Crowded transport within networked representations of complex geometries

The efficacy of many cellular processes, from protein synthesis to T-cell immune response, depends on timely transport of macromolecules through crowded intracellular environments. Transport in complex environments spans fields beyond cell biology (e.g., robotics, nanoparticle drug delivery), yet quantifying and characterising relationships between environmental geometry, crowding, and transport in real-life heterogeneous environments remains challenging. Analytical results exist for idealised domains (e.g., infinite lattices, fractals, comb lattices, higher-dimensional regular lattices with obstacles or directional disorder), but are difficult to extend to finite, heterogeneous, non-fractal environments typical of intracellular spaces. Numerical approaches on high-resolution meshes incur substantial computational costs, especially for stochastic models requiring repeated simulations, and it is unclear how to relate high-dimensional geometric descriptions to transport statistics. To circumvent mesh-based limitations, the authors adopt networked representations (“balls and sticks”) of complex geometries, where reservoirs (nodes) are connected by narrow channels (edges or throats). Such networks provide low-dimensional characterisations and enable use of topological descriptors predictive of transport (degree-based and spectral statistics). Prior networked transport studies often consider single, non-crowded individuals or uni-directional exclusion processes (TASEP). In contrast, this work focuses on unbiased transport where individuals are locally well-mixed in reservoirs and experience crowding only along edges (narrow channels), being purely constrained to the network. The authors present a hierarchy of diffusive transport models with increasing computational scalability to elucidate governing principles linking transport, crowding, and geometry. They concentrate on how crowding and topology affect equilibration times (reciprocal of the spectral gap), highlighting that heterogeneity in microscopic structure enables low-connectivity networks to achieve minimal equilibration times. They further extend the framework to infer the dynamics of a single motile (tagged) individual, relevant to intracellular signalling where single-molecule dynamics can determine cellular outcomes.

Prior work on crowded transport includes: (i) one-dimensional lattices where excluded-volume interactions hinder tracer motion and can shift mean squared displacement (MSD) from diffusive to subdiffusive; (ii) diffusion on fractal environments where MSD exponents can be unchanged by crowding; (iii) comb lattices where backbone exclusion can speed tracer transport and yield transient superdiffusion; (iv) higher-dimensional regular lattices with obstacles or directional disorder (e.g., Manhattan lattices). Analytical results sometimes exploit infinite domains, symmetry, and scaling arguments. For finite heterogeneous environments, analytical progress is rarer (e.g., protein transport in heterochromatin via fractal results; lattice gas cellular automata coupled to vector fields for cell migration). Advances in imaging (light-sheet microscopy, x-ray tomography) have enabled reconstructions but require computationally expensive mesh-based simulations. Network-based representations (pore-network models) extracted by algorithms like maximal ball and SNOW segment free space into reservoirs and throats, providing tractable, low-dimensional models and rich topological descriptors (degree, spectral/Laplacian properties). Networked exclusion processes have been studied primarily for TASEP on lines, small graphs, and large networks, revealing shocks and jams; applications include protein dynamics on filament networks assuming a single global reservoir. The present work departs by modelling unbiased, symmetric exclusion confined to edges with crowding only in channels and well-mixed reservoirs, developing scalable models to connect geometry, crowding, and transport.

The study develops a hierarchical framework:

