Heterogeneous Mean First-Passage Time Scaling in Fractal Media.

The mean first passage time (MFPT) of random walks is a key quantity characterizing dynamic processes on disordered media. In a random fractal embedded in the Euclidean space, the MFPT is known to obey the power law scaling with the distance between a source and a target site with a universal exponent. We find that the scaling law for the MFPT is not determined solely by the distance between a source and a target but also by their locations. The role of a site in the first passage processes is quantified by the random walk centrality. It turns out that the site of highest random walk centrality, dubbed as a hub, intervenes in first passage processes. We show that the MFPT from a departure site to a target site is determined by a competition between direct paths and indirect paths detouring via the hub. Consequently, the MFPT displays a crossover scaling between a short distance regime, where direct paths are dominant, and a long distance regime, where indirect paths are dominant. The two regimes are characterized by power laws with different scaling exponents. The crossover scaling behavior is confirmed by extensive numerical calculations of the MFPTs on the critical percolation cluster in two dimensional square lattices.

Stochastic model of T cell repolarization during target elimination (II).

Cytotoxic T lymphocytes (T cells) and natural killer cells form a tight contact, the immunological synapse (IS), with target cells, where they release their lytic granules containing perforin/granzyme and cytokine-containing vesicles. During this process the cell repolarizes and moves the microtubule organizing center (MTOC) toward the IS. In the first part of our work we developed a computational model for the molecular-motor-driven motion of the microtubule cytoskeleton during T cell polarization and analyzed the effects of cortical-sliding and capture-shrinkage mechanisms. Here we use this model to analyze the dynamics of the MTOC repositioning in situations in which 1) the IS is in an arbitrary position with respect to the initial position of the MTOC and 2) the T cell has two IS at two arbitrary positions. In the case of one IS, we found that the initial position determines which mechanism is dominant and that the time of repositioning does not rise monotonously with the MTOC-IS distance. In the case of two IS, we observe several scenarios that have also been reported experimentally: the MTOC alternates stochastically (but with a well-defined average transition time) between the two IS; it wiggles in between the two IS without transiting to one of the two; or it is at some point pulled to one of the two IS and stays there. Our model allows one to predict which scenario emerges in dependency of the mechanisms in action and the number of dyneins present. We report that the presence of capture-shrinkage mechanism in at least one IS is necessary to assure the transitions in every cell configuration. Moreover, the frequency of transitions does not decrease with the distance between the two IS and is the highest when both mechanisms are present in both IS.

Ameboid cell migration through regular arrays of micropillars under confinement.

Migrating cells often encounter a wide variety of topographic features-including the presence of obstacles-when navigating through crowded biological environments. Unraveling the impact of topography and crowding on the dynamics of cells is key to better understand many essential physiological processes such as the immune response. We study the impact of geometrical cues on ameboid migration of HL-60 cells differentiated into neutrophils. A microfluidic device is designed to track the cells in confining geometries between two parallel plates with distance h, in which identical micropillars are arranged in regular pillar forests with pillar spacing e. We observe that the cells are temporarily captured near pillars, with a mean contact time that is independent of h and e. By decreasing the vertical confinement h, we find that the cell velocity is not affected, while the persistence reduces; thus, cells are able to preserve their velocity when highly squeezed but lose the ability to control their direction of motion. At a given h, we show that by decreasing the pillar spacing e in the weak lateral confinement regime, the mean escape time of cells from effective local traps between neighboring pillars grows. This effect, together with the increase of cell-pillar contact frequency, leads to the reduction of diffusion constant D. By disentangling the contributions of these two effects on D in numerical simulations, we verify that the impact of cell-pillar contacts on cell diffusivity is more pronounced at smaller pillar spacing.

Interorganelle Tethering to Endocytic Organelles Determines Directional Cytokine Transport in CD4+ T Cells.

