Asymmetric stem cell division maintains the genetic heterogeneity of tissue cells.

Within a given tissue, the stem cell niche provides the microenvironment for stem cells suitable for their self-renewal. Conceptually, the niche space constrains the size of a stem-cell pool, as the cells sharing the niche compete for its space. It has been suggested that either neutral- or non-neutral-competition of stem cells changes the clone dynamics of stem cells. Theoretically, if the rate of asymmetric division is high, the stem cell competition is limited, thus suppressing clonal expansion. However, the effects of asymmetric division on clone dynamics have never been experimentally tested. Here, using the Drosophila germline stem cell (GSC) system, as a simple model of the in-vivo niche, we examine the effect of division modes (asymmetric or symmetric) on clonal dynamics by combining experimental approaches with mathematical modeling. Our experimental data and computational model both suggest that the rate of asymmetric division is proportional to the time a stem cell clone takes to expand. Taken together, our data suggests that asymmetric division is essential for maintaining the genetic variation of stem cells and thus serves as a critical mechanism for safeguarding fertility over the animal age or preventing multiple disorders caused by the clonal expansion of stem cells.

A model of actin-driven endocytosis explains differences of endocytic motility in budding and fission yeast.

A comparative study (Sun et al., 2019) showed that the abundance of proteins at sites of endocytosis in fission and budding yeast is more similar in the two species than previously thought, yet membrane invaginations in fission yeast elongate twofold faster and are nearly twice as long as in budding yeast. Here we use a three-dimensional model of a motile endocytic invagination (Nickaeen et al., 2019) to investigate factors affecting elongation of the invaginations. We found that differences in turgor pressure in the two yeast species can largely explain the paradoxical differences observed experimentally in endocytic motility.

Mad dephosphorylation at the nuclear pore is essential for asymmetric stem cell division.

Stem cells divide asymmetrically to generate a stem cell and a differentiating daughter cell. Yet, it remains poorly understood how a stem cell and a differentiating daughter cell can receive distinct levels of niche signal and thus acquire different cell fates (self-renewal versus differentiation), despite being adjacent to each other and thus seemingly exposed to similar levels of niche signaling. In the Drosophila ovary, germline stem cells (GSCs) are maintained by short range bone morphogenetic protein (BMP) signaling; the BMP ligands activate a receptor that phosphorylates the downstream molecule mothers against decapentaplegic (Mad). Phosphorylated Mad (pMad) accumulates in the GSC nucleus and activates the stem cell transcription program. Here, we demonstrate that pMad is highly concentrated in the nucleus of the GSC, while it quickly decreases in the nucleus of the differentiating daughter cell, the precystoblast (preCB), before the completion of cytokinesis. We show that a known Mad phosphatase, Dullard (Dd), is required for the asymmetric partitioning of pMad. Our mathematical modeling recapitulates the high sensitivity of the ratio of pMad levels to the Mad phosphatase activity and explains how the asymmetry arises in a shared cytoplasm. Together, these studies reveal a mechanism for breaking the symmetry of daughter cells during asymmetric stem cell division.

Actin assembly produces sufficient forces for endocytosis in yeast.

We formulated a spatially resolved model to estimate forces exerted by a polymerizing actin meshwork on an invagination of the plasma membrane during endocytosis in yeast cells. The model, which approximates the actin meshwork as a visco-active gel exerting forces on a rigid spherocylinder representing the endocytic invagination, is tightly constrained by experimental data. Simulations of the model produce forces that can overcome resistance of turgor pressure in yeast cells. Strong forces emerge due to the high density of polymerized actin in the vicinity of the invagination and because of entanglement of the meshwork due to its dendritic structure and cross-linking. The model predicts forces orthogonal to the invagination that are consistent with formation of a flask shape, which would diminish the net force due to turgor pressure. Simulations of the model with either two rings of nucleation-promoting factors (NPFs) as in fission yeast or a single ring of NPFs as in budding yeast produce enough force to elongate the invagination against the turgor pressure.

A free-boundary model of a motile cell explains turning behavior.