1. Full Markov Model (FMM): A continuous-time Markov chain on a network G=(V,E). Each edge (i,j) is a narrow channel discretised as a 1D lattice of integer length K_ij. Individuals (total N) undergo a symmetric simple exclusion process (SSEP) along channels: at rate α, an individual attempts to jump to adjacent lattice site or reservoir; jumps to occupied sites are aborted (exclusion). Reservoirs are large and assumed well-mixed with negligible crowding; individuals in reservoir i attempt to enter any adjacent channel’s first site at rate γ_i=τ_i^−1, where τ_i is the mean (geometry-dependent) exit time from reservoir i. The FMM state S(t) tracks reservoir occupancies and binary occupancies of all channel sites under exclusion. Equilibration time is the reciprocal of the spectral gap of the FMM generator. • Equilibrium occupancy approximation: Using single- and two-site probability functions and detailed balance, the equilibrium occupancy probability p̃ of channel sites is constant along a channel. Mean-field closure leads to a quadratic for p̃: K⟨τ⟩ p̃^2 − (N+K + α|V|⟨τ⟩) p̃ + N = 0 (taking the physically relevant root 0<p̃<1). High-density regime condition arises by setting p̃≈1−ε (ε≪1), giving ⟨τ⟩ ≲ (N−K)/(α|V|). Here ⟨τ⟩ is the average reservoir exit time, K the total number of channel sites.
2. Reduced Markov Model (RMM): To reduce dimensionality, individuals hop directly between connected reservoirs with state vector n (n_i individuals in reservoir i). In the high-density regime, transition rate k_ij^HD(n) is derived via particle-hole duality by analysing the vacancy switching time: the time for a vacancy to enter internal channel sites and be absorbed at the opposite end, which corresponds to exchanging a background individual between reservoirs. Combining geometric distribution of attempts and first-passage times on finite lattices yields explicit expressions for k_ij^HD(n) as functions of α, τ_i, τ_j, and K_ij (see Eq. (1) and Methods Eq. (24)). The RMM is a CTMC over reservoir occupancies with dimensionality N−1.
3. Ornstein–Uhlenbeck (OU) coarse-graining: Introducing continuous fractions x_i = n_i/N_HD (N_HD=N−K_tot), a Kramers–Moyal expansion of the RMM master equation leads to a Fokker–Planck equation with drift g_i^HD(x) and diffusion ε_i^HD(x). Linearising about the equilibrium distribution x* with x_i = τ_i/∑_j τ_j (from global balance) yields a multivariate OU process: dQ/dt = −∂_z(F^HD z Q) + (1/2)∂_z∂_z(D^HD Q). The mean evolves as μ(t;x0)=exp(−t F^HD) x0 + (I−exp(−t F^HD)) x, and covariance Σ(t)=∫_0^t e^{−uF^HD} D^HD e^{−u(F^HD)^T} du. The drift matrix F^HD is a weighted graph Laplacian depending on adjacency A_ij, reservoir exit times τ_i, and channel lengths K_ij via the continuous extension of k_ij^HD; equilibration time is governed by the spectral gap of F^HD.
4. Optimal topology analysis: Using the spectral gap of F^HD as an efficient proxy for equilibration time, the authors enumerate connected topologies (e.g., all 728 5-node networks), compute total edge length (sum of K_ij), and identify optimal frontiers under connectivity constraints (rescaled total edge length between minimum spanning tree and complete graph). They compare homogeneous vs heterogeneous τ vectors (τ(φ) with fixed mean) to assess heterogeneity effects.
5. Tagged individual dynamics: Within the FMM, a tagged individual entering a channel undergoes a biased random walk driven by background exchanges. The tagged individual crossing probability P_T^{(i→j)}(x_i,x_j;K) (probability to reach reservoir j before i when starting from second site) is derived via recurrence relations with state-dependent forward probabilities q_k and solved using products reducible to Beta functions; asymptotics yield expressions P_T ∼ [f_i(x_i,x_j)/∑_l f_l(x_i,x_j)]^{−1}. The mean exit time m_T^{(i,j)}(x_i,x_j;K) obeys a similar recurrence with time increments T_k and has a closed-form involving P_T and functions h_k, f_k. These results inform a discrete network random walk model for tagged individuals on large random geometric networks to compute first-passage statistics and path properties with and without crowding. Implementation notes: Network extraction from images (e.g., SNOW) yields low-dimensional networks (e.g., 81 reservoirs vs 16,774 voxels for a cardiomyocyte). Spectral computations on F^HD are inexpensive (e.g., FMM matrix dimension 6968 vs F^HD dimension 3 in a 3-reservoir example).