Delivery of vesicles to their desired destinations plays a central role in maintaining proper cell functionality. In certain scenarios, depending on loaded cargos, the vesicles have spatially distinct destinations. For example, in T cells, some cytokines (e.g., IL-2) are polarized to the T cell-target cell interface, whereas the other cytokines are delivered multidirectionally (e.g., TNF-α). In this study, we show that in primary human CD4+ T cells, both TNF-α+ and IL-2+ vesicles can tether with endocytic organelles (lysosomes/late endosomes) by forming membrane contact sites. Tethered cytokine-containing vesicle (CytV)-endocytic organelle pairs are released sequentially. Only endocytic organelle-tethered CytVs are preferentially transported to their desired destination. Mathematical models suggest that endocytic organelle tethering could regulate the direction of cytokine transport by selectively attaching different microtubule motor proteins (such as kinesin and dynein) to the corresponding CytVs. These findings establish the previously unknown interorganelle tethering to endocytic organelles as a universal solution for directional cytokine transport in CD4+ T cells. Modulating tethering to endocytic organelles can, therefore, coordinately control directionally distinct cytokine transport.

Optimal Non-Markovian Search Strategies with n-Step Memory.

Stochastic search processes are ubiquitous in nature and are expected to become more efficient when equipped with a memory, where the searcher has been before. A natural realization of a search process with long-lasting memory is a migrating cell that is repelled from the diffusive chemotactic signal that it secretes on its way, denoted as an autochemotactic searcher. To analyze the efficiency of this class of non-Markovian search processes, we present a general formalism that allows one to compute the mean first-passage time (MFPT) for a given set of conditional transition probabilities for non-Markovian random walks on a lattice. We show that the optimal choice of the n-step transition probabilities decreases the MFPT systematically and substantially with an increasing number of steps. It turns out that the optimal search strategies can be reduced to simple cycles defined by a small parameter set and that mirror-asymmetric walks are more efficient. For the autochemotactic searcher, we show that an optimal coupling between the searcher and the chemical reduces the MFPT to 1/3 of the one for a Markovian random walk.

Stochastic Model of T Cell Repolarization during Target Elimination I.

Cytotoxic T lymphocytes (T) and natural killer cells are the main cytotoxic killer cells of the human body to eliminate pathogen-infected or tumorigenic cells (i.e., target cells). Once a natural killer or T cell has identified a target cell, they form a tight contact zone, the immunological synapse (IS). One then observes a repolarization of the cell involving the rotation of the microtubule (MT) cytoskeleton and a movement of the MT organizing center (MTOC) to a position that is just underneath the plasma membrane at the center of the IS. Concomitantly, a massive relocation of organelles attached to MTs is observed, including the Golgi apparatus, lytic granules, and mitochondria. Because the mechanism of this relocation is still elusive, we devise a theoretical model for the molecular-motor-driven motion of the MT cytoskeleton confined between plasma membrane and nucleus during T cell polarization. We analyze different scenarios currently discussed in the literature, the cortical sliding and capture-shrinkage mechanisms, and compare quantitative predictions about the spatiotemporal evolution of MTOC position and MT cytoskeleton morphology with experimental observations. The model predicts the experimentally observed biphasic nature of the repositioning due to an interplay between MT cytoskeleton geometry and motor forces and confirms the dominance of the capture-shrinkage over the cortical sliding mechanism when the MTOC and IS are initially diametrically opposed. We also find that the two mechanisms act synergistically, thereby reducing the resources necessary for repositioning. Moreover, it turns out that the localization of dyneins in the peripheral supramolecular activation cluster facilitates their interaction with the MTs. Our model also opens a way to infer details of the dynein distribution from the experimentally observed features of the MT cytoskeleton dynamics. In a subsequent publication, we will address the issue of general initial configurations and situations in which the T cell established two ISs.

Migration of Cytotoxic T Lymphocytes in 3D Collagen Matrices.

CD8+ cytotoxic T lymphocytes (CTL) and natural killer cells are the main cytotoxic killer cells of the human body to eliminate pathogen-infected or tumorigenic cells (also known as target cells). To find their targets, they have to navigate and migrate through complex biological microenvironments, a key component of which is the extracellular matrix (ECM). The mechanisms underlying killer cell's navigation are not well understood. To mimic an ECM, we use a matrix formed by different collagen concentrations and analyze migration trajectories of primary human CTLs. Different migration patterns are observed and can be grouped into three motility types: slow, fast, and mixed. The dynamics are well described by a two-state persistent random walk model, which allows cells to switch between slow motion with low persistence and fast motion with high persistence. We hypothesize that the slow motility mode describes CTLs creating channels through the collagen matrix by deforming and tearing apart collagen fibers and that the fast motility mode describes CTLs moving within these channels. Experimental evidence supporting this scenario is presented by visualizing migrating T cells following each other on exactly the same track and showing cells moving quickly in channel-like cavities within the surrounding collagen matrix. Consequently, the efficiency of the stochastic search process of CTLs in the ECM should strongly be influenced by a dynamically changing channel network produced by the killer cells themselves.

