Smoldyn: particle-based simulation with rule-based modeling, improved molecular interaction and a library interface

Parallelization of Particle-Based Reaction-Diffusion Simulations Using MPI

Particle-based reaction-diffusion offers a high-resolution alternative to the continuum reaction-diffusion approach, capturing the volume-excluding nature of molecules undergoing stochastic dynamics. This is essential for simulating self-assembly into higher-order structures like filaments, lattices, or macromolecular complexes. Applications of self-assembly are ubiquitous in chemistry, biology, and materials science, but these higher-resolution methods increase computational cost. Here, we present a parallel implementation of the particle-based NERDSS software using the message passing interface (MPI), achieving close to linear scaling for up to 96 processors. By using a spatial decomposition of the system across processors, our approach extends to very large simulation volumes. The scalability of parallel NERDSS is evaluated for reversible reactions and several examples of higher-order self-assembly in 3D and 2D, with all test cases producing accurate solutions. Parallel efficiency depends on the system size, timescales, and reaction network, showing optimal scaling for smaller assemblies with slower timescales. We provide parallel NERDSS code open-source, supporting development and extension to other particle-based reaction-diffusion software.

Phosphorylation-Dependent Dispersion of the Response Regulator in Bacterial Chemotaxis

Protein phosphorylation is a fundamental cellular regulatory mechanism that governs the activation and deactivation of numerous proteins. In two-component signaling transduction pathways, the phosphorylation of response regulator proteins and their subsequent diffusion play pivotal roles in signal transmission. However, the impact of protein phosphorylation on their dispersion properties remains elusive. In this study, using the response regulator CheY in bacterial chemotaxis as a model, we performed comprehensive measurements of the spatial distributions and diffusion characteristics of CheY and phosphorylated CheY through single-molecule tracking within live cells. We discovered that phosphorylation significantly enhances diffusion and mitigates the constraining influence of the cell membrane on these proteins. Moreover, we observed that ATP-dependent fluctuations also promote protein diffusion and reduce the restraining effect of the cell membrane. These findings highlight important effects of phosphorylation beyond protein activation.

Particle-based MR modeling with diffusion, microstructure, and enzymatic reactions

PURPOSE

To develop a novel particle-based in silico MR model and demonstrate applications of this model to signal mechanisms which are affected by the spatial organization of particles, including metabolic reaction kinetics, microstructural effects on diffusion, and radiofrequency (RF) refocusing effects in gradient-echo sequences.

METHODS

The model was developed by integrating a forward solution of the Bloch equations with a Brownian dynamics simulator. Simulation configurations were then designed to model MR signal dynamics of interest, with a primary focus on hyperpolarized 13C MRI methods. Phantom scans and spectrophotometric assays were conducted to validate model results in vitro.

RESULTS

The model accurately reproduced the reaction kinetics of enzyme-mediated conversion of pyruvate to lactate. When varying proportions of restrictive structure were added to the reaction volume, nonlinear changes in the reaction rate measured in vitro were replicated in silico. Modeling of RF refocusing effects characterized the degree of diffusion-weighted contribution from preserved residual magnetization in nonspoiled gradient-echo sequences.

CONCLUSIONS

These results show accurate reproduction of a range of MR signal mechanisms, establishing the model's capability to investigate the multifactorial signal dynamics such as those underlying hyperpolarized 13C MRI data.

Parallelization of particle-based reaction-diffusion simulations using MPI

Particle-based reaction-diffusion models offer a high-resolution alternative to the continuum reaction-diffusion approach, capturing the discrete and volume-excluding nature of molecules undergoing stochastic dynamics. These methods are thus uniquely capable of simulating explicit self-assembly of particles into higher-order structures like filaments, spherical cages, or heterogeneous macromolecular complexes, which are ubiquitous across living systems and in materials design. The disadvantage of these high-resolution methods is their increased computational cost. Here we present a parallel implementation of the particle-based NERDSS software using the Message Passing Interface (MPI) and spatial domain decomposition, achieving close to linear scaling for up to 96 processors in the largest simulation systems. The scalability of parallel NERDSS is evaluated for bimolecular reactions in 3D and 2D, for self-assembly of trimeric and hexameric complexes, and for protein lattice assembly from 3D to 2D, with all parallel test cases producing accurate solutions. We demonstrate how parallel efficiency depends on the system size, the reaction network, and the limiting timescales of the system, showing optimal scaling only for smaller assemblies with slower timescales. The formation of very large assemblies represents a challenge in evaluating reaction updates across processors, and here we restrict assembly sizes to below the spatial decomposition size. We provide the parallel NERDSS code open source, with detailed documentation for developers and extension to other particle-based reaction-diffusion software.

Phospho-signaling couples polar asymmetry and proteolysis within a membraneless microdomain in Caulobacter crescentus

Asymmetric cell division in bacteria is achieved through cell polarization, where regulatory proteins are directed to specific cell poles. In Caulobacter crescentus, both poles contain a membraneless microdomain, established by the polar assembly hub PopZ, through most of the cell cycle, yet many PopZ clients are unipolar and transiently localized. We find that PopZ's interaction with the response regulator CpdR is controlled by phosphorylation, via the histidine kinase CckA. Phosphorylated CpdR does not interact with PopZ and is not localized to cell poles. At poles where CckA acts as a phosphatase, dephosphorylated CpdR binds directly with PopZ and subsequently recruits ClpX, substrates, and other members of a protease complex to the cell pole. We also find that co-recruitment of protease components and substrates to polar microdomains enhances their coordinated activity. This study connects phospho-signaling with polar assembly and the activity of a protease that triggers cell cycle progression and cell differentiation.

bnglViz: online visualization of rule-based models

MOTIVATION

Rule-based modeling is a powerful method to describe and simulate interactions among multi-site molecules and multi-molecular species, accounting for the internal connectivity of molecules in chemical species. This modeling technique is implemented in BioNetGen software that is used by various tools and software frameworks, such as BioNetGen stand-alone software, NFSim simulation engine, Virtual Cell simulation and modeling framework, SmolDyn and PySB software tools. These tools exchange models using BioNetGen scripting language (BNGL). Until now, there was no online visualization of such rule-based models. Modelers and researchers reading the manuscripts describing rule-based models had to learn BNGL scripting or master one of these tools to understand the models.

RESULTS

Here, we introduce bnglViz, an online platform for visualizing BNGL files as graphical cartoons, empowering researchers to grasp the nuances of rule-based models swiftly and efficiently, and making the exploration of complex biological systems more accessible than ever before. The produced visualizations can be used as supplemental figures in publications or as a way to annotate BNGL models on web repositories.

AVAILABILITY AND IMPLEMENTATION

Available at https://bnglviz.github.io/.

