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Transport in crowded, complex environments occurs across many spatial scales. Geometric restrictions can hinder the motion of individuals and, combined with crowding, can have drastic effects on global transport phenomena. However, in general, the interplay between crowding and geometry in complex real-life environments is poorly understood. Existing analytical methodologies are not always readily extendable to heterogeneous environments and, in these situations, predictions of crowded transport behaviour rely on computationally intensive mesh-based approaches. Here, we take a different approach based on networked representations of complex environments in order to provide an efficient framework to explore the interactions between environments’ geometry and crowding. We demonstrate how this framework can be used to extract detailed information both at the level of the individual as well as of the whole population, identify the environments’ topological features that enable accurate prediction of transport phenomena, and provide insights into the design of optimal environments.

 

We introduce Catalyst.jl, a flexible and feature-filled Julia library for modeling and high-performance simulation of chemical reaction networks (CRNs). Catalyst supports simulating stochastic chemical kinetics (jump process), chemical Langevin equation (stochastic differential equation), and reaction rate equation (ordinary differential equation) representations for CRNs. Through comprehensive benchmarks, we demonstrate that Catalyst simulation runtimes are often one to two orders of magnitude faster than other popular tools. More broadly, Catalyst acts as both a domain-specific language and an intermediate representation for symbolically encoding CRN models as Julia-native objects. This enables a pipeline of symbolically specifying, analyzing, and modifying CRNs; converting Catalyst models to symbolic representations of concrete mathematical models; and generating compiled code for numerical solvers. Leveraging ModelingToolkit.jl and Symbolics.jl, Catalyst models can be analyzed, simplified, and compiled into optimized representations for use in numerical solvers. Finally, we demonstrate Catalyst’s broad extensibility and composability by highlighting how it can compose with a variety of Julia libraries, and how existing open-source biological modeling projects have extended its intermediate representation. 

In particle-based stochastic reaction–diffusion models, reaction rates and placement kernels are used to decide the probability per time a reaction can occur between reactant particles and to decide where product particles should be placed. When choosing kernels to use in reversible reactions, a key constraint is to ensure that detailed balance of spatial reaction fluxes holds at all points at equilibrium. In this work, we formulate a general partial-integral differential equation model that encompasses several of the commonly used contact reactivity (e.g., Smoluchowski-Collins-Kimball) and volume reactivity (e.g., Doi) particle models. From these equations, we derive a detailed balance condition for the reversible A + B ⇆ C reaction. In bounded domains with no-flux boundary conditions, when choosing unbinding kernels consistent with several commonly used binding kernels, we show that preserving detailed balance of spatial reaction fluxes at all points requires spatially varying unbinding rate functions near the domain boundary. Brownian dynamics simulation algorithms can realize such varying rates through ignoring domain boundaries during unbinding and rejecting unbinding events that result in product particles being placed outside the domain. 

Protein–protein binding domains are critical in signaling networks. Src homology 2 (SH2) domains are binding domains that interact with sequences containing phosphorylated tyrosines. A subset of SH2 domain–containing proteins has tandem domains, which are thought to enhance binding affinity and specificity. However, a trade-off exists between long-lived binding and the ability to rapidly reverse signaling, which is a critical requirement of noise-filtering mechanisms such as kinetic proofreading. Here, we use modeling to show that the unbinding rate of tandem, but not single, SH2 domains can be accelerated by phosphatases. Using surface plasmon resonance, we show that the phosphatase CD45 can accelerate the unbinding rate of zeta chain–associated protein kinase 70 (ZAP70), a tandem SH2 domain–containing kinase, from biphosphorylated peptides from the T cell receptor (TCR). An important functional prediction of accelerated unbinding is that the intracellular ZAP70–TCR half-life in T cells will not be fixed but rather, dependent on the extracellular TCR–antigen half-life, and we show that this is the case in both cell lines and primary T cells. The work highlights that tandem SH2 domains can break the trade-off between signal fidelity (requiring long half-life) and signal reversibility (requiring short half-life), which is a key requirement for T cell antigen discrimination mediated by kinetic proofreading. 

T cells use their T cell receptors (TCRs) to discriminate between lower-affinity self and higher-affinity non-self peptides presented on major histocompatibility complex (pMHC) antigens. Although the discriminatory power of the TCR is widely believed to be near-perfect, technical difficulties have hampered efforts to precisely quantify it. Here, we describe a method for measuring very low TCR/pMHC affinities and use it to measure the discriminatory power of the TCR and the factors affecting it. We find that TCR discrimination, although enhanced compared with conventional cell-surface receptors, is imperfect: primary human T cells can respond to pMHC with affinities as low as K D ∼ 1 mM. The kinetic proofreading mechanism fit our data, providing the first estimates of both the time delay (2.8 s) and number of biochemical steps (2.67) that are consistent with the extraordinary sensitivity of antigen recognition. Our findings explain why self pMHC frequently induce autoimmune diseases and anti-tumour responses, and suggest ways to modify TCR discrimination.