Capillary Action in Scalar Active Matter.

We study the capacity of active matter to rise in thin tubes against gravity and other related phenomena like wetting of vertical plates and spontaneous imbibition, where a wetting liquid is drawn into a porous medium. This capillary action or capillarity is well known in classical fluids and originates from attractive interactions between the liquid molecules and the container walls, and from the attraction of the liquid molecules among each other. We observe capillarity in a minimal model for scalar active matter with purely repulsive interactions, where an effective attraction emerges due to slowdown during collisions between active particles and between active particles and walls. Simulations indicate that the capillary rise in thin tubes is approximately proportional to the active sedimentation length λ and that the wetting height of a vertical plate grows superlinear with λ. In a disordered porous medium the imbibition height scales as ⟨h⟩∝λϕ_{m}, where ϕ_{m} is its packing fraction. These predictions are highly relevant for suspensions of sedimenting active colloids or motile bacteria in a porous medium under the influence of a constant force field.

Oxygen in the Tumor Microenvironment: Mathematical and Numerical Modeling.

There are many reasons to try to achieve a good grasp of the distribution of oxygen in the tumor microenvironment. The lack of oxygen - hypoxia - is a main actor in the evolution of tumors and in their growth and appears to be just as important in tumor invasion and metastasis. Mathematical models of the distribution of oxygen in tumors which are based on reaction-diffusion equations provide partial but qualitatively significant descriptions of the measured oxygen concentrations in the tumor microenvironment, especially when they incorporate important elements of the blood vessel network such as the blood vessel size and spatial distribution and the pulsation of local pressure due to blood circulation. Here, we review our mathematical and numerical approaches to the distribution of oxygen that yield insights both on the role of the distribution of blood vessel density and size and on the fluctuations of blood pressure.

Search and Capture Efficiency of Dynamic Microtubules for Centrosome Relocation during IS Formation.

Upon contact with antigen-presenting cells, cytotoxic T lymphocytes (T cells) establish a highly organized contact zone denoted as the immunological synapse (IS). The formation of the IS implies relocation of the microtubule organizing center (MTOC) toward the contact zone, which necessitates a proper connection between the MTOC and the IS via dynamic microtubules (MTs). The efficiency of the MTs finding the IS within the relevant timescale is, however, still illusive. We investigate how MTs search the three-dimensional constrained cellular volume for the IS and bind upon encounter to dynein anchored at the IS cortex. The search efficiency is estimated by calculating the time required for the MTs to reach the dynein-enriched region of the IS. In this study, we develop simple mathematical and numerical models incorporating relevant components of a cell and propose an optimal search strategy. Using the mathematical model, we have quantified the average search time for a wide range of model parameters and proposed an optimized set of values leading to the minimal capture time. Our results show that search times are minimal when the IS formed at the nearest or at the farthest sites on the cell surface with respect to the perinuclear MTOC. The search time increases monotonically away from these two specific sites and is maximal at an intermediate position near the equator of the cell. We observed that search time strongly depends on the number of searching MTs and distance of the MTOC from the nuclear surface.

Dynamic vessel adaptation in synthetic arteriovenous networks.

Blood vessel networks of living organisms continuously adapt their structure under the influence of hemodynamic and metabolic stimuli. For a fixed vessel arrangement, blood flow characteristics still depend crucially on the morphology of each vessel. Vessel diameters adapt dynamically according to internal and external stimuli: Endothelial wall shear stress, intravascular pressure, flow-dependent metabolic stimuli, and electrical stimuli conducted from distal to proximal segments along vascular walls. Pries et al. formulated a theoretical model involving these four local stimuli to simulate long-term changes of vessel diameters during structural adaption of microvascular networks. Here we apply this vessel adaptation algorithm to synthetic arteriovenous blood vessel networks generated by our simulation framework "Tumorcode". We fixed the free model parameters by an optimization method combined with the requirement of homogeneous flow in the capillary bed. We find that the local blood volume, surface to volume ratio and branching ratio differs from networks with radii fulfilling Murray's law exactly to networks with radii obtained by the adaptation algorithm although their relation is close to Murray's law.