To understand shapes and movements of cells undergoing lamellipodial motility, we systematically explore minimal free-boundary models of actin-myosin contractility consisting of the force-balance and myosin transport equations. The models account for isotropic contraction proportional to myosin density, viscous stresses in the actin network, and constant-strength viscous-like adhesion. The contraction generates a spatially graded centripetal actin flow, which in turn reinforces the contraction via myosin redistribution and causes retraction of the lamellipodial boundary. Actin protrusion at the boundary counters the retraction, and the balance of the protrusion and retraction shapes the lamellipodium. The model analysis shows that initiation of motility critically depends on three dimensionless parameter combinations, which represent myosin-dependent contractility, a characteristic viscosity-adhesion length, and a rate of actin protrusion. When the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectories, and the motile behavior is sensitive to conditions at the cell boundary. Scanning of a model parameter space shows that the contractile mechanism of motility supports robust cell turning in conditions where short viscosity-adhesion lengths and fast protrusion cause an accumulation of myosin in a small region at the cell rear, destabilizing the axial symmetry of a moving cell.

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology.

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium 'sparks' as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.

A conservative algorithm for parabolic problems in domains with moving boundaries.

We describe a novel conservative algorithm for parabolic problems in domains with moving boundaries developed for modeling in cell biology. The spatial discretization is accomplished by applying Voronoi decomposition to a fixed rectangular grid. In the vicinity of the boundary, the procedure generates irregular Voronoi cells that conform to the domain shape and merge seamlessly with regular control volumes in the domain interior. Consequently, our algorithm is free of the CFL stability issue due to moving interfaces and does not involve cell-merging or mass redistribution. Local mass conservation is ensured by finite-volume discretization and natural-neighbor interpolation. Numerical experiments with two-dimensional geometries demonstrate exact mass conservation and indicate an order of convergence in space between one and two. The use of standard meshing techniques makes extension of the method to three dimensions conceptually straightforward.

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.

Spatial modeling of cell signaling networks.

The shape of a cell, the sizes of subcellular compartments, and the spatial distribution of molecules within the cytoplasm can all control how molecules interact to produce a cellular behavior. This chapter describes how these spatial features can be included in mechanistic mathematical models of cell signaling. The Virtual Cell computational modeling and simulation software is used to illustrate the considerations required to build a spatial model. An explanation of how to appropriately choose between physical formulations that implicitly or explicitly account for cell geometry and between deterministic versus stochastic formulations for molecular dynamics is provided, along with a discussion of their respective strengths and weaknesses. As a first step toward constructing a spatial model, the geometry needs to be specified and associated with the molecules, reactions, and membrane flux processes of the network. Initial conditions, diffusion coefficients, velocities, and boundary conditions complete the specifications required to define the mathematics of the model. The numerical methods used to solve reaction-diffusion problems both deterministically and stochastically are then described and some guidance is provided in how to set up and run simulations. A study of cAMP signaling in neurons ends the chapter, providing an example of the insights that can be gained in interpreting experimental results through the application of spatial modeling.

Virtual Cell: computational tools for modeling in cell biology.

The Virtual Cell (VCell) is a general computational framework for modeling physicochemical and electrophysiological processes in living cells. Developed by the National Resource for Cell Analysis and Modeling at the University of Connecticut Health Center, it provides automated tools for simulating a wide range of cellular phenomena in space and time, both deterministically and stochastically. These computational tools allow one to couple electrophysiology and reaction kinetics with transport mechanisms, such as diffusion and directed transport, and map them onto spatial domains of various shapes, including irregular three-dimensional geometries derived from experimental images. In this article, we review new robust computational tools recently deployed in VCell for treating spatially resolved models.

Diffusion amid random overlapping obstacles: similarities, invariants, approximations.

Efficient and accurate numerical techniques are used to examine similarities of effective diffusion in a void between random overlapping obstacles: essential invariance of effective diffusion coefficients (D(eff)) with respect to obstacle shapes and applicability of a two-parameter power law over nearly entire range of excluded volume fractions (φ), except for a small vicinity of a percolation threshold. It is shown that while neither of the properties is exact, deviations from them are remarkably small. This allows for quick estimation of void percolation thresholds and approximate reconstruction of D(eff) (φ) for obstacles of any given shape. In 3D, the similarities of effective diffusion yield a simple multiplication "rule" that provides a fast means of estimating D(eff) for a mixture of overlapping obstacles of different shapes with comparable sizes.

Use of virtual cell in studies of cellular dynamics.