Contribution of different macromolecules to the diffusion of a 40 nm particle in Escherichia coli

Due to the high concentration of proteins, nucleic acids, and other macromolecules, the bacterial cytoplasm is typically described as a crowded environment. However, the extent to which each of these macromolecules individually affects the mobility of macromolecular complexes, and how this depends on growth conditions, is presently unclear. In this study, we sought to quantify the crowding experienced by an exogenous 40 nm fluorescent particle in the cytoplasm of E. coli under different growth conditions. By performing single-particle tracking measurements in cells selectively depleted of DNA and/or mRNA, we determined the contribution to crowding of mRNA, DNA, and remaining cellular components, i.e., mostly proteins and ribosomes. To estimate this contribution to crowding, we quantified the difference of the particle's diffusion coefficient in conditions with and without those macromolecules. We found that the contributions of the three classes of components were of comparable magnitude, being largest in the case of proteins and ribosomes. We further found that the contributions of mRNA and DNA to crowding were significantly larger than expected based on their volumetric fractions alone. Finally, we found that the crowding contributions change only slightly with the growth conditions. These results reveal how various cellular components partake in crowding of the cytoplasm and the consequences this has for the mobility of large macromolecular complexes.

MCell4 with BioNetGen: A Monte Carlo simulator of rule-based reaction-diffusion systems with Python interface

Biochemical signaling pathways in living cells are often highly organized into spatially segregated volumes, membranes, scaffolds, subcellular compartments, and organelles comprising small numbers of interacting molecules. At this level of granularity stochastic behavior dominates, well-mixed continuum approximations based on concentrations break down and a particle-based approach is more accurate and more efficient. We describe and validate a new version of the open-source MCell simulation program (MCell4), which supports generalized 3D Monte Carlo modeling of diffusion and chemical reaction of discrete molecules and macromolecular complexes in solution, on surfaces representing membranes, and combinations thereof. The main improvements in MCell4 compared to the previous versions, MCell3 and MCell3-R, include a Python interface and native BioNetGen reaction language (BNGL) support. MCell4's Python interface opens up completely new possibilities for interfacing with external simulators to allow creation of sophisticated event-driven multiscale/multiphysics simulations. The native BNGL support, implemented through a new open-source library libBNG (also introduced in this paper), provides the capability to run a given BNGL model spatially resolved in MCell4 and, with appropriate simplifying assumptions, also in the BioNetGen simulation environment, greatly accelerating and simplifying model validation and comparison.

Cell4D: a general purpose spatial stochastic simulator for cellular pathways

BACKGROUND

With the generation of vast compendia of biological datasets, the challenge is how best to interpret 'omics data alongside biochemical and other small-scale experiments to gain meaningful biological insights. Key to this challenge are computational methods that enable domain-users to generate novel hypotheses that can be used to guide future experiments. Of particular interest are flexible modeling platforms, capable of simulating a diverse range of biological systems with low barriers of adoption to those with limited computational expertise.

RESULTS

We introduce Cell4D, a spatial-temporal modeling platform combining a robust simulation engine with integrated graphics visualization, a model design editor, and an underlying XML data model capable of capturing a variety of cellular functions. Cell4D provides an interactive visualization mode, allowing intuitive feedback on model behavior and exploration of novel hypotheses, together with a non-graphics mode, compatible with high performance cloud compute solutions, to facilitate generation of statistical data. To demonstrate the flexibility and effectiveness of Cell4D, we investigate the dynamics of CEACAM1 localization in T-cell activation. We confirm the importance of Ca2+ microdomains in activating calmodulin and highlight a key role of activated calmodulin on the surface expression of CEACAM1. We further show how lymphocyte-specific protein tyrosine kinase can help regulate this cell surface expression and exploit spatial modeling features of Cell4D to test the hypothesis that lipid rafts regulate clustering of CEACAM1 to promote trans-binding to neighbouring cells.

CONCLUSIONS

Through demonstrating its ability to test and generate hypotheses, Cell4D represents an effective tool to help integrate knowledge across diverse, large and small-scale datasets.

Single-protein Diffusion in the Periplasm of Escherichia coli

The width of the periplasmic space of Gram-negative bacteria is only about 25-30 nm along the long axis of the cell, which affects free diffusion of (macro)molecules. We have performed single-particle displacement measurements and diffusion simulation studies to determine the impact of confinement on the apparent mobility of proteins in the periplasm of Escherichia coli. The diffusion of a reporter protein and of OsmY, an osmotically regulated periplasmic protein, is characterized by a fast and slow component regardless of the osmotic conditions. The diffusion coefficient of the fast fraction increases upon osmotic upshift, in agreement with a decrease in macromolecular crowding of the periplasm, but the mobility of the slow (immobile) fraction is not affected by the osmotic stress. We observe that the confinement created by the inner and outer membranes results in a lower apparent diffusion coefficient, but this can only partially explain the slow component of diffusion in the particle displacement measurements, suggesting that a fraction of the proteins is hindered in its mobility by large periplasmic structures. Using particle-based simulations, we have determined the confinement effect on the apparent diffusion coefficient of the particles for geometries akin the periplasmic space of Gram-negative bacteria.

Technologies for whole-cell modeling: Genome-wide reconstruction of a cell in silico

With advances in high-throughput, large-scale in vivo measurement and genome modification techniques at the single-nucleotide level, there is an increasing demand for the development of new technologies for the flexible design and control of cellular systems. Computer-aided design is a powerful tool to design new cells. Whole-cell modeling aims to integrate various cellular subsystems, determine their interactions and cooperative mechanisms, and predict comprehensive cellular behaviors by computational simulations on a genome-wide scale. It has been applied to prokaryotes, yeasts, and higher eukaryotic cells, and utilized in a wide range of applications, including production of valuable substances, drug discovery, and controlled differentiation. Whole-cell modeling, consisting of several thousand elements with diverse scales and properties, requires innovative model construction, simulation, and analysis techniques. Furthermore, whole-cell modeling has been extended to multiple scales, including high-resolution modeling at the single-nucleotide and single-amino acid levels and multicellular modeling of tissues and organs. This review presents an overview of the current state of whole-cell modeling, discusses the novel computational and experimental technologies driving it, and introduces further developments toward multihierarchical modeling on a whole-genome scale.

Design principles of Cdr2 node patterns in fission yeast cells

Pattern-forming networks have diverse roles in cell biology. Rod-shaped fission yeast cells use pattern formation to control the localization of mitotic signaling proteins and the cytokinetic ring. During interphase, the kinase Cdr2 forms membrane-bound multiprotein complexes termed nodes, which are positioned in the cell middle due in part to the node inhibitor Pom1 enriched at cell tips. Node positioning is important for timely cell cycle progression and positioning of the cytokinetic ring. Here, we combined experimental and modeling approaches to investigate pattern formation by the Pom1-Cdr2 system. We found that Cdr2 nodes accumulate near the nucleus, and Cdr2 undergoes nucleocytoplasmic shuttling when cortical anchoring is reduced. We generated particle-based simulations based on tip inhibition, nuclear positioning, and cortical anchoring. We tested model predictions by investigating Pom1-Cdr2 localization patterns after perturbing each positioning mechanism, including in both anucleate and multinucleated cells. Experiments show that tip inhibition and cortical anchoring alone are sufficient for the assembly and positioning of nodes in the absence of the nucleus, but that the nucleus and Pom1 facilitate the formation of unexpected node patterns in multinucleated cells. These findings have implications for spatial control of cytokinesis by nodes and for spatial patterning in other biological systems.

