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Complementarity, Complexity and the Fokker–Planck Equation; from the Microscale Quantum Stochastic Events to Fractal Dynamics of Cancer

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Cancer, Complexity, Computation

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 46))

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Abstract

Tumourigenesis possesses no equivalent among known physical phenomena. It is initiated at the quantum level by thermodynamic fluctuations of macromolecules. Accumulation of non-lethal alterations in quasi-deterministic dynamic cellular network of genes and their regulatory protein elements facilitated by changes in microenvironment results in a weak emergence of non-complementary, malignant phenotype. The Weibull distribution of cancer incidence suggests that neuro-immuno-hormonal network modifies that process. Eucaryotic cells are supramolecular objects. They make use of quantum entanglement, quantum tunneling, coherence, and chirality in formation of novel molecular couplings with both multiple feedbacks, synergy, and hysteresis. Complementarity at each integration level and non-ergodicity are their distinguishing features. Quantum effects may contribute to the conjugated appearance of cancer mutations. Connectivity, that is, coupling between integration levels is associated with the emergence of at least three features: fractal geometry of space–time, in which growth occurs, conditional probability of events, which reduces sensitivity to the initial conditions, and entropy. The latter one determines both a capability of the supramolecular system for transfer of biologically relevant information and evolution of intercellular interactions. There is a limit for self-organization of cells into structures of higher order defined by the Fibonacci constant. A relationship between sigmoidal dynamics and the Feigenbaum diagram suggests that both growth and self-organization occur with parameters within the Mandelbrot set. The set of non-interacting, infiltrating cancer cells becomes topologically dense. It has the highest entropy. The global spatial fractal dimension approaches the integer value. Hence, the coefficient of cellular expansion is a novel quantitative measure of biological tumour aggressiveness. It is based on complexity of intercellular interactions. Neither biological complexity can be reduced to physical one, nor be fully mathematized. Computer simulations may help to elucidate details of tumourigenesis. The mathematical models should be expressed in the algebraic form of fractal sheaves and fractional equations.

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  1. Department of Urology, Ameos Clinics, Landrat-Beushausen-Strasse 26, 31061, Alfeld, Germany

    Przemyslaw Waliszewski

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  1. University of Novi Sad, Novi Sad, Serbia

    Igor Balaz

  2. Unconventional Computing Laboratory, University of the West of England, Bristol, UK

    Andrew Adamatzky

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Waliszewski, P. (2022). Complementarity, Complexity and the Fokker–Planck Equation; from the Microscale Quantum Stochastic Events to Fractal Dynamics of Cancer. In: Balaz, I., Adamatzky, A. (eds) Cancer, Complexity, Computation. Emergence, Complexity and Computation, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-031-04379-6_2

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