Session B03: Statistical and Nonlinear Physics of Complex Systems

Mathematical Modeling of Drug-Induced Persistence in Cancer

Drug-tolerant persistence in tumor cells remains a significant challenge in oncology. Unlike resistance driven by genetic mutations, persistence is a reversible state arising from phenotypic shifts in response to external stress, such as cytotoxic agents. This state is often characterized by quiescence and low proliferation, although some cells may resume cycling. In this study, we develop a mathematical model to describe the emergence of drug-induced persistence and its implications. We assume an initial steady-state distribution of tumor cells across two variables: x (chance-to-persist) and s (phenotypic state related to drug survivability) before drug treatment. The variable x reflects epigenetic states conferring specific persistence potentials to subpopulations, while s represents the degree to which different phenotypic states enable survival under drug exposure. The population probability density is then subjected to dynamics governed by our modified Fokker-Planck equation with an advection term dependent on s and drug concentration. The model captures adaptations through advection and diffusion in s and selection pressure from drug-induced death. Simulations of various treatment schemes provide quantitative insights into the emergence of persistence, and our theory can be generalized to study phenotypic plasticity and investigate cellular adaptation mechanisms under different external stresses and stimuli.    

Network theory explain aging dynamics

Dysregulation of gene expression patterns leads to the loss of cellular function during aging. The exact characteristics of the cellular loss of function depend on the specific up/down-regulated genes and vary between pathways and cell types. However, the mechanism behind this dysregulation remains elusive. Here we develop an analytical framework using network theory to characterize the impact of aging on the gene regulatory networks. We find that the accumulation of somatic mutations and changes in methylation levels result in a global loss of network stability. This leads to an increase in cellular susceptibility to changes which can explain cell-to-cell variability. We also find a slowdown in the response time to biochemical signals during aging. This could explain the inflammaging phenomena where the healthy process of inflammation does not shut down and turns chronic. It can also explain the slowing down in the healing process of wounds in old age. Our results highlight the importance of the complex gene regulation network in aging and provide an analytical framework bridging between the biology of aging and nonlinear dynamics.   

Exceptions to the Ratchet Principle: Inhomogeneous Fluctuations with and without Rectification

The "ratchet principle" is one of the few generic results that hold outside thermal equilibrium. It states that non-equilibrium systems which violate parity symmetry generically exhibit steady-state currents. This has proven extremely useful in explaining many non-equilibrium transport processes, such as the motion of dynein and kinesin along microtubules in the cell. However, people have recently noticed some apparent exceptions to this principle. In this talk, I will show that the reason for these exceptions is an emergent bulk momentum conservation which must also be broken to produce a ratchet current. As a result, the ratchet principle should be amended accordingly.   

Design principles of gene regulatory networks underlying low-dimensional cell-fate decision systems

Cell-fate decisions, driven by changes in gene expression, are often binary switches. This is reflected in transcriptomic data, where the first principal component (PC1) explains a significant percentage of variance. Interestingly, PC1 variance remains stably high despite errors in gene signature, a phenomenon we term "PC1 stability." While binary decisions are often attributed to "core toggle switches" of two mutually inhibiting transcription factors, actual Gene Regulatory Networks (GRNs) are far more complex. The role of this complexity in maintaining low dimensionality and PC1 stability is unclear.

We analyzed GRNs across various cell-fate contexts and simulated their behavior over diverse parameters. Our findings reveal that GRNs often feature two mutually inhibiting "teams" of genes, where intra-team members positively influence each other, and cross-team members exert negative influences. Perturbing these networks showed strong teams promote both high PC1 variance and PC1 stability. Using artificial networks, we found that multiple toggle-switch-based networks can give rise to high PC1 variance, teams are necessary for PC1 stability. We thus propose that team-based topology is key to robust, low-dimensional cell-fate canalization, and the absence of PC1 stability in transcriptomic data for a given gene signature can indicate a lack of strong, cohesive interactions in the underlying gene regulatory network.   

Finding signatures of low-dimensional geometric landscapes in high-dimensional cell fate transitions

Hundreds of highly specialized cell phenotypes cooperate together to enable healthy functioning in many animals. When growing or injured, cells can self-organize and transition between these cell types. The consistency and robustness of developmental cell fate trajectories suggests that complex gene regulatory networks effectively act as low-dimensional cell fate landscapes. We introduce a phenomenological model of cell fate transitions that predicts signatures of these landscapes observable in gene expression measurements. By combining low-dimensional gradient dynamical systems and high-dimensional Hopfield networks, our model captures the interplay between cell fate, gene expression, and signals. Using existing single-cell RNA-sequencing time-series data, we compare experimental observations to theoretical landscape candidates belonging to different bifurcation classes. These results show that a geometric landscape approach can reveal new insights in time series single-cell RNA-sequencing data of cell fate transitions.   

Reduced-order modeling and analysis of fluid flows: from wall-bounded shear flows to convection

The study of problems ranging from wall-bounded shear flows to convection is complicated by the large range of spatiotemporal scales that need to be resolved. Reduced-order models can help explore computationally challenging parameter regimes and facilitate iterative optimization computation. The first part of this talk develops a structured input-output analysis to analyze transitional wall-bounded shear flows. This framework captures the key nonlinear effect that enables the prediction of a wider range of known transitional flow structures within a linear analytical modelling paradigm. The results for plane Couette flow closely match those obtained from extensive direct numerical simulations (DNS) and nonlinear optimal perturbation analysis but are achieved with vastly reduced computational cost. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large domains. The second part of this talk employs single-mode equations reducing three spatial dimensions into one vertical dimension to analyze well-organized columnar structures. The computational efficiency of this reduced description allows one to reach computationally challenging and oceanographically relevant parameter regimes for salt-finger convection. Single-mode equations were also applied to convection in a porous medium and fixed-flux convection and capture the essential physics in a wide parameter regime.   

Current-current relationship in non-equilibrium networks: linearity and applications

The paradigm of irreversible thermodynamics largely revolves around pairs of conjugated observables (current and force) and linear relations among them, close to equilibrium. This study takes a different path, by examining how currents respond to changes in other currents, without directly addressing driving forces. Using continuous-time Markov chains, we establish a linear-affine relationship between any two network currents, valid even far from equilibrium, and derive a formula for the current-to-current susceptibility via spanning-tree ensembles. Finally, we discuss how inherent linearity among network currents can be exploited in complex metabolic networks, in order to map the internal reaction currents of a cell, from the knowledge of input/output currents.