Simulation-based Reconstructed Diffusion unveils the effect of aging on protein diffusion in Escherichia coli

We have developed Simulation-based Reconstructed Diffusion (SbRD) to determine diffusion coefficients corrected for confinement effects and for the bias introduced by two-dimensional models describing a three-dimensional motion. We validate the method on simulated diffusion data in three-dimensional cell-shaped compartments. We use SbRD, combined with a new cell detection method, to determine the diffusion coefficients of a set of native proteins in Escherichia coli. We observe slower diffusion at the cell poles than in the nucleoid region of exponentially growing cells, which is independent of the presence of polysomes. Furthermore, we show that the newly formed pole of dividing cells exhibits a faster diffusion than the old one. We hypothesize that the observed slowdown at the cell poles is caused by the accumulation of aggregated or damaged proteins, and that the effect is asymmetric due to cell aging.

Phospho-signaling couples polar asymmetry and proteolysis within a membraneless microdomain in C. crescentus

Asymmetric cell division in bacteria is achieved through cell polarization, where regulatory proteins are directed to specific cell poles. Curiously, both poles contain a membraneless microdomain, established by the polar assembly hub PopZ, through most of the cell cycle, yet many PopZ clients are unipolar and transiently localized. We find that PopZ's interaction with the response regulator CpdR is controlled by phosphorylation, via the histidine kinase CckA. Phosphorylated CpdR does not interact with PopZ and is not localized to cell poles. At poles where CckA acts as a phosphatase, de-phosphorylated CpdR binds directly with PopZ and subsequently recruits ClpX, substrates, and other members of a protease complex to the cell pole. We also find that co-recruitment of protease components and substrates to polar microdomains enhances their coordinated activity. This study connects phosphosignaling with polar assembly and the activity of a protease that triggers cell cycle progression and cell differentiation.

Design principles of Cdr2 node patterns in fission yeast cells

Pattern forming networks have diverse roles in cell biology. Rod-shaped fission yeast cells use pattern formation to control the localization of mitotic signaling proteins and the cytokinetic ring. During interphase, the kinase Cdr2 forms membrane-bound multiprotein complexes termed nodes, which are positioned in the cell middle due in part to the node inhibitor Pom1 enriched at cell tips. Node positioning is important for timely cell cycle progression and positioning of the cytokinetic ring. Here, we combined experimental and modeling approaches to investigate pattern formation by the Pom1-Cdr2 system. We found that Cdr2 nodes accumulate near the nucleus, and Cdr2 undergoes nucleocytoplasmic shuttling when cortical anchoring is reduced. We generated particle-based simulations based on tip inhibition, nuclear positioning, and cortical anchoring. We tested model predictions by investigating Pom1-Cdr2 localization patterns after perturbing each positioning mechanism, including in both anucleate and multinucleated cells. Experiments show that tip inhibition and cortical anchoring alone are sufficient for the assembly and positioning of nodes in the absence of the nucleus, but that the nucleus and Pom1 facilitate the formation of unexpected node patterns in multinucleated cells. These findings have implications for spatial control of cytokinesis by nodes and for spatial patterning in other biological systems.

Implications of different membrane compartmentalization models in particle-based in silico studies

Studying membrane dynamics is important to understand the cellular response to environmental stimuli. A decisive spatial characteristic of the plasma membrane is its compartmental structure created by the actin-based membrane-skeleton (fences) and anchored transmembrane proteins (pickets). Particle-based reaction-diffusion simulation of the membrane offers a suitable temporal and spatial resolution to analyse its spatially heterogeneous and stochastic dynamics. Fences have been modelled via hop probabilities, potentials or explicit picket fences. Our study analyses the different approaches' constraints and their impact on simulation results and performance. Each of the methods comes with its own constraints; the picket fences require small timesteps, potential fences might induce a bias in diffusion in crowded systems, and probabilistic fences, in addition to carefully scaling the probability with the timesteps, induce higher computational costs for each propagation step.

GB1 Dimerization in Crowders: A Multiple Resolution Approach

In-cell protein-protein association, which is crucial in enzyme catalysis and polymerization, occurs in an environment that is highly heterogeneous and crowded. The crowder molecules exclude the reactant molecules from occupying certain regions of the cell, resulting in changes in the reaction thermodynamics and kinetics. Recent studies, both experiment and simulations, revealed that the nature of the interaction between crowder and protein species, in particular the soft interactions, plays an important role in crowder induced effects on protein association. To this end, from a simulation perspective, it is important to decipher the level of structural resolution in a protein-crowder model that can faithfully capture the influence of crowding on protein association. Here, we investigate the dimerization of model system GB1 in the presence of lysozyme crowders at two structural resolutions. The lower resolution model assumes both protein and crowder species as spherical beads, similar to the analytical scaled particle theory model, whereas the higher resolution model retains residue specific structural details for protein and crowder species. From the higher resolution model, it is found that GB1 dimer formation is destabilized in the presence of lysozyme crowders, and the destabilization is more for the side-by-side dimer compared to the domain-swapped dimer, in qualitative agreement with experimental findings. However, the low resolution CG model predicts stabilization of the dimers in the presence of the lysozyme crowder, similar to the SPT model. Our results indicate a nontrivial role of the choice of model resolution in computer simulation studies investigating crowder induced effects.

Rapid Particle-Based Simulations of Cellular Signalling with the FLAME-Accelerated Signalling Tool (FaST) and GPUs

Cellular signalling is a vital process in living organisms for coordinating highly diverse responses to various stimuli. Particle-based modelling excels in its ability to model complex features of cellular signalling pathways including stochasticity, spatial effects, and heterogeneity, thus improving our understanding of critical decision processes in biology. Yet, particle-based modelling is computationally prohibitive to implement. We recently developed FaST (FLAME-accelerated signalling tool), a software tool that harnesses the power of high-performance computation to reduce the computational burden of particle-based modelling. In particular, employing the unique massively parallel architecture of graphic processing units (GPUs) provided extreme speed ups of simulations by >650-fold. In this chapter, we provide a step-by-step walkthrough of how to use FaST to create GPU-accelerated simulations of a simple cellular signalling network. We further explore how the flexibility of FaST can be used to implement entirely customized simulations while still including the intrinsic speed up advantages of GPU-based parallelization.

