Quantifying intermittent transport in cell cytoplasm

Survival probabilities and first-passage distributions of self-propelled particles in spherical cavities

A model of self-propelled motion in a closed compartment containing simple or complex fluids is formulated in this paper in terms of the dynamics of a point particle moving in a spherical cavity under the action of random thermal forces and exponentially correlated noise. The particle's time evolution is governed by a generalized Langevin equation (GLE) in which the memory function, connected to the thermal forces by a fluctuation-dissipation relation, is described by Jeffrey's model of viscoelasticity (which reduces to a model of ordinary viscous dynamics in a suitable limit). The GLE is transformed exactly to a Fokker-Planck equation that in spherical polar coordinates is in turn found to admit of an exact solution for the particle's probability density function under absorbing boundary conditions at the surface of the sphere. The solution is used to derive an expression (that is also exact) for the survival probability of the particle in the sphere, starting from its center, which is then used to calculate the distribution of the particle's first-passage times to the boundary. The behavior of these quantities is investigated as a function of the Péclet number and the persistence time of the athermal forces, providing insight into the effects of nonequilibrium fluctuations on confined particle motion in three dimensions.

Simulating rolling paths and reorientation behavior of ball-rolling dung beetles

Ball-rolling dung beetles show a remarkable ability to maintain a straight path while rolling dung balls away from a dung pile. Rolling in a straight line is beneficial, as it enables beetles to efficiently escape competition near the dung pile. Research has shown that beetles use the sky to choose and maintain an initial rolling direction, and to reorient (correct their direction) when pushed off their intended course by obstacles or uneven ground. While beetles' mechanisms for navigation are well understood, it remains unclear how beetles regulate the timing of reorientation and under what circumstances reorientation is beneficial. Previous studies have focused only on the observable data from the movement of real dung beetles, in the field and simulated environments. In this paper, we formulate a mathematical model based on a persistent random walk to simulate a dung beetle's movement in a circular arena. We simulate two possible reorientation strategies and analyze the impact when reorientation is not perfect. We show that our model provides an approximation of actual dung ball rolling paths, analyze the benefits of each reorientation technique under varying conditions, and show that when the sky is obscured, rolling without reorientation can be a beetle's optimal strategy.

Kinetic Modeling of Virus Growth in Cells

When a virus infects a host cell, it hijacks the biosynthetic capacity of the cell to produce virus progeny, a process that may take less than an hour or more than a week. The overall time required for a virus to reproduce depends collectively on the rates of multiple steps in the infection process, including initial binding of the virus particle to the surface of the cell, virus internalization and release of the viral genome within the cell, decoding of the genome to make viral proteins, replication of the genome, assembly of progeny virus particles, and release of these particles into the extracellular environment. For a large number of virus types, much has been learned about the molecular mechanisms and rates of the various steps. However, in only relatively few cases during the last 50 years has an attempt been made-using mathematical modeling-to account for how the different steps contribute to the overall timing and productivity of the infection cycle in a cell. Here we review the initial case studies, which include studies of the one-step growth behavior of viruses that infect bacteria (Qβ, T7, and M13), human immunodeficiency virus, influenza A virus, poliovirus, vesicular stomatitis virus, baculovirus, hepatitis B and C viruses, and herpes simplex virus. Further, we consider how such models enable one to explore how cellular resources are utilized and how antiviral strategies might be designed to resist escape. Finally, we highlight challenges and opportunities at the frontiers of cell-level modeling of virus infections.

Imaging, Tracking and Computational Analyses of Virus Entry and Egress with the Cytoskeleton

Viruses have a dual nature: particles are “passive substances” lacking chemical energy transformation, whereas infected cells are “active substances” turning-over energy. How passive viral substances convert to active substances, comprising viral replication and assembly compartments has been of intense interest to virologists, cell and molecular biologists and immunologists. Infection starts with virus entry into a susceptible cell and delivers the viral genome to the replication site. This is a multi-step process, and involves the cytoskeleton and associated motor proteins. Likewise, the egress of progeny virus particles from the replication site to the extracellular space is enhanced by the cytoskeleton and associated motor proteins. This overcomes the limitation of thermal diffusion, and transports virions and virion components, often in association with cellular organelles. This review explores how the analysis of viral trajectories informs about mechanisms of infection. We discuss the methodology enabling researchers to visualize single virions in cells by fluorescence imaging and tracking. Virus visualization and tracking are increasingly enhanced by computational analyses of virus trajectories as well as in silico modeling. Combined approaches reveal previously unrecognized features of virus-infected cells. Using select examples of complementary methodology, we highlight the role of actin filaments and microtubules, and their associated motors in virus infections. In-depth studies of single virion dynamics at high temporal and spatial resolutions thereby provide deep insight into virus infection processes, and are a basis for uncovering underlying mechanisms of how cells function.

Viral highway to nucleus exposed by image correlation analyses

Parvoviral genome translocation from the plasma membrane into the nucleus is a coordinated multistep process mediated by capsid proteins. We used fast confocal microscopy line scan imaging combined with image correlation methods including auto-, pair- and cross-correlation, and number and brightness analysis, to study the parvovirus entry pathway at the single-particle level in living cells. Our results show that the endosome-associated movement of virus particles fluctuates from fast to slow. Fast transit of single cytoplasmic capsids to the nuclear envelope is followed by slow movement of capsids and fast diffusion of capsid fragments in the nucleoplasm. The unique combination of image analyses allowed us to follow the fate of intracellular single virus particles and their interactions with importin β revealing previously unknown dynamics of the entry pathway.

