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.

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.

To understand shapes and movements of simple motile cells, we systematically explore minimal models describing a cell as a two-dimensional actin-myosin gel with a free boundary. The models account for actin-myosin contraction balanced by viscous stresses in the actin gel and uniform adhesion. The myosin contraction causes the lamellipodial boundary to retract. Actin protrusion at the boundary counters the retraction, and the balance of protrusion and retraction shapes the cell. The models reproduce a variety of motile shapes observed experimentally. The analysis shows that the mechanical state of a cell depends on a small number of parameters. We find that when the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectory. Scanning model parameters shows that the contractile mechanism of motility supports robust cell turning behavior in conditions where deformable actin gel and fast protrusion destabilize 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.

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.

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.

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.

Virtual Cell modelling and simulation software environment

The Virtual Cell (VCell; http://vcell.org/) is a problem solving environment, built on a central database, for analysis, modelling and simulation of cell biological processes. VCell integrates a growing range of molecular mechanisms, including reaction kinetics, diffusion, flow, membrane transport, lateral membrane diffusion and electrophysiology, and can associate these with geometries derived from experimental microscope images. It has been developed and deployed as a web-based, distributed, client-server system, with more than a thousand world-wide users. VCell provides a separation of layers (core technologies and abstractions) representing biological models, physical mechanisms, geometry, mathematical models and numerical methods. This separation clarifies the impact of modelling decisions, assumptions and approximations. The result is a physically consistent, mathematically rigorous, spatial modelling and simulation framework. Users create biological models and VCell will automatically (i) generate the appropriate mathematical encoding for running a simulation and (ii) generate and compile the appropriate computer code. Both deterministic and stochastic algorithms are supported for describing and running non-spatial simulations; a full partial differential equation solver using the finite volume numerical algorithm is available for reaction-diffusion-advection simulations in complex cell geometries including 3D geometries derived from microscope images. Using the VCell database, models and model components can be reused and updated, as well as privately shared among collaborating groups, or published. Exchange of models with other tools is possible via import/export of SBML, CellML and MatLab formats. Furthermore, curation of models is facilitated by external database binding mechanisms for unique identification of components and by standardised annotations compliant with the MIRIAM standard. VCell is now open source, with its native model encoding language (VCML) being a public specification, which stands as the basis for a new generation of more customised, experiment-centric modelling tools using a new plug-in based platform.

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.

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.

Diffusion on a Curved Surface Coupled to Diffusion in the Volume: Application to Cell Biology

An algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume, a problem often arising in cell biology. It applies to pixilated surfaces obtained from experimental images and performs at low computational cost. In the method, the Laplace-Beltrami operator is approximated locally by the Laplacian on the tangential plane and then a finite volume discretization scheme based on a Voronoi decomposition is applied. Convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2. The method is applied to a cell-biological problem where a signaling molecule, G-protein Rac, cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior. Simulations on realistic cell geometry are performed to validate, and determine the accuracy of, a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching. The method is implemented within the Virtual Cell computational framework freely accessible at www.vcell.org.

In vivo dynamics of Rac-membrane interactions

The small GTPase Rac cycles between the membrane and the cytosol as it is activated by nucleotide exchange factors (GEFs) and inactivated by GTPase-activating proteins (GAPs). Solubility in the cytosol is conferred by binding of Rac to guanine-nucleotide dissociation inhibitors (GDIs). To analyze the in vivo dynamics of Rac, we developed a photobleaching method to measure the dissociation rate constant (k(off)) of membrane-bound GFP-Rac. We find that k(off) is 0.048 s(-1) for wtRac and approximately 10-fold less (0.004 s(-1)) for G12VRac. Thus, the major route for dissociation is conversion of membrane-bound GTP-Rac to GDP-Rac; however, dissociation of GTP-Rac occurs at a detectable rate. Overexpression of the GEF Tiam1 unexpectedly decreased k(off) for wtRac, most likely by converting membrane-bound GDP-Rac back to GTP-Rac. Both overexpression and small hairpin RNA-mediated suppression of RhoGDI strongly affected the amount of membrane-bound Rac but surprisingly had only slight effects on k(off). These results indicate that RhoGDI controls Rac function mainly through effects on activation and/or membrane association.