The dawn of interoperating spatial models in cell biology

Spatial simulations are becoming an increasingly ubiquitous component in the cycle of discovery, experimentation, and communication across the sciences. In cell biology, many researchers share a vision of developing multiscale models that recapitulate observable behaviors spanning from atoms to cells to tissues. For this dream to become a reality, however, simulation technologies must provide a means for integration and interoperability as they advance. Already, the field has developed numerous methods that span scales of length, time, and complexity to create an extensive body of effective simulation approaches, and although these approaches rarely interoperate, they collectively cover a large spectrum of knowledge that future models may handle in a more unified manner. Here, we discuss the importance of making the data, workflows, and outputs of spatial simulations shareable and interoperable; and how democratization could encourage diverse biologists to participate more easily in developing models to advance our understanding of biological systems.

Protein diffusion in Escherichia coli cytoplasm scales with the mass of the complexes and is location dependent

We analyze the structure of the cytoplasm by performing single-molecule displacement mapping on a diverse set of native cytoplasmic proteins in exponentially growing Escherichia coli. We evaluate the method for application in small compartments and find that confining effects of the cell membrane affect the diffusion maps. Our analysis reveals that protein diffusion at the poles is consistently slower than in the center of the cell, i.e., to an extent greater than the confining effect of the cell membrane. We also show that the diffusion coefficient scales with the mass of the used probes, taking into account the oligomeric state of the proteins, while parameters such as native protein abundance or the number of protein-protein interactions do not correlate with the mobility of the proteins. We argue that our data paint the prokaryotic cytoplasm as a compartment with subdomains in which the diffusion of macromolecules changes with the perceived viscosity.

Stochastic model of ERK-mediated progesterone receptor translocation, clustering and transcriptional activity

Progesterone receptor (PR) transcriptional activity is a key factor in the differentiation of the uterine endometrium. By consequence, progestin has been identified as an important treatment modality for endometrial cancer. PR transcriptional activity is controlled by extracellular-signal-regulated kinase (ERK) mediated phosphorylation, downstream of growth factor receptors such as EGFR. However, phosphorylation of PR also targets it for ubiquitination and destruction in the proteasome. Quantitative studies of these opposing roles are much needed toward validation of potential new progestin-based therapeutics. In this work, we propose a spatial stochastic model to study the effects of the opposing roles for PR phosphorylation on the levels of active transcription factor. Our numerical simulations confirm earlier in vitro experiments in endometrial cancer cell lines, identifying clustering as a mechanism that amplifies the ability of progesterone receptors to influence gene transcription. We additionally show the usefulness of a statistical method we developed to quantify and control variations in stochastic simulations in general biochemical systems, assisting modelers in defining minimal but meaningful numbers of simulations while guaranteeing outputs remain within a pre-defined confidence level.

Detailed balance for particle models of reversible reactions in bounded domains

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.

GeomBD3: Brownian Dynamics Simulation Software for Biological and Engineered Systems

GeomBD3 is a robust Brownian dynamics simulation package designed to easily handle natural or engineered systems in diverse environments and arrangements. The software package described herein allows users to design, execute, and analyze BD simulations. The simulations use all-atom, rigid molecular models that diffuse according to overdamped Langevin dynamics and interact through electrostatic, Lennard-Jones, and ligand desolvation potentials. The program automatically calculates molecular association rates, surface residence times, and association statistics for any number of user-defined association criteria. Users can also extract molecular association pathways, diffusion coefficients, intermolecular interaction energies, intermolecular contact probability maps, and more using the provided supplementary analysis scripts. We detail the use of the package from start to finish and apply it to a protein-ligand system and a large nucleic acid biosensor. GeomBD3 provides a versatile tool for researchers from various disciplines that can aid in rational design of engineered systems or play an explanatory role as a complement to experiments. GeomBD version 3 is available on our website at http://chemcha-gpu0.ucr.edu/geombd3/ and KBbox at https://kbbox.h-its.org/toolbox/methods/molecular-simulation/geombd/.

The Role of in silico Research in Developing Nanoparticle-Based Therapeutics

Nanoparticles (NPs) hold great potential as therapeutics, particularly in the realm of drug delivery. They are effective at functional cargo delivery and offer a great degree of amenability that can be used to offset toxic side effects or to target drugs to specific regions in the body. However, there are many challenges associated with the development of NP-based drug formulations that hamper their successful clinical translation. Arguably, the most significant barrier in the way of efficacious NP-based drug delivery systems is the tedious and time-consuming nature of NP formulation—a process that needs to account for downstream effects, such as the onset of potential toxicity or immunogenicity, in vivo biodistribution and overall pharmacokinetic profiles, all while maintaining desirable therapeutic outcomes. Computational and AI-based approaches have shown promise in alleviating some of these restrictions. Via predictive modeling and deep learning, in silico approaches have shown the ability to accurately model NP-membrane interactions and cellular uptake based on minimal data, such as the physicochemical characteristics of a given NP. More importantly, machine learning allows computational models to predict how specific changes could be made to the physicochemical characteristics of a NP to improve functional aspects, such as drug retention or endocytosis. On a larger scale, they are also able to predict the in vivo pharmacokinetics of NP-encapsulated drugs, predicting aspects such as circulatory half-life, toxicity, and biodistribution. However, the convergence of nanomedicine and computational approaches is still in its infancy and limited in its applicability. The interactions between NPs, the encapsulated drug and the body form an intricate network of interactions that cannot be modeled with absolute certainty. Despite this, rapid advancements in the area promise to deliver increasingly powerful tools capable of accelerating the development of advanced nanoscale therapeutics. Here, we describe computational approaches that have been utilized in the field of nanomedicine, focusing on approaches for NP design and engineering.

Simulation of receptor triggering by kinetic segregation shows role of oligomers and close-contacts

The activation of T cells, key players of the immune system, involves local evacuation of phosphatase CD45 from a region of the T cell's surface, segregating it from the T cell receptor. What drives this evacuation? In the presence of antigen, what ensures evacuation happens in the subsecond timescales necessary to initiate signaling? In the absence of antigen, what mechanisms ensure that evacuation does not happen spontaneously, which could cause signaling errors? Phenomena known to influence spatial organization of CD45 or similar surface molecules include diffusive motion in the lipid bilayer, oligomerization reactions, and mechanical compression against a nearby surface, such as that of the cell presenting the antigen. Computer simulations can investigate hypothesized spatiotemporal mechanisms of T cell signaling. The challenge to computational studies of evacuation is that the base process, spontaneous evacuation by simple diffusion, is in the extreme rare event limit, meaning direct stochastic simulation is unfeasible. Here, we combine particle-based spatial stochastic simulation with the weighted ensemble method for rare events to compute the mean first passage time for cell surface availability by surface reorganization of CD45. We confirm mathematical estimates that, at physiological concentrations, spontaneous evacuation is extremely rare, roughly 300 years. We find that dimerization decreases the time required for evacuation. A weak bimolecular interaction (dissociation constant estimate 460 μM) is sufficient for an order of magnitude reduction of spontaneous evacuation times, and oligomerization to hexamers reduces times to below 1 s. This introduces a mechanism whereby externally induced CD45 oligomerization could significantly modify T cell function. For large regions of close contact, such as those induced by large microvilli, molecular size and compressibility imply a nonzero reentry probability of 60%, decreasing evacuation times. Simulations show that these reduced evacuation times are still unrealistically long (even with a fourfold variation centered around previous estimates of parameters), suggesting that a yet-to-be-described mechanism, besides compressional exclusion at a close contact, drives evacuation.

