typically, we use K = 3 and k from 0 to any, but k = 0 normally happens only in er (Erdős-Rényi) network. In sf and ss networks, they are absent. When removing nodes, such nodes with k = 0 may emerge. In the entire autonomous network, there are K × N links.

Figure 2

“Crocodiles” of half-chaotic (a) and chaotic (b) networks. The plot of A(t)—the number of different node states than in a network without disturbance. Shown plots contain 1200 processes each; it is a complete set of available disturbances of a network of N = 400 nodes and s = 4 when disturbance is a permanent change in function for the initial input state of one node. Red curve (bold for chaotic case b) is a q(t)—part of all processes that stayed still ordered (its damage, d(t), is small, i.e., A(t) < 150; 150 is an arbitrarily taken threshold for a small avalanche). For q and d, it is denoted by the same red scale on the left. As observed, the jump from order to chaos is short, and we call it an “explosion to chaos”. All processes of chaotic network (b) exploded before tmx = 1000, but up to tmx, the attractor for the initial network was not found. For a half-chaotic network (a), such an attractor was 8, but the teeth in the crocodile’s lower jaw indicate that the attractor of this one particular process is slightly larger but still a short attractor. Explosions into chaos could have happened here until the end of the first attractor cycle. On plot (a), a transparent blue “Derrida plot” [9] p. 200, [22] p. 296 is observed. It is d_t+1(d_t) calculated from a Derrida-annealed approximation model (constructed for fully random network). The intersection of the curves with the diagonal d_t+1 = d_t is the point of equilibrium called “chaotic Derrida equilibrium”. It is marked for s = 4 and K = 3 by a dashed line, and it also is the maximum of the right peak. For s = 2 and K = 2, there is no such equilibrium; therefore, such parameters lead to an ordered network, and this point is precisely the edge of chaos and order.

Figure 3

Effects in q (q—the degree of order) from an increase in negative feedbacks and classic modularity, but without the control of a short attractor (see met2&3 in [2,47]). The share of negative feedbacks was increased in method 2 by changing some positive feedbacks to negative ones in a fully random network. The side effect of this method was the narrowing of the function, which also had a significant effect on the increase in stability. Both of these factors and the state without these changes are marked. Three types of networks were investigated: sf, ss and er (second letter in diagram). For the er network level of k = 0, it is marked by the green line. In the right column, the results of method 3 are shown, in which modularity was additionally introduced; here, “wild” denotes the effect of introducing modularity itself. N = 400, tmx = 20,000. As observed, a significant increase in stability is observed only for sf 2,4 in met2 and for met3 2,4 even without met2. Parameters s,K = 2,4 and 4,3 result in significant differences in the share of met2 and the narrowing of the function; for 2,4, of the increases in stability is mostly a result of the narrowing of function. Conclusion: of the increase in negative feedback and classic modularity can increase stability in the long term even without the control of a short attractor, but it is an exceptional case. It may be that function narrowing is an important condition in such cases. These tests should be considered as a preliminary diagnosis.

Figure 4

Two peaks in the damage size distribution of P(d) and the gap between for network types sf (scale-free) and er (Erdős-Rényi) (second letter of shortcut). Networks parameters: n = 400, s = 4, K = 3, initially PAS. Influence of regulation by feedbacks. First letter (a, b and c) indicates the model: A—strong regulation by feedback; b—small regulation by feedback; c—without any regulation (detailed descriptions in [2,47]. The gap and the right peak are blurred only for “af”, i.e., sf network with strong regulation.