Model of chromosomal loci dynamics in bacteria as fractional diffusion with intermittent transport

The short-time dynamics of bacterial chromosomal loci is a mixture of subdiffusive and active motion, in the form of rapid relocations with near-ballistic dynamics. While previous work has shown that such rapid motions are ubiquitous, we still have little grasp on their physical nature, and no positive model is available that describes them. Here, we propose a minimal theoretical model for loci movements as a fractional Brownian motion subject to a constant but intermittent driving force, and compare simulations and analytical calculations to data from high-resolution dynamic tracking in E. coli. This analysis yields the characteristic time scales for intermittency. Finally, we discuss the possible shortcomings of this model, and show that an increase in the effective local noise felt by the chromosome associates to the active relocations.

From mobility to crosstalk. A model of intracellular miRNAs motion may explain the RNAs interaction mechanism on the basis of target subcellular localization

MicroRNAs (miRNAs), 22 nucleotides long molecules with the function to reduce gene expression by inhibiting mRNA translation through partial complementary to one or more messenger RNA (mRNA) molecules. A single miRNA can reduce the expression levels of hundreds of genes and one mRNA can be a target for many miRNAs. Despite the study models used so far, miRNAs and mRNAs cannot be seen as acting in an isolated manner or even "in pairs". They most likely exert their complex actions through numerous overlapping interrelations. One of the models depicting interdependence of intracytoplasmic RNAs is the crosstalk model. It is based on a competition between several target mRNAs which are regulated by the same miRNA. In this paper, we will discuss the mobility mechanism of miRNAs, recently suggested by data from "single particle tracking" experiments. These data suggests that miRNA intracellular mobility may be of "intermittent active transport"(IAT) type. IAT is a mobility model composed by alternation of active transport (AT) and Brownian motion (BM). Based on a mathematical model, we concluded that, AT phase may explain the efficiency in reaching far targets and the BM phase may explain the competition. Furthermore, we suggest that the interaction between miRNAs and their targets depends on the concentration of the molecules, the affinity between the molecules and also on the intracellular localization of the molecules. Hence, the probability that a miRNA interacts with its target depends also on the distance to the target and the macromolecular crowding. Taken together, our data proposes an intracytoplasmic mobility mechanism for miRNA and shows that this model can partially explain the RNA crosstalk.

Coarse-graining intermittent intracellular transport: Two- and three-dimensional models

Viruses and other cellular cargo that lack locomotion must rely on diffusion and cellular transport systems to navigate through a biological cell. Indeed, advances in single particle tracking have revealed that viral motion alternates between (a) diffusion in the cytoplasm and (b) active transport along microtubules. This intermittency makes quantitative analysis of trajectories difficult. Therefore, the purpose of this paper is to construct mathematical methods to approximate intermittent dynamics by effective stochastic differential equations. The coarse-graining method that we develop is more accurate than existing techniques and applicable to a wide range of intermittent transport models. In particular, we apply our method to two- and three-dimensional cell geometries (disk, sphere, and cylinder) and demonstrate its accuracy. In addition to these specific applications, we also explain our method in full generality for use on future intermittent models.

Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity

We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells.

Recycling of kinesin-1 motors by diffusion after transport

Kinesin motors drive the long-distance anterograde transport of cellular components along microtubule tracks. Kinesin-dependent transport plays a critical role in neurogenesis and neuronal function due to the large distance separating the soma and nerve terminal. The fate of kinesin motors after delivery of their cargoes is unknown but has been postulated to involve degradation at the nerve terminal, recycling via retrograde motors, and/or recycling via diffusion. We set out to test these models concerning the fate of kinesin-1 motors after completion of transport in neuronal cells. We find that kinesin-1 motors are neither degraded nor returned by retrograde motors. By combining mathematical modeling and experimental analysis, we propose a model in which the distribution and recycling of kinesin-1 motors fits a “loose bucket brigade” where individual motors alter between periods of active transport and free diffusion within neuronal processes. These results suggest that individual kinesin-1 motors are utilized for multiple rounds of transport.

Physical principles and models describing intracellular virus particle dynamics

Modeling in cellular biology benefits greatly from quantitative analysis that arise from the theory of diffusion and chemical reactions. Recent progress in single particle imaging enables the visualization of viral trajectories evolving in the cytoplasm. Biophysical models and mathematical analysis have been developed to unravel the complexity of single viral trajectories. We review here models of active motion of viruses along the cytoskeleton as well as their diffusion. We present resent efforts to estimate global trafficking properties, such as the probability and the mean time for a viral particle to reach a small nuclear pore. However, most signaling pathways involved in controlling viral motion remain undescribed and should be the goal of future modeling efforts.

Quantitative analysis of virus and plasmid trafficking in cells

Intracellular transport of DNA carriers is a fundamental step of gene delivery. By combining both theoretical and numerical approaches we study here single and several viruses and DNA particles trafficking in the cell cytoplasm to a small nuclear pore. We present a physical model to account for certain aspects of cellular organization, starting with the observation that a viral trajectory consists of epochs of pure diffusion and epochs of active transport along microtubules. We define a general degradation rate to describe the limitations of the delivery of plasmid or viral particles to a nuclear pore imposed by various types of direct and indirect hydrolysis activity inside the cytoplasm. By replacing the switching dynamics by a single steady state stochastic description, we obtain estimates for the probability and the mean time for the first one of many particles to go from the cell membrane to a small nuclear pore. Computational simulations confirm that our model can be used to analyze and interpret viral trajectories and estimate quantitatively the success of nuclear delivery.