Reaction-diffusion model for STIM-ORAI interaction: The role of ROS and mutations.

Release of Ca2+ from endoplasmatic retriculum (ER) Ca2+ stores causes stromal interaction molecules (STIM) in the ER membrane and ORAI proteins in the plasma membrane (PM) to interact and form the Ca2+ release activated Ca2+ (CRAC) channels, which represent a major Ca2+ entry route in non-excitable cells and thus control various cell functions. It is experimentally possible to mutate ORAI1 proteins and therefore modify, especially block, the Ca2+ influx into the cell. On the basis of the model of Hoover and Lewis (2011), we formulate a reaction-diffusion model to quantify the STIM1-ORAI1 interaction during CRAC channel formation and analyze different ORAI1 channel stoichiometries and different ratios of STIM1 and ORAI1 in comparison with experimental data. We incorporate the inhibition of ORAI1 channels by ROS into our model and calculate its contribution to the CRAC channel amplitude. We observe a large decrease of the CRAC channel amplitude evoked by mutations of ORAI1 proteins.

Spatial Cytoskeleton Organization Supports Targeted Intracellular Transport.

The efficiency of intracellular cargo transport from specific sources to target locations is strongly dependent upon molecular motor-assisted motion along the cytoskeleton. Radial transport along microtubules and lateral transport along the filaments of the actin cortex underneath the cell membrane are characteristic for cells with a centrosome. The interplay between the specific cytoskeleton organization and the motor performance results in a spatially inhomogeneous intermittent search strategy. To analyze the efficiency of such intracellular search strategies, we formulate a random velocity model with intermittent arrest states. We evaluate efficiency in terms of mean first passage times for three different, frequently encountered intracellular transport tasks: 1) the narrow escape problem, which emerges during cargo transport to a synapse or other specific region of the cell membrane; 2) the reaction problem, which considers the binding time of two particles within the cell; and 3) the reaction-escape problem, which arises when cargo must be released at a synapse only after pairing with another particle. Our results indicate that cells are able to realize efficient search strategies for various intracellular transport tasks economically through a spatial cytoskeleton organization that involves only a narrow actin cortex rather than a cell body filled with randomly oriented actin filaments.

Quantum Relaxation and Metastability of Lattice Bosons with Cavity-Induced Long-Range Interactions.

The coupling of cold atoms to the radiation field within a high-finesse optical resonator, an optical cavity, induces long-range interactions which can compete with an underlying optical lattice. The interplay between short- and long-range interactions gives rise to new phases of matter including supersolidity (SS) and density waves (DW), and interesting quantum dynamics. Here it is shown that for hard-core bosons in one dimension the ground state phase diagram and the quantum relaxation after sudden quenches can be calculated exactly in the thermodynamic limit. Remanent DW order is observed for quenches from a DW ground state into the superfluid (SF) phase below a dynamical transition line. After sufficiently strong SF to DW quenches beyond a static metastability line DW order emerges on top of remanent SF order, giving rise to a dynamically generated SS state. Our method to handle infinite- and short-range interactions in the infinite system size limit opens a way to solve exactly other Hamiltonians with infinite- and short-range interactions as well.

Tumorcode : A framework to simulate vascularized tumors.

During the past years our group published several articles using computer simulations to address the complex interaction of tumors and the vasculature as underlying transport network. Advances in imaging and lab techniques pushed in vitro research of tumor spheroids forward and animal models as well as clinical studies provided more insights to single processes taking part in tumor growth, however, an overall picture is still missing. Computer simulations are a non-invasive option to cumulate current knowledge and form a quasi in vivo system. In our software, several known models were assembled into a multi-scale approach which allows to study length scales relevant for clinical applications. We release our code to the public domain, together with a detailed description of the implementation and several examples, with the hope of usage and futher development by the community. A justification for the included algorithms and the biological models was obtained in previous publications, here we summarize the technical aspects following the workflow of a typical simulation procedure.

Spatial organization of the cytoskeleton enhances cargo delivery to specific target areas on the plasma membrane of spherical cells.