The Virtual Cell (VCell) is a unique computational environment for modeling and simulation of cell biology. It has been specifically designed to be a tool for a wide range of scientists, from experimental cell biologists to theoretical biophysicists. The models created with VCell can range from the simple, to evaluate hypotheses or to interpret experimental data, to complex multilayered models used to probe the predicted behavior of spatially resolved, highly nonlinear systems. In this chapter, we discuss modeling capabilities of VCell and demonstrate representative examples of the models published by the VCell users.

Quantifying a pathway: kinetic analysis of actin dendritic nucleation.

Progress in uncovering the reaction networks that underlie important cell functions is laying the groundwork for quantitative identification of protein-interaction pathways. Since direct measurement of rate constants is not always feasible, the parameters are often inferred from multiple pieces of data using kinetic analyses based on appropriate mathematical models. The success of this approach relies on the sufficiency of available experimental data for a unique parameterization of the network. The concept of a rate-limiting step is applied to the analysis of experimental data that are usually used to quantify a pathway of actin dendritic nucleation, the Arp2/3-mediated mechanism that enables rapid changes of cell shape in response to external cues. The method yields analytical descriptions of the dynamics of polymerized actin and provides insights into how the experimental curves should be analyzed. It is shown that dynamics measured by pyrene-labeled actin assays with varying Arp2/3 concentrations are equally well described by two different rate-limiting steps: 1), binding of a nucleating complex to the side of a preexisting filament; or 2), its subsequent activation. To distinguish between the alternatives, we propose experiments with varying concentrations of actin monomers, taking advantage of the fact that the number of branches in the two cases depends differently on the initial monomer concentration. The idea is tested by simulating the proposed experiments with the use of spatial stochastic modeling.

CLIP-170-dependent capture of membrane organelles by microtubules initiates minus-end directed transport.

Cytoplasmic microtubules (MTs) continuously grow and shorten at free plus ends. During mitosis, this dynamic behavior allows MTs to capture chromosomes to initiate their movement to the spindle poles; however, the role of MT dynamics in capturing organelles for transport in interphase cells has not been demonstrated. Here we use Xenopus melanophores to test the hypothesis that MT dynamics significantly contribute to the efficiency of MT minus-end directed transport of membrane organelles. We demonstrate that initiation of transport of membrane-bounded melanosomes (pigment granules) to the cell center involves their capture by MT plus ends, and that inhibition of MT dynamics or loss of the MT plus-end tracking protein CLIP-170 from MT tips dramatically inhibits pigment aggregation. We conclude that MT dynamics are required for the initiation of MT transport of membrane organelles in interphase cells, and that +TIPs such as CLIP-170 play an important role in this process.

Diffusion in cytoplasm: effects of excluded volume due to internal membranes and cytoskeletal structures.

The intricate geometry of cytoskeletal networks and internal membranes causes the space available for diffusion in cytoplasm to be convoluted, thereby affecting macromolecule diffusivity. We present a first systematic computational study of this effect by approximating intracellular structures as mixtures of random overlapping obstacles of various shapes. Effective diffusion coefficients are computed using a fast homogenization technique. It is found that a simple two-parameter power law provides a remarkably accurate description of effective diffusion over the entire range of volume fractions and for any given composition of structures. This universality allows for fast computation of diffusion coefficients, once the obstacle shapes and volume fractions are specified. We demonstrate that the excluded volume effect alone can account for a four-to-sixfold reduction in diffusive transport in cells, relative to diffusion in vitro. The study lays the foundation for an accurate coarse-grain formulation that would account for cytoplasm heterogeneity on a micron scale and binding of tracers to intracellular structures.

Actin bundling: initiation mechanisms and kinetics.

Bundling of rapidly polymerizing actin filaments underlies the dynamics of filopodial protrusions that play an important role in cell migration and cell-cell interaction. Recently, the formation of actin bundles has been reconstituted in vitro, and two scenarios of bundle initiation, involving binding of two filament tips and, alternatively, linking of the tip of one filament to the side of the other, have been discussed. A first theoretical analysis is presented indicating that the two mechanisms can be distinguished experimentally. While both of them result counterintuitively in comparable numbers of bundles, these numbers scale differently with the average bundle length. We propose an experiment for determining which of the two mechanisms is involved in the in vitro bundle formation.

On sensitivity amplification in intracellular signaling cascades.

