The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an over-damped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of ATP hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review we present a wide range of analytical methods and models of intracellular transport. In the case of diffusive transport, we consider narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion. In the case of active transport, we consider Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasisteady-state reduction methods, and mean field approximations. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self-organization of subcellular structures.

The RNA binding protein TDP-43 forms intranuclear or cytoplasmic aggregates in age-related neurodegenerative diseases. In this study, we found that RNA binding–deficient TDP-43 (produced by neurodegeneration-causing mutations or posttranslational acetylation in its RNA recognition motifs) drove TDP-43 demixing into intranuclear liquid spherical shells with liquid cores. These droplets, which we named “anisosomes”, have shells that exhibit birefringence, thus indicating liquid crystal formation. Guided by mathematical modeling, we identified the primary components of the liquid core to be HSP70 family chaperones, whose adenosine triphosphate (ATP)–dependent activity maintained the liquidity of shells and cores. In vivo proteasome inhibition within neurons, to mimic aging-related reduction of proteasome activity, induced TDP-43–containing anisosomes. These structures converted to aggregates when ATP levels were reduced. Thus, acetylation, HSP70, and proteasome activities regulate TDP-43 phase separation and conversion into a gel or solid phase.

The binding site barrier (BSB) was originally proposed to describe the binding behavior of antibodies to cells peripheral to blood vessels, preventing their further penetration into the tumors. Yet, it is revisited herein to describe the intratumoral cellular disposition of nanoparticles (NPs). Specifically, the BSB limits NP diffusion and results in unintended internalization of NPs by stroma cells localized near blood vessels. This not only limits the therapeutic outcome but also promotes adverse off-target effects. In the current study, it was shown that tumor-associated fibroblast cells (TAFs) are the major component of the BSB, particularly in tumors with a stroma-vessel architecture where the location of TAFs aligns with blood vessels. Specifically, TAF distance to blood vessels, expression of receptor proteins, and binding affinity affect the intensity of the BSB. The physical barrier elicited by extracellular matrix also prolongs the retention of NPs in the stroma, potentially contributing to the BSB. The influence of particle size on the BSB was also investigated. The strongest BSB effect was found with small (∼18 nm) NPs targeted with the anisamide ligand. The uptake of these NPs by TAFs was about 7-fold higher than that of the other cells 16 h post-intravenous injection. This was because TAFs also expressed the sigma receptor under the influence of TGF-β secreted by the tumor cells. Overall, the current study underscores the importance of BSBs in the delivery of nanotherapeutics and provides a rationale for exploiting BSBs to target TAFs.

A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently developed quasistationary perturbation method. Using the fact that the activating ionic channel's random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.

We present a quasi-steady state reduction of a linear reaction-hyperbolic master equation describing the directed intermittent search for a hidden target by a motor-driven particle moving on a one-dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile nonsearch phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker-Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V, diffusivity D, and target absorption rate λ of the random search. The quasi-steady state reduction determines V, D, and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3-state model and show that our quasi-steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a "tug-of-war", and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets.

Particle tracking is a powerful biophysical tool that requires conversion of large video files into position time series, i.e. traces of the species of interest for data analysis. Current tracking methods, based on a limited set of input parameters to identify bright objects, are ill-equipped to handle the spectrum of spatiotemporal heterogeneity and poor signal-to-noise ratios typically presented by submicron species in complex biological environments. Extensive user involvement is frequently necessary to optimize and execute tracking methods, which is not only inefficient but introduces user bias. To develop a fully automated tracking method, we developed a convolutional neural network for particle localization from image data, comprised of over 6,000 parameters, and employed machine learning techniques to train the network on a diverse portfolio of video conditions. The neural network tracker provides unprecedented automation and accuracy, with exceptionally low false positive and false negative rates on both 2D and 3D simulated videos and 2D experimental videos of difficult-to-track species. Significance StatementThe increasing availability of powerful light microscopes capable of collecting terabytes of high-resolution 2D and 3D videos in a single day has created a great demand for automated image analysis tools. Tracking the movement of nanometer scale particles (e.g., virus, proteins, synthetic drug particles) is critical for understanding how pathogens breach mucosal barriers and for the design of new drug therapies. Our advancement is to use an artificial neural network that provides, first and foremost, substantially improved automation. Additionally, our method improves accuracy compared to current methods and reproducibility across users and labs.

We construct a path-integral representation of solutions to a stochastic hybrid system, consisting of one or more continuous variables evolving according to a piecewise-deterministic dynamics. The differential equations for the continuous variables are coupled to a set of discrete variables that satisfy a continuous-time Markov process, which means that the differential equations are only valid between jumps in the discrete variables. Examples of stochastic hybrid systems arise in biophysical models of stochastic ion channels, motor-driven intracellular transport, gene networks, and stochastic neural networks. We use the path-integral representation to derive a large deviation action principle for a stochastic hybrid system. Minimizing the associated action functional with respect to the set of all trajectories emanating from a metastable state (assuming that such a minimization scheme exists) then determines the most probable paths of escape. Moreover, evaluating the action functional along a most probable path generates the so-called quasipotential used in the calculation of mean first passage times. We illustrate the theory by considering the optimal paths of escape from a metastable state in a bistable neural network.

We consider a stochastic version of an excitable system based on the Morris-Lecar model of a neuron, in which the noise originates from stochastic sodium and potassium ion channels opening and closing. One can analyze neural excitability in the deterministic model by using a separation of time scales involving a fast voltage variable and a slow recovery variable, which represents the fraction of open potassium channels. In the stochastic setting, spontaneous excitation is initiated by ion channel noise. If the recovery variable is constant during initiation, the spontaneous activity rate can be calculated using Kramer's rate theory. The validity of this assumption in the stochastic model is examined using a systematic perturbation analysis. We find that, in most physically relevant cases, this assumption breaks down, requiring an alternative to Kramer's theory for excitable systems with one deterministic fixed point. We also show that an exit time problem can be formulated in an excitable system by considering maximum likelihood trajectories of the stochastic process.