Intracellular transport is vital for the proper functioning and survival of a cell. Cargo (proteins, vesicles, organelles, etc) is transferred from its place of creation to its target locations via molecular motor assisted transport along cytoskeletal filaments. The transport efficiency is strongly affected by the spatial organization of the cytoskeleton, which constitutes an inhomogeneous, complex network. In cells with a centrosome microtubules grow radially from the central microtubule organizing center towards the cell periphery whereas actin filaments form a dense meshwork, the actin cortex, underneath the cell membrane with a broad range of orientations. The emerging ballistic motion along filaments is frequently interrupted due to constricting intersection nodes or cycles of detachment and reattachment processes in the crowded cytoplasm. In order to investigate the efficiency of search strategies established by the cell's specific spatial organization of the cytoskeleton we formulate a random velocity model with intermittent arrest states. With extensive computer simulations we analyze the dependence of the mean first passage times for narrow escape problems on the structural characteristics of the cytoskeleton, the motor properties and the fraction of time spent in each state. We find that an inhomogeneous architecture with a small width of the actin cortex constitutes an efficient intracellular search strategy.

Optimality of Spatially Inhomogeneous Search Strategies.

We consider random search processes alternating stochastically between diffusion and ballistic motion, in which the distribution function of ballistic motion directions varies from point to point in space. The specific space dependence of the directional distribution together with the switching rates between the two modes of motion establishes a spatially inhomogeneous search strategy. We show that the mean first passage times for several standard search problems-narrow escape, reaction partner finding, reaction escape-can be minimized with a directional distribution that is reminiscent of the spatial organization of the cytoskeleton filaments of cells with a centrosome: radial ballistic transport from the center to the periphery and back, and ballistic transport in random directions within a concentric shell of thickness Δ_{opt} along the domain boundary. The results suggest that living cells realize efficient search strategies for various intracellular transport problems economically through a spatial cytoskeleton organization that involves radial microtubules in the central region and only a narrow actin cortex rather than a cell body filled with randomly oriented actin filaments.

Co-chaperones of the mammalian endoplasmic reticulum.

In mammalian cells, the rough endoplasmic reticulum or ER plays a central role in the biogenesis of most extracellular plus many organellar proteins and in cellular calcium homeostasis. Therefore, this organelle comprises molecular chaperones that are involved in import, folding/assembly, export, and degradation of polypeptides in millimolar concentrations. In addition, there are calcium channels/pumps and signal transduction components present in the ER membrane that affect and are affected by these processes. The ER lumenal Hsp70, termed immunoglobulin-heavy chain binding protein or BiP, is the central player in all these activities and involves up to seven different co-chaperones, i.e. ER-membrane integrated as well as ER-lumenal Hsp40s, which are termed ERj or ERdj, and two nucleotide exchange factors.

Computer Simulations of the Tumor Vasculature: Applications to Interstitial Fluid Flow, Drug Delivery, and Oxygen Supply.

Tumor vasculature, the blood vessel network supplying a growing tumor with nutrients such as oxygen or glucose, is in many respects different from the hierarchically organized arterio-venous blood vessel network in normal tissues. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature. Integrative models, based on detailed experimental data and physical laws, implement, in silico, the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. This chapter provides an overview over the current status of computer simulations of vascular remodeling during tumor growth including interstitial fluid flow, drug delivery, and oxygen supply within the tumor. The model predictions are compared with experimental and clinical data and a number of longstanding physiological paradigms about tumor vasculature and intratumoral solute transport are critically scrutinized.

Calcium microdomains at the immunological synapse: how ORAI channels, mitochondria and calcium pumps generate local calcium signals for efficient T-cell activation.

Cell polarization enables restriction of signalling into microdomains. Polarization of lymphocytes following formation of a mature immunological synapse (IS) is essential for calcium-dependent T-cell activation. Here, we analyse calcium microdomains at the IS with total internal reflection fluorescence microscopy. We find that the subplasmalemmal calcium signal following IS formation is sufficiently low to prevent calcium-dependent inactivation of ORAI channels. This is achieved by localizing mitochondria close to ORAI channels. Furthermore, we find that plasma membrane calcium ATPases (PMCAs) are re-distributed into areas beneath mitochondria, which prevented PMCA up-modulation and decreased calcium export locally. This nano-scale distribution-only induced following IS formation-maximizes the efficiency of calcium influx through ORAI channels while it decreases calcium clearance by PMCA, resulting in a more sustained NFAT activity and subsequent activation of T cells.