Sensitivity amplification has long been regarded as a virtually universal property of signal transduction cascades, yet a comprehensive parameter analysis remains a challenge even for relatively simple networks. We use a fast and accurate method to compute properties of multilevel cascades of activation-inactivation cycles and show that the monocyclic cascades amplify sensitivity only under specific conditions. In particular, it is found that efficient sensitivity amplification in a cascade, relative to the sensitivities of individual cycles, requires asymmetry in saturation of converter enzymes, with inhibitors much more saturated than activators.

A model of the kinetics of lanthanum in human bone, using data collected during the clinical development of the phosphate binder lanthanum carbonate.

OBJECTIVE: Lanthanum carbonate (Fosrenol) is a non-calcium phosphate binder that controls hyperphosphataemia without increasing calcium intake above guideline targets. The biological fate and bone load of lanthanum were modelled with the aid of a four-compartment kinetic model, analogous to that of calcium. METHODS: The model used data from healthy subjects who received intravenous lanthanum chloride or oral lanthanum carbonate, and bone lanthanum concentration data collected from dialysis patients during three long-term trials (up to 5 years). RESULTS: Infusion of lanthanum chloride or ingestion of lanthanum carbonate led to a rapid rise in plasma lanthanum concentrations, followed by an exponential decrease. Comparison of oral and intravenous exposure confirmed that lanthanum is very poorly absorbed. On a typical intake of lanthanum (3000 mg/day as lanthanum carbonate), the rate of absorption was calculated as 2.2 microg/h, with a urinary excretion rate constant of 0.004-0.01 h(-1). The faecal content of endogenous lanthanum was estimated to be 8- to 20-fold greater than that of urine, compared with a ratio of only about 1 for calcium. The model predicts that upon multiple dosing, plasma lanthanum concentrations rise rapidly to a near plateau and then increase by about 3% per year. However, this small change is obscured by the variability of the study data, which show that a plateau is rapidly attained by 2 weeks and is thereafter maintained for at least 2 years. The initial deposition rate of lanthanum in bone was 1 microg/g/year and, after 10 years of lanthanum carbonate treatment, the model predicts a 7-fold increase in total bone lanthanum (from 10 mg to 69 mg [from 1 microg/g wet weight to 6.6 microg/g wet weight]), with lanthanum cleared after cessation of treatment at 13% per year. The model indicates that lanthanum flow from bone surface to bone interior is much lower than that of calcium. CONCLUSION: Bone is the major reservoir for metals, but bone lanthanum concentrations are predicted to remain low after long-term treatment because of very poor intestinal absorption.

Quantitative analysis of G-actin transport in motile cells.

Cell migration is based on an actin treadmill, which in turn depends on recycling of G-actin across the cell, from the rear where F-actin disassembles, to the front, where F-actin polymerizes. To analyze the rates of the actin transport, we used the Virtual Cell software to solve the diffusion-drift-reaction equations for the G-actin concentration in a realistic three-dimensional geometry of the motile cell. Numerical solutions demonstrate that F-actin disassembly at the cell rear and assembly at the front, along with diffusion, establish a G-actin gradient that transports G-actin forward "globally" across the lamellipod. Alternatively, if the F-actin assembly and disassembly are distributed throughout the lamellipod, F-/G-actin turnover is local, and diffusion plays little role. Chemical reactions and/or convective flow of cytoplasm of plausible magnitude affect the transport very little. Spatial distribution of G-actin is smooth and not sensitive to F-actin density fluctuations. Finally, we conclude that the cell body volume slows characteristic diffusion-related relaxation time in motile cell from approximately 10 to approximately 100 s. We discuss biological implications of the local and global regimes of the G-actin transport.

Switching of membrane organelles between cytoskeletal transport systems is determined by regulation of the microtubule-based transport.

Intracellular transport of membrane organelles occurs along microtubules (MTs) and actin filaments (AFs). Although transport along each type of the cytoskeletal tracks is well characterized, the switching between the two types of transport is poorly understood because it cannot be observed directly in living cells. To gain insight into the regulation of the switching of membrane organelles between the two major transport systems, we developed a novel approach that combines live cell imaging with computational modeling. Using this approach, we measured the parameters that determine how fast membrane organelles switch back and forth between MTs and AFs (the switching rate constants) and compared these parameters during different signaling states. We show that regulation involves a major change in a single parameter: the transferring rate from AFs onto MTs. This result suggests that MT transport is the defining factor whose regulation determines the choice of the cytoskeletal tracks during the transport of membrane organelles.