• Crowding plus geometry can drastically impede equilibration: In a 3-reservoir network, without crowding the equilibration time decreases monotonically with decreasing average reservoir exit time ⟨τ⟩, whereas with crowding this monotonicity is lost; bottlenecks form as channel occupancies increase (Fig. 2b). High-density regime criterion: ⟨τ⟩ < N_HD/(α|V|), where N_HD=N−K_tot (effective number of particles in reservoirs).
• Massive dimensionality reduction with accuracy: The OU drift matrix F^HD (a weighted graph Laplacian) accurately predicts FMM equilibration times. Example: FMM transition matrix dimension 6968 vs F^HD dimension 3, with close agreement of spectral gaps (Fig. 2b).
• Topology strongly affects equilibration: Across all 728 connected 5-node topologies (channels lengths K_ij set by 3D positions), equilibration times span several orders of magnitude (Fig. 3a). Complete networks equilibrate fastest but are often unrealistic due to spatial constraints; imposing a total edge length constraint yields an optimal frontier of non-complete topologies (Fig. 3a–b).
• Global optimal frontier: After rescaling total edge length (0–1 between MST and complete graph) and normalising equilibration times by the complete graph’s time, numerics indicate an approximately universal optimal frontier independent of the ensemble of channel lengths K, increasingly tight as |V| grows (Supplementary Fig. 6). This provides a benchmark for optimal network design and algorithm assessment.
• Reservoir heterogeneity enables near-minimal equilibration at low connectivity: Increasing heterogeneity in τ (keeping mean ⟨τ⟩ fixed) significantly lowers the connectivity required to achieve equilibration factors close to 1 (i.e., close to the complete network) (Fig. 3c). Heterogeneous environments can thus achieve globally efficient equilibration under connectivity constraints.
• Distinct topological signatures of optimal networks: Weighted degree distributions (from diagonal entries of F^HD) show that, with homogeneous τ, optimal low-connectivity networks over-represent reservoirs with high potential transition rates R_i, whereas highly connected optimal networks may over-represent low R_i; mid-connectivity regimes show mixed over-representation. With sufficient heterogeneity, reservoirs with high R_i are over-represented across connectivity levels (Fig. 3f). Missing a single critical edge can increase equilibration time by >4× in heterogeneous settings (Supplementary Fig. 8).
• Tagged individual transport is highly sensitive to channel length and occupancy bias: The crossing probability P_T from i to j depends on fractional occupancy x_i/(x_i+x_j); as K increases, P_T becomes extremely sensitive, becoming effectively zero near equilibrium occupancy for long channels (Fig. 4b). For homogeneous τ_i=τ_j, P_T≈0 when K exceeds a threshold scaling with N/|V| (Supplementary Fig. 5).
• Tagged individual mean exit times m_T increase by many orders of magnitude with longer channels and near-equilibrium occupancies; m_T can exceed the population equilibration time (Fig. 4c, Supplementary Fig. 3b).
• Crowding alters paths and increases path lengths in large networks: In random geometric networks with 1000 reservoirs (N=3×10^4, α=1, τ_i=0.1), crowding broadens the distribution of expected channel crossings and biases paths toward short channels, favouring indirect routes and increasing expected path length between cellular and nuclear membranes (Fig. 5a–d).
• Efficient geometric compression: Network extraction from a cardiomyocyte image yields 81 reservoirs vs 16,774 voxels in a Cartesian mesh, enabling tractable analysis (Supplementary Note 4).

The network-based framework reveals and quantifies how microscopic geometry (reservoir exit times, channel lengths) and crowding interact to control transport. Dimensionality reduction from FMM to RMM and ultimately to an OU process retains key physics and provides an efficient surrogate (spectral gap of F^HD) for equilibration times on large, heterogeneous networks. Analysis uncovers a strong dependence of equilibration on network topology, identifies optimal networks under connectivity constraints, and suggests the existence of a near-universal optimal frontier when rescaled appropriately. Reservoir heterogeneity, commonly present in real complex environments, can substantially reduce the connectivity required to achieve near-minimal equilibration times, informing design principles for efficient environments. At the individual level, crowding within channels creates state-dependent biases that severely limit traversals across long channels and inflate first-passage times, reshaping paths in intracellular-like networks and potentially impacting signalling efficiency. Weighted degree distributions and Laplacian structure connect geometric features to optimal transport strategies, offering interpretable design rules. The approach paves the way for analysing biological systems (e.g., cardiomyocyte bioenergetics under mitochondrial reorganisation) and engineering applications (synthetic porous design) where evaluating many geometries is essential.

This work introduces a scalable hierarchy of models for crowded transport in complex geometries using network representations: a detailed FMM with exclusion along channels, a reduced exchange RMM, and an OU coarse-graining whose drift matrix’s spectral gap accurately predicts equilibration times. Key contributions include: (i) identifying a high-density regime where crowding and geometry non-trivially slow equilibration; (ii) establishing an efficient spectral predictor enabling exploration of large topological ensembles; (iii) discovering optimal frontiers of network topologies under connectivity constraints and evidence for a global rescaled optimal curve; (iv) demonstrating that reservoir heterogeneity enables minimal equilibration at reduced connectivity; and (v) deriving analytical expressions for tagged individual crossing probabilities and mean exit times, showing drastic sensitivity to channel length and occupancy that reshapes paths and increases first-passage distances. Future directions include allowing joint fluctuations in reservoir and channel occupancies, validating the universality of the optimal frontier, incorporating reactions within the RMM/OU framework for geometry-controlled kinetics, and extending to active/biased transport (e.g., PASEP) to study crowding in cytoskeletal and intercellular transport, as well as applications in porous media and synthetic design.

• The OU approximation relies on linearisation near equilibrium and large N_HD; far-from-equilibrium dynamics and finite-size effects may deviate.
• Narrow channel occupancies are averaged/assumed constant in the coarse-grained models; prior work indicates potentially strong channel-density fluctuations that are not captured.
• Reservoir mean exit times τ_i are treated as abstract parameters; while computable from specific geometries (narrow escape problems), uncertainties in τ_i mapping from images may affect predictions.
• The high-density transition rates k_ij^HD(n) assume exponential approximation of vacancy switching times; accuracy can vary with parameters (validated numerically but still an approximation).
• Optimal frontier universality is supported by numerical evidence, not a formal proof.
• Network extraction (e.g., SNOW) introduces modelling choices; results may depend on segmentation quality and mapping from continuous geometries to discrete networks.