Python interfaces for the Smoldyn simulator

MOTIVATION

Smoldyn is a particle-based biochemical simulator that is frequently used for systems biology and biophysics research. Previously, users could only define models using text-based input or a C/C++ application programming interface (API), which were convenient, but limited extensibility.

RESULTS

We added a Python API to Smoldyn to improve integration with other software tools, such as Jupyter notebooks, other Python code libraries and other simulators. It includes low-level functions that closely mimic the existing C/C++ API and higher-level functions that are more convenient to use. These latter functions follow modern object-oriented Python conventions.

AVAILABILITY AND IMPLEMENTATION

Smoldyn is open source and free, available at http://www.smoldyn.org and can be installed with the Python package manager pip. It runs on Mac, Windows and Linux.

Documentation is available at http://www.smoldyn.org/SmoldynManual.pdf and https://smoldyn.readthedocs.io/en/latest/python/api.html.

Trapping or slowing the diffusion of T cell receptors at close contacts initiates T cell signaling

T cell activation is initiated by T cell receptor (TCR) phosphorylation. This requires the local depletion of large receptor-type phosphatases from "close contacts" formed when T cells interact with surfaces presenting agonistic TCR ligands, but exactly how the ligands potentiate signaling is unclear. It has been proposed that TCR ligands could enhance receptor phosphorylation and signaling just by holding TCRs in phosphatase-depleted close contacts, but this has not been directly tested. We devised simple methods to move the TCR in and out of close contacts formed by T cells interacting with supported lipid bilayers (SLBs) and to slow the receptor's diffusion in the contacts, using a series of anti-CD3ε Fab- and ligand-based adducts of the receptor. TCRs engaging a Fab extended with the large extracellular region of CD45 were excluded from contacts and produced no signaling. Conversely, allowing the extended Fab to become tethered to the SLB trapped the TCR in the close contacts, leading to very strong signaling. Importantly, attaching untethered anti-CD3ε Fab or peptide/MHC ligands, each of which were largely inactive in solution but both of which reduced TCR diffusion in close contacts approximately fivefold, also initiated signaling during cell/SLB contact. Our findings indicate that holding TCRs in close contacts or simply slowing their diffusion in phosphatase-depleted regions of the cell surface suffices to initiate signaling, effects we could reproduce in single-particle stochastic simulations. Our study shows that the TCR is preconfigured for signaling in a way that allows it to be triggered by ligands acting simply as receptor "traps."

Programmatic modeling for biological systems

Computational modeling has become an established technique to encode mathematical representations of cellular processes and gain mechanistic insights that drive testable predictions. These models are often constructed using graphical user interfaces or domain-specific languages, with community standards used for interchange. Models undergo steady state or dynamic analysis, which can include simulation and calibration within a single application, or transfer across various tools. Here, we describe a novel programmatic modeling paradigm, whereby modeling is augmented with software engineering best practices. We focus on Python - a popular programming language with a large scientific package ecosystem. Models can be encoded as programs, adding benefits such as modularity, testing, and automated documentation generators, while still being extensible and exportable to standardized formats for use with external tools if desired. Programmatic modeling is a key technology to enable collaborative model development and enhance dissemination, transparency, and reproducibility.

Molecular Mechanics Study of Flow and Surface Influence in Ligand-Protein Association

Ligand–protein association is the first and critical step for many biological and chemical processes. This study investigated the molecular association processes under different environments. In biology, cells have different compartments where ligand–protein binding may occur on a membrane. In experiments involving ligand–protein binding, such as the surface plasmon resonance and continuous flow biosynthesis, a substrate flow and surface are required in experimental settings. As compared with a simple binding condition, which includes only the ligand, protein, and solvent, the association rate and processes may be affected by additional ligand transporting forces and other intermolecular interactions between the ligand and environmental objects. We evaluated these environmental factors by using a ligand xk263 binding to HIV protease (HIVp) with atomistic details. Using Brownian dynamics simulations, we modeled xk263 and HIVp association time and probability when a system has xk263 diffusion flux and a non-polar self-assembled monolayer surface. We also examined different protein orientations and accessible surfaces for xk263. To allow xk263 to access to the dimer interface of immobilized HIVp, we simulated the system by placing the protein 20Å above the surface because immobilizing HIVp on a surface prevented xk263 from contacting with the interface. The non-specific interactions increased the binding probability while the association time remained unchanged. When the xk263 diffusion flux increased, the effective xk263 concentration around HIVp, xk263–HIVp association time and binding probability decreased non-linearly regardless of interacting with the self-assembled monolayer surface or not. The work sheds light on the effects of the solvent flow and surface environment on ligand–protein associations and provides a perspective on experimental design.

Mechanistic dissection of spatial organization in NF-κB signaling pathways by hybrid simulations

The nuclear factor kappa-light-chain-enhancer of activated B cells (NF-κB) is one of the most important transcription factors involved in the regulation of inflammatory signaling pathways. Inappropriate activation of these pathways has been linked to autoimmunity and cancers. Emerging experimental evidences have been showing the existence of elaborate spatial organizations for various molecular components in the pathways. One example is the scaffold protein tumor necrosis factor receptor associated factor (TRAF). While most TRAF proteins form trimeric quaternary structure through their coiled-coil regions, the N-terminal region of some members in the family can further be dimerized. This dimerization of TRAF trimers can drive them into higher-order clusters as a response to receptor stimulation, which functions as a spatial platform to mediate the downstream poly-ubiquitination. However, the molecular mechanism underlying the TRAF protein clustering and its functional impacts are not well-understood. In this article, we developed a hybrid simulation method to tackle this problem. The assembly of TRAF-based signaling platform at the membrane-proximal region is modeled with spatial resolution, while the dynamics of downstream signaling network, including the negative feedbacks through various signaling inhibitors, is simulated as stochastic chemical reactions. These two algorithms are further synchronized under a multiscale simulation framework. Using this computational model, we illustrated that the formation of TRAF signaling platform can trigger an oscillatory NF-κB response. We further demonstrated that the temporal patterns of downstream signal oscillations are closely regulated by the spatial factors of TRAF clustering, such as the geometry and energy of dimerization between TRAF trimers. In general, our study sheds light on the basic mechanism of NF-κB signaling pathway and highlights the functional importance of spatial regulation within the pathway. The simulation framework also showcases its potential of application to other signaling pathways in cells.

