A review of dynamical systems approaches for the detection of chaotic attractors in cancer networks

Abstract

Cancers are complex dynamical systems. They remain the leading cause of disease-related pediatric mortality in North America. To overcome this burden, we must decipher the state-space attractor dynamics of gene expression patterns and protein oscillations orchestrated by cancer stemness networks. The review provides an overview of dynamical systems theory to steer cancer research in pattern science. While most of our current tools in network medicine rely on statistical correlation methods, causality inference remains primitively developed. As such, a survey of attractor reconstruction methods and machine algorithms for the detection of causal structures applicable in experimentally derived time series cancer datasets is presented. A toolbox of complex systems approaches are discussed for reconstructing the signaling state space of cancer networks, interpreting causal relationships in their time series gene expression patterns, and assisting clinical decision making in computational oncology. As a proof of concept, the applicability of some algorithms are demonstrated on pediatric brain cancer datasets and the requirement of their time series analysis is highlighted. ll

DeepVelo: Single-cell Transcriptomic Deep Velocity Field Learning with Neural Ordinary Differential Equations

“…More interestingly, single-cell processes have long been hypothesized to exhibit properties similar to chaotic systems ( 8 , 36 , 37 ). By recovering single-cell gene expression dynamics with scDVF, we observed chaotic behaviors in the in silico gene perturbation studies, where a small change in the initial gene expression state may result in a large difference in the future states, also known as the butterfly effect.…”

“…However, linear systems may fail to capture the non-linearity of single-cell dynamics. Moreover, single-cell dynamical systems have a high degrees-of-freedom due to the high dimensionality of the data, which could lead to errors in any dimension ( 7 ).…”

“…For example, heterodimeric cooperation in transcription factors is a known non-linear relationship, where two different transcription factors must bind to activate another gene ( 7 ). Moreover, single-cell dynamical systems have a high degrees-of-freedom due to the high dimensionality of the data, which could lead to errors in any dimension ( 8 ).…”

“…Moreover, single-cell dynamical systems have a high degrees-of-freedom due to the high dimensionality of the data, which could lead to errors in any dimension (7).…”

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“…There may also be more complicated trajectories in the state space that have completely irregular shapes, called strange attractors. These characterize chaotic dynamics, resembling a completely irregular stochastic behavior, although determined by deterministic non-linear rules [ 70 ]. A system can have more than one attractor, e.g., two different stable steady states (a condition called bistability) and evolve toward one or the other depending on the starting state of the system.…”