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Conventional cancer drugs are small molecules that target specific pathways. We introduced PCMS, a 26 kDa supramolecule combining sensors (S), molecular motors (M), and switching molecules (C), integrated within a fourth-generation PAMAM structure (P). PCMS identifies and deactivates cancer cell nucleus dynamics. A decade ago, we demonstrated programmable, clock-like interactions among the S-C-M components. In this study, we captured images of fractal patterns formed by chromosomal compartments and developed a theoretical model of their fractal dynamics. We showed that the nucleus behaves like a cavity, producing resonance effects similar to Chladni patterns. When the external agent, PCMS, interacts with this cavity, it generates a fractal pattern. We identified and mapped five key phase transitions that ultimately lead to the breakdown of cancer cell nuclei.

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.