Exploratory polarization facilitates mating partner selection in Saccharomyces cerevisiae

Yeast decode pheromone gradients to locate mating partners, providing a model for chemotropism. How yeast polarize toward a single partner in crowded environments is unclear. Initially, cells often polarize in unproductive directions, but then they relocate the polarity site until two partners’ polarity sites align, whereupon the cells “commit” to each other by stabilizing polarity to promote fusion. Here we address the role of the early mobile polarity sites. We found that commitment by either partner failed if just one partner was defective in generating, orienting, or stabilizing its mobile polarity sites. Mobile polarity sites were enriched for pheromone receptors and G proteins, and we suggest that such sites engage in an exploratory search of the local pheromone landscape, stabilizing only when they detect elevated pheromone levels. Mobile polarity sites were also enriched for pheromone secretion factors, and simulations suggest that only focal secretion at polarity sites would produce high pheromone concentrations at the partner’s polarity site, triggering commitment.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

Executable biochemical space for specification and analysis of biochemical systems

Computational systems biology provides multiple formalisms for modelling of biochemical processes among which the rule-based approach is one of the most suitable. Its main advantage is a compact and precise mechanistic description of complex processes. However, state-of-the-art rule-based languages still suffer several shortcomings that limit their use in practice. In particular, the elementary (low-level) syntax and semantics of rule-based languages complicate model construction and maintenance for users outside computer science. On the other hand, mathematical models based on differential equations (ODEs) still make the most typical used modelling framework. In consequence, robust re-interpretation and integration of models are difficult, thus making the systems biology paradigm technically challenging. Though several high-level languages have been developed at the top of rule-based principles, none of them provides a satisfactory and complete solution for semi-automated description and annotation of heterogeneous biophysical processes integrated at the cellular level. We present the second generation of a rule-based language called Biochemical Space Language (BCSL) that combines the advantages of different approaches and thus makes an effort to overcome several problems of existing solutions. BCSL relies on the formal basis of the rule-based methodology while preserving user-friendly syntax of plain chemical equations. BCSL combines the following aspects: the level of abstraction that hides structural and quantitative details but yet gives a precise mechanistic view of systems dynamics; executable semantics allowing formal analysis and consistency checking at the level of the language; universality allowing the integration of different biochemical mechanisms; scalability and compactness of the specification; hierarchical specification and composability of chemical entities; and support for genome-scale annotation.

NERDSS: A Nonequilibrium Simulator for Multibody Self-Assembly at the Cellular Scale

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reaction-diffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.

The FLAME-accelerated signalling tool (FaST) for facile parallelisation of flexible agent-based models of cell signalling

Agent-based modelling is particularly adept at modelling complex features of cell signalling pathways, where heterogeneity, stochastic and spatial effects are important, thus increasing our understanding of decision processes in biology in such scenarios. However, agent-based modelling often is computationally prohibitive to implement. Parallel computing, either on central processing units (CPUs) or graphical processing units (GPUs), can provide a means to improve computational feasibility of agent-based applications but generally requires specialist coding knowledge and extensive optimisation. In this paper, we address these challenges through the development and implementation of the FLAME-accelerated signalling tool (FaST), a software that permits easy creation and parallelisation of agent-based models of cell signalling, on CPUs or GPUs. FaST incorporates validated new agent-based methods, for accurate modelling of reaction kinetics and, as proof of concept, successfully converted an ordinary differential equation (ODE) model of apoptosis execution into an agent-based model. We finally parallelised this model through FaST on CPUs and GPUs resulting in an increase in performance of 5.8× (16 CPUs) and 53.9×, respectively. The FaST takes advantage of the communicating X-machine approach used by FLAME and FLAME GPU to allow easy alteration or addition of functionality to parallel applications, but still includes inherent parallelisation optimisation. The FaST, therefore, represents a new and innovative tool to easily create and parallelise bespoke, robust, agent-based models of cell signalling.

pSpatiocyte: a high-performance simulator for intracellular reaction-diffusion systems

Background

Studies using quantitative experimental methods have shown that intracellular spatial distribution of molecules plays a central role in many cellular systems. Spatially resolved computer simulations can integrate quantitative data from these experiments to construct physically accurate models of the systems. Although computationally expensive, microscopic resolution reaction-diffusion simulators, such as Spatiocyte can directly capture intracellular effects comprising diffusion-limited reactions and volume exclusion from crowded molecules by explicitly representing individual diffusing molecules in space. To alleviate the steep computational cost typically associated with the simulation of large or crowded intracellular compartments, we present a parallelized Spatiocyte method called pSpatiocyte.

Results

The new high-performance method employs unique parallelization schemes on hexagonal close-packed (HCP) lattice to efficiently exploit the resources of common workstations and large distributed memory parallel computers. We introduce a coordinate system for fast accesses to HCP lattice voxels, a parallelized event scheduler, a parallelized Gillespie’s direct-method for unimolecular reactions, and a parallelized event for diffusion and bimolecular reaction processes. We verified the correctness of pSpatiocyte reaction and diffusion processes by comparison to theory. To evaluate the performance of pSpatiocyte, we performed a series of parallelized diffusion runs on the RIKEN K computer. In the case of fine lattice discretization with low voxel occupancy, pSpatiocyte exhibited 74% parallel efficiency and achieved a speedup of 7686 times with 663552 cores compared to the runtime with 64 cores. In the weak scaling performance, pSpatiocyte obtained efficiencies of at least 60% with up to 663552 cores. When executing the Michaelis-Menten benchmark model on an eight-core workstation, pSpatiocyte required 45- and 55-fold shorter runtimes than Smoldyn and the parallel version of ReaDDy, respectively. As a high-performance application example, we study the dual phosphorylation-dephosphorylation cycle of the MAPK system, a typical reaction network motif in cell signaling pathways.

Conclusions

pSpatiocyte demonstrates good accuracies, fast runtimes and a significant performance advantage over well-known microscopic particle methods in large-scale simulations of intracellular reaction-diffusion systems. The source code of pSpatiocyte is available at https://spatiocyte.org.

Potential based, spatial simulation of dynamically nested particles

BACKGROUND

To study cell biological phenomena which depend on diffusion, active transport processes, or the locations of species, modeling and simulation studies need to take space into account. To describe the system as a collection of discrete objects moving and interacting in continuous space, various particle-based reaction diffusion simulators for cell-biological system have been developed. So far the focus has been on particles as solid spheres or points. However, spatial dynamics might happen at different organizational levels, such as proteins, vesicles or cells with interrelated dynamics which requires spatial approaches that take this multi-levelness of cell biological systems into account.

