Crowded transport within networked representations of complex geometries

The efficient transport of macromolecules within crowded, complex environments is crucial for various biological processes and technological applications. Examples range from protein synthesis and T-cell immune responses to the design of autonomous robotic motion and targeted drug delivery. However, understanding the relationship between environmental geometry, crowding, and transport remains challenging. Existing analytical methods often struggle with heterogeneous environments, relying instead on computationally expensive mesh-based approaches. This research proposes a more efficient framework using networked representations of complex geometries to overcome these limitations. The approach simplifies complex geometries into networks of interconnected reservoirs (nodes) and narrow channels (edges), representing regions of high crowding. This dimensionality reduction allows for the efficient analysis of transport dynamics and the identification of topological features impacting transport behavior. The framework is particularly relevant to studying intracellular transport, where high-resolution images of complex geometries are increasingly available through advanced imaging techniques, but computationally intensive analysis is currently a limiting factor.

Existing research on crowded transport has explored various scenarios. Studies on one-dimensional lattices showed that excluded volume interactions significantly hinder tracer particle motion, leading to anomalous diffusion. However, in some fractal environments, crowding effects on the mean squared displacement exponent were less pronounced. On comb lattices, exclusion interactions surprisingly sped up transport. Higher-dimensional regular lattices have also been studied, focusing on diffusion in the presence of obstacles and crowded transport on Manhattan lattices. Analytical progress in these studies often relies on symmetry, scaling arguments, and infinite domain assumptions. While some analytical work addresses finite, heterogeneous environments, such as protein transport within heterochromatin or cell migration, these methods are often not readily applicable to detailed real-life environments like the intracellular space. Consequently, computationally intensive numerical methods using high-resolution meshes are commonly employed, which limit the exploration of a large ensemble of geometries.

The authors introduce a hierarchy of diffusive transport models with increasing computational scalability. The core model is a continuous-time Markov chain (CTMC) termed the Full Markov Model (FMM). The FMM explicitly models crowding within narrow channels as a symmetric simple exclusion process (SSEP), where at most one individual occupies each lattice site. Individuals are assumed well-mixed within the larger reservoirs. Jumps between lattice sites and reservoirs occur at rates α and γᵢ (inverse of the mean reservoir exit time τᵢ), respectively. The equilibration time, calculated as the reciprocal of the spectral gap of the transition matrix, serves as the key statistic to quantify transport. However, the high dimensionality of the FMM’s transition matrix (2^K*N*V − 1) limits its applicability to large networks. To address this, a Reduced Markov Model (RMM) is introduced, which allows for direct exchange between reservoirs. In the high-density regime (where crowding effects are significant), the transition rates between reservoirs are derived by considering the dynamics of interacting individuals within the channels. The RMM transition rates are given by a formula considering the average time for a vacancy to switch between the reservoirs. Further dimensionality reduction is achieved by employing a continuous mean-field approximation of reservoir occupancy, leading to an Ornstein-Uhlenbeck (OU) process, described by a Fokker-Planck equation. The OU process provides a computationally efficient way to approximate the equilibration time of the FMM using the spectral gap of a networked graph Laplacian. Finally, the framework is extended to analyze the dynamics of a tagged individual, providing insights into individual-level transport behavior. The tagged individual's dynamics are modeled via a discrete networked random walk model, enabling the calculation of the tagged individual crossing probability (P_T) and mean exit time (m_T).

The study reveals that individual crowding combined with geometry-induced crowding significantly slows down equilibration. In the absence of crowding, networks with shorter average reservoir exit times equilibrate faster. However, with crowding, this monotonicity is lost; the equilibration time increases when volume-excluding interactions create bottlenecks. An analytical condition, 〈τ〉 < N_HD/(α|V|), determines the high-density regime where the combined effect of geometry and crowding is most prominent. The RMM and OU process approximations accurately predict equilibration times, significantly reducing computational cost. Analysis of a toy network with five reservoirs demonstrates that equilibration times vary drastically based on topology. While complete networks facilitate the quickest equilibration, they are unrealistic for many complex environments. Optimal networks, minimizing equilibration time under a connectivity constraint, exhibit distinct topological structures characterized by a weighted degree distribution. This distribution reflects the relative importance of different reservoirs in achieving optimal transport. Heterogeneity in reservoir exit times significantly influences the optimal frontier, enabling efficient equilibration even in sparsely connected networks. Finally, the analysis of tagged individual dynamics shows that narrow channel length drastically alters the probability of crossing and mean exit time, making transport highly sensitive to local network topology in crowded environments. Crowding significantly alters the paths taken by tagged individuals, favoring shorter channels and leading to longer overall path lengths.

This work provides a powerful framework for understanding the interplay between geometry, crowding, and transport in complex environments. The use of network representations offers significant computational advantages over traditional mesh-based approaches. The identified global relationships between topology and equilibration times, coupled with the analysis of individual-level dynamics, provide valuable insights for optimizing transport efficiency in various settings. The findings are particularly relevant to understanding intracellular transport processes, such as those in cardiomyocytes. The ability to analyze a large number of experimental geometries facilitates a deeper understanding of how intracellular restructuring affects cellular processes. The framework's generalizability extends to include active transport and reaction-diffusion processes, broadening its applicability to various biological and engineering systems.

This study introduces a versatile and computationally efficient framework for modeling crowded transport in complex geometries using network representations. The framework reveals crucial relationships between network topology, crowding, and both population-level and individual-level transport dynamics. The findings provide valuable insights for designing optimal environments and offer significant potential for applications in biology and engineering. Future research could investigate the effects of fluctuating narrow channel occupancies and extend the framework to incorporate more sophisticated models of individual behavior and interactions.

The study makes certain assumptions and simplifications, including the assumption of well-mixed individuals within reservoirs and the use of simplified models for crowding within channels. These simplifications, while improving computational efficiency, might limit the accuracy of the model for some specific systems. Furthermore, the study primarily focuses on symmetric random walks. While the inclusion of the PASEP expands the applicability to active transport, more realistic models of active transport might be necessary for specific biological applications. The study’s findings on optimal networks are based on specific network generation methods, and further investigation is needed to determine the generalizability of these results to other network structures.