A Review of Mathematical and Computational Methods in Cancer Dynamics

Abstract

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Keywords: algorithms; cancer; complex networks; complexity science; dynamical systems; information theory; inverse problems; systems oncology.

Figure 1

Biological inverse problem. The workflow summarizes a blueprint of causal inference methods and measures discussed in the review for systems oncology. Given time-resolved cancer data (e.g., live-cell imaging of protein flows, time-sequential transcriptomic profiling, etc.), we can employ complex systems tools such as dynamical systems modelling or statistical machine learning algorithms for pattern discovery. Dynamical systems approaches include attractor embedding followed by chaotic behavior detection tools as discussed, or complex networks inference. Chaotic behavior detection tools comprises of many approaches discussed in the paper including attractor embedding, fractal analysis, frequency spectra, and Lyapunov exponents. However, these approaches may have dimensionality limits and hence, AI-driven causal inference algorithms are proposed as promising tools for causal pattern discovery in single-cell time-sequential analyses, which include algorithmic information dynamics (i.e., measuring the algorithmic complexity of complex graph networks via perturbation analysis in software space), recurrent neural networks (e.g., RC networks, liquid neural networks, etc.), and model-driven AI (e.g., turbulence modelling/multiscale computational fluid dynamics).

Figure 2

Attractors and oscillations. (A) Time-delay Coordinate Embedding. A schematic of attractor reconstruction from a time-series signal of some variable X(t) is shown by time-delay embedding (i.e., Convergent Cross Mapping). τ represent the time-delay. However, for complex large-scale datasets, machine learning algorithms such as reservoir computing (RC) and deep learning architectures are suggested (Image was adapted from 37). (B) Three different types of attractors which can self-organize in the signaling/expression state-space of cancer processes are shown: a limit cycle (periodic oscillation), quasi-periodic attractor, and a strange attractor (chaotic). The simplest of attractors, a fixed-point, is not shown herein. Their corresponding frequency spectra are shown below, with the oscillator’s angular frequency as the independent variable and the amplitude of the oscillations as the dependent variable. The oscillation of a limit cycle attractor has a defined amplitude (A) and peak in the frequency spectrum at a frequency (ω). A broad frequency spectrum is observed for the strange attractor, which exhibits a fractal-dimension in state-space. However, the frequency/power spectrum can be more complex depending on the system. For instance, complex attractors, such as those observed fluid turbulence, exhibit a broad frequency spectrum with an anomalous power-law scaling (i.e., multifractality) due to intermittency.

Figure 3

Differentiation dynamics in pediatric glioma systems. On the left, a schematic of the discussed epigenetic variants of pediatric high-grade gliomas (pHGGs) are shown with their corresponding brain regions recapitulating altered neurodevelopmental differentiation circuits. The corresponding Waddington landscape for their stalled differentiation dynamics is shown to the right. The cancer cell fates are shown as stalled attractors on the landscape (gene expression or signalling state-space) resembling stem cell states. Below, a string of the amino acid sequence of the histone tail H3 code is provided with the sites of the recurrent epigenetic mutations in these pHGGs. Some of these epigenetic modifications correspond to active chromatin marks with transcriptional activity while others are repressive marks (inhibited gene expression). The polycomb system is an essential regulator of pHGG differentiation dynamics. The toy-model system provides the biological insights underlying the complex dynamics and mathematical concepts discussed in the paper.