RESULTS

Based on the perception of particles forming hollow spheres, ML-Force contributes to the family of particle-based simulation approaches: in addition to excluded volumes and forces, it also supports compartmental dynamics and relating dynamics between different organizational levels explicitly. Thereby, compartmental dynamics, e.g., particles entering and leaving other particles, and bimolecular reactions are modeled using pair-wise potentials (forces) and the Langevin equation. In addition, forces that act independently of other particles can be applied to direct the movement of particles. Attributes and the possibility to define arbitrary functions on particles, their attributes and content, to determine the results and kinetics of reactions add to the expressiveness of ML-Force. Its implementation comprises a rudimentary rule-based embedded domain-specific modeling language for specifying models and a simulator for executing models continuously. Applications inspired by cell biological models from literature, such as vesicle transport or yeast growth, show the value of the realized features. They facilitate capturing more complex spatial dynamics, such as the fission of compartments or the directed movement of particles, and enable the integration of non-spatial intra-compartmental dynamics as stochastic events.

CONCLUSIONS

By handling all dynamics based on potentials (forces) and the Langevin equation, compartmental dynamics, such as dynamic nesting, fusion and fission of compartmental structures are handled continuously and are seamlessly integrated with traditional particle-based reaction-diffusion dynamics within the cell. Thereby, attributes and arbitrary functions allow to flexibly describe diverse spatial phenomena, and relate dynamics across organizational levels. Also they prove crucial in modeling intra-cellular or intra-compartmental dynamics in a non-spatial manner, and, thus, to abstract from spatial dynamics, on demand which increases the range of multi-compartmental processes that can be captured.

Diffusion-influenced reaction rates in the presence of pair interactions

The kinetics of bimolecular reactions in solution depends, among other factors, on intermolecular forces such as steric repulsion or electrostatic interaction. Microscopically, a pair of molecules first has to meet by diffusion before the reaction can take place. In this work, we establish an extension of Doi's volume reaction model to molecules interacting via pair potentials, which is a key ingredient for interacting-particle-based reaction-diffusion (iPRD) simulations. As a central result, we relate model parameters and macroscopic reaction rate constants in this situation. We solve the corresponding reaction-diffusion equation in the steady state and derive semi-analytical expressions for the reaction rate constant and the local concentration profiles. Our results apply to the full spectrum from well-mixed to diffusion-limited kinetics. For limiting cases, we give explicit formulas, and we provide a computationally inexpensive numerical scheme for the general case, including the intermediate, diffusion-influenced regime. The obtained rate constants decompose uniquely into encounter and formation rates, and we discuss the effect of the potential on both subprocesses, exemplified for a soft harmonic repulsion and a Lennard-Jones potential. The analysis is complemented by extensive stochastic iPRD simulations, and we find excellent agreement with the theoretical predictions.

Ratiometric GPCR signaling enables directional sensing in yeast

Accurate detection of extracellular chemical gradients is essential for many cellular behaviors. Gradient sensing is challenging for small cells, which can experience little difference in ligand concentrations on the up-gradient and down-gradient sides of the cell. Nevertheless, the tiny cells of the yeast Saccharomyces cerevisiae reliably decode gradients of extracellular pheromones to find their mates. By imaging the behavior of polarity factors and pheromone receptors, we quantified the accuracy of initial polarization during mating encounters. We found that cells bias the orientation of initial polarity up-gradient, even though they have unevenly distributed receptors. Uneven receptor density means that the gradient of ligand-bound receptors does not accurately reflect the external pheromone gradient. Nevertheless, yeast cells appear to avoid being misled by responding to the fraction of occupied receptors rather than simply the concentration of ligand-bound receptors. Such ratiometric sensing also serves to amplify the gradient of active G protein. However, this process is quite error-prone, and initial errors are corrected during a subsequent indecisive phase in which polarity clusters exhibit erratic mobile behavior.

Cells use surface receptors to decode spatial information from chemical gradients, but accurate decoding is hampered by small cell size and the presence of molecular noise. This study shows that yeast cells decode pheromone gradients by measuring the local ratio of bound to unbound receptors. This mechanism corrects for uneven receptor density at the surface and amplifies the gradient transmitted to downstream components.

An implicit lipid model for efficient reaction-diffusion simulations of protein binding to surfaces of arbitrary topology

Localization of proteins to a membrane is an essential step in a broad range of biological processes such as signaling, virion formation, and clathrin-mediated endocytosis. The strength and specificity of proteins binding to a membrane depend on the lipid composition. Single-particle reaction-diffusion methods offer a powerful tool for capturing lipid-specific binding to membrane surfaces by treating lipids explicitly as individual diffusible binding sites. However, modeling lipid particle populations is expensive. Here, we present an algorithm for reversible binding of proteins to continuum surfaces with implicit lipids, providing dramatic speed-ups to many body simulations. Our algorithm can be readily integrated into most reaction-diffusion software packages. We characterize changes to kinetics that emerge from explicit vs implicit lipids as well as surface adsorption models, showing excellent agreement between our method and the full explicit lipid model. Compared to models of surface adsorption, which couple together binding affinity and lipid concentration, our implicit lipid model decouples them to provide more flexibility for controlling surface binding properties and lipid inhomogeneity, thus reproducing binding kinetics and equilibria. Crucially, we demonstrate our method's application to membranes of arbitrary curvature and topology, modeled via a subdivision limit surface, again showing excellent agreement with explicit lipid simulations. Unlike adsorption models, our method retains the ability to bind lipids after proteins are localized to the surface (through, e.g., a protein-protein interaction), which can greatly increase the stability of multiprotein complexes on the surface. Our method will enable efficient cell-scale simulations involving proteins localizing to realistic membrane models, which is a critical step for predictive modeling and quantification of in vitro and in vivo dynamics.

Brownian Dynamics Simulations of Biological Molecules

Brownian dynamics (BD) is a technique for carrying out computer simulations of physical systems that are driven by thermal fluctuations. Biological systems at the macromolecular and cellular level, while falling in the gap between well-established atomic-level models and continuum models, are especially suitable for such simulations. We present a brief history, examples of important biological processes that are driven by thermal motion, and those that have been profitably studied by BD. We also present some of the challenges facing developers of algorithms and software, especially in the attempt to simulate larger systems more accurately and for longer times.

Rule-Based Modeling Using Wildcards in the Smoldyn Simulator

Many biological molecules exist in multiple variants, such as proteins with different posttranslational modifications, DNAs with different sequences, and phospholipids with different chain lengths. Representing these variants as distinct species, as most biochemical simulators do, leads to the problem that the number of species, and chemical reactions that interconvert them, typically increase combinatorially with the number of ways that the molecules can vary. This can be alleviated by "rule-based modeling methods," in which software generates the chemical reaction network from relatively simple "rules." This chapter presents a new approach to rule-based modeling. It is based on wildcards that match to species names, much as wildcards can match to file names in computer operating systems. It is much simpler to use than the formal rule-based modeling approaches developed previously but can lead to unintended consequences if not used carefully. This chapter demonstrates rule-based modeling with wildcards through examples for signaling systems, protein complexation, polymerization, nucleic acid sequence copying and mutation, the "SMILES" chemical notation, and others. The method is implemented in Smoldyn, a spatial and stochastic biochemical simulator, for both generate-first and on-the-fly expansion, meaning whether the reaction network is generated before or during the simulation.

Surface reaction-diffusion kinetics on lattice at the microscopic scale

Microscopic models of reaction-diffusion processes on the cell membrane can link local spatiotemporal effects to macroscopic self-organized patterns often observed on the membrane. Simulation schemes based on the microscopic lattice method (MLM) can model these processes at the microscopic scale by tracking individual molecules, represented as hard spheres, on fine lattice voxels. Although MLM is simple to implement and is generally less computationally demanding than off-lattice approaches, its accuracy and consistency in modeling surface reactions have not been fully verified. Using the Spatiocyte scheme, we study the accuracy of MLM in diffusion-influenced surface reactions. We derive the lattice-based bimolecular association rates for two-dimensional (2D) surface-surface reaction and one-dimensional (1D) volume-surface adsorption according to the Smoluchowski-Collins-Kimball model and random walk theory. We match the time-dependent rates on lattice with off-lattice counterparts to obtain the correct expressions for MLM parameters in terms of physical constants. The expressions indicate that the voxel size needs to be at least 0.6% larger than the molecule to accurately simulate surface reactions on triangular lattice. On square lattice, the minimum voxel size should be even larger, at 5%. We also demonstrate the ability of MLM-based schemes such as Spatiocyte to simulate a reaction-diffusion model that involves all dimensions: three-dimensional (3D) diffusion in the cytoplasm, 2D diffusion on the cell membrane, and 1D cytoplasm-membrane adsorption. With the model, we examine the contribution of the 2D reaction pathway to the overall reaction rate at different reactant diffusivity, reactivity, and concentrations.

ReaDDy 2: Fast and flexible software framework for interacting-particle reaction dynamics

Interacting-particle reaction dynamics (iPRD) combines the simulation of dynamical trajectories of interacting particles as in molecular dynamics (MD) simulations with reaction kinetics, in which particles appear, disappear, or change their type and interactions based on a set of reaction rules. This combination facilitates the simulation of reaction kinetics in crowded environments, involving complex molecular geometries such as polymers, and employing complex reaction mechanisms such as breaking and fusion of polymers. iPRD simulations are ideal to simulate the detailed spatiotemporal reaction mechanism in complex and dense environments, such as in signalling processes at cellular membranes, or in nano- to microscale chemical reactors. Here we introduce the iPRD software ReaDDy 2, which provides a Python interface in which the simulation environment, particle interactions and reaction rules can be conveniently defined and the simulation can be run, stored and analyzed. A C++ interface is available to enable deeper and more flexible interactions with the framework. The main computational work of ReaDDy 2 is done in hardware-specific simulation kernels. While the version introduced here provides single- and multi-threading CPU kernels, the architecture is ready to implement GPU and multi-node kernels. We demonstrate the efficiency and validity of ReaDDy 2 using several benchmark examples. ReaDDy 2 is available at the https://readdy.github.io/ website.

Center Finding in E. coli and the Role of Mathematical Modeling: Past, Present and Future

We review the key role played by mathematical modeling in elucidating two center-finding patterning systems in Escherichia coli: midcell division positioning by the MinCDE system and DNA partitioning by the ParABS system. We focus particularly on how, despite much experimental effort, these systems were simply too complex to unravel by experiments alone, and instead required key injections of quantitative, mathematical thinking. We conclude the review by analyzing the frequency of modeling approaches in microbiology over time. We find that while such methods are increasing in popularity, they are still probably heavily under-utilized for optimal progress on complex biological questions.

From membrane receptors to protein synthesis and actin cytoskeleton: Mechanisms underlying long lasting forms of synaptic plasticity

Synaptic plasticity, the activity dependent change in synaptic strength, forms the molecular foundation of learning and memory. Synaptic plasticity includes structural changes, with spines changing their size to accomodate insertion and removal of postynaptic receptors, which are correlated with functional changes. Of particular relevance for memory storage are the long lasting forms of synaptic plasticity which are protein synthesis dependent. Due to the importance of spine structural plasticity and protein synthesis, this review focuses on the signaling pathways that connect synaptic stimulation with regulation of protein synthesis and remodeling of the actin cytoskeleton. We also review computational models that implement novel aspects of molecular signaling in synaptic plasticity, such as the role of neuromodulators and spatial microdomains, as well as highlight the need for computational models that connect activation of memory kinases with spine actin dynamics.

The Interplay of Structural and Cellular Biophysics Controls Clustering of Multivalent Molecules

Dynamic molecular clusters are assembled through weak multivalent interactions and are platforms for cellular functions, especially receptor-mediated signaling. Clustering is also a prerequisite for liquid-liquid phase separation. It is not well understood, however, how molecular structure and cellular organization control clustering. Using coarse-grained kinetic Langevin dynamics, we performed computational experiments on a prototypical ternary system modeled after membrane-bound nephrin, the adaptor Nck1, and the actin nucleation promoting factor NWASP. Steady-state cluster size distributions favored stoichiometries that optimized binding (stoichiometry matching) but still were quite broad. At high concentrations, the system can be driven beyond the saturation boundary such that cluster size is limited only by the number of available molecules. This behavior would be predictive of phase separation. Domains close to binding sites sterically inhibited clustering much less than terminal domains because the latter effectively restrict access to the cluster interior. Increased flexibility of interacting molecules diminished clustering by shielding binding sites within compact conformations. Membrane association of nephrin increased the cluster size distribution in a density-dependent manner. These properties provide insights into how molecular ensembles function to localize and amplify cell signaling.

MCell-R: A Particle-Resolution Network-Free Spatial Modeling Framework

Spatial heterogeneity can have dramatic effects on the biochemical networks that drive cell regulation and decision-making. For this reason, a number of methods have been developed to model spatial heterogeneity and incorporated into widely used modeling platforms. Unfortunately, the standard approaches for specifying and simulating chemical reaction networks become untenable when dealing with multistate, multicomponent systems that are characterized by combinatorial complexity. To address this issue, we developed MCell-R, a framework that extends the particle-based spatial Monte Carlo simulator, MCell, with the rule-based model specification and simulation capabilities provided by BioNetGen and NFsim. The BioNetGen syntax enables the specification of biomolecules as structured objects whose components can have different internal states that represent such features as covalent modification and conformation and which can bind components of other molecules to form molecular complexes. The network-free simulation algorithm used by NFsim enables efficient simulation of rule-based models even when the size of the network implied by the biochemical rules is too large to enumerate explicitly, which frequently occurs in detailed models of biochemical signaling. The result is a framework that can efficiently simulate systems characterized by combinatorial complexity at the level of spatially resolved individual molecules over biologically relevant time and length scales.