Modeling and analysis of calcium signaling events leading to long-term depression in cerebellar Purkinje cells

Modeling and simulation of the calcium signaling events that precede long-term depression of synaptic activity in cerebellar Purkinje cells are performed using the Virtual Cell biological modeling framework. It is found that the unusually high density and low sensitivity of inositol-1,4,5-trisphosphate receptors (IP3R) are critical to the ability of the cell to generate and localize a calcium spike in a single dendritic spine. The results also demonstrate the model's capability to simulate the supralinear calcium spike observed experimentally during coincident activation of the parallel and climbing fibers. The sensitivity of the calcium spikes to certain biological and geometrical effects is investigated as well as the mechanisms that underlie the cell's ability to generate the supralinear spike. The sensitivity of calcium release rates from the IP3R to calcium concentrations, as well as IP3 concentrations, allows the calcium spike to form. The diffusion barrier caused by the small radius of the spine neck is shown to be important, as a threshold radius is observed above which a spike cannot be formed. Additionally, the calcium buffer capacity and diffusion rates from the spine are found to be important parameters in shaping the calcium spike.

Cooperativity between cell contractility and adhesion

In living cells, the cytoskeleton connects to the extracellular environment through focal adhesions, multimolecular structures that can sense applied force. A model is presented that for the first time explains why the focal adhesions tend to high-curvature regions at the cell periphery. It is based on experimental evidence for positive feedback between adhesion formation and assembly of actomyosin bundles (stress fibers). The model predicts that the focal adhesions propagate by treadmilling with a velocity proportional to the integrin diffusion coefficient.

Bio-switches: what makes them robust?

Ideas of how a system of interacting enzymes can act as a switch are based on the concept of bistability of a biochemical network. This means that, because of the very structure of a signaling pathway, the system can be in one of two stable steady states: active or inactive. Switching from one state to another may then occur in response to external stimuli or as a result of internal development. However, the bistability of a biochemical network might not be robust enough to be the sole mechanism behind bio-switching. On the basis of recent experimental data on the cell-cycle G2/M transition during starfish oocyte meiotic maturation, it is shown that cooperative phenomena--such as phase changes associated with clustering, dissolution of aggregates and so on--may play central roles in providing a decisive and irreversible transition.

Quantitative cell biology with the Virtual Cell

Cell biological processes are controlled by an interacting set of biochemical and electrophysiological events that are distributed within complex cellular structures. Computational models, comprising quantitative data on the interacting molecular participants in these events, provide a means for applying the scientific method to these complex systems. The Virtual Cell is a computational environment designed for cell biologists, to facilitate the construction of models and the generation of predictive simulations from them. This review summarizes how a Virtual Cell model is assembled and describes the physical principles underlying the calculations that are performed. Applications to problems in nucleocytoplasmic transport and intracellular calcium dynamics will illustrate the power of this paradigm for elucidating cell biology.

Cyclin aggregation and robustness of bio-switching

During the cell cycle, Cdc2-cyclin B kinase abruptly becomes active and triggers the entry into mitosis/meiosis. Recently, it was found that inactive Cdc2-cyclin B is present in aggregates in immature starfish oocytes and becomes disaggregated at the time of its activation during maturation. We discuss a possible scenario in which aggregation of Cdc2-cyclin B dramatically enhances robustness of this activation. In this scenario, only inactive Cdc2-cyclin B can form aggregates, and the aggregates are in equilibrium with inactive Cdc2-cyclin B in solution. During maturation, the hormone-triggered inactivation of Myt1 depletes the soluble inactive Cdc2-cyclin B and the turnover leads to dissolution of the aggregates. This phase change, when coupled with the instability of the signaling network, provides a robust bio-switch.

Computational cell biology: spatiotemporal simulation of cellular events

The field of computational cell biology has emerged within the past 5 years because of the need to apply disciplined computational approaches to build and test complex hypotheses on the interacting structural, physical, and chemical features that underlie intracellular processes. To meet this need, newly developed software tools allow cell biologists and biophysicists to build models and generate simulations from them. The construction of general-purpose computational approaches is especially challenging if the spatial complexity of cellular systems is to be explicitly treated. This review surveys some of the existing efforts in this field with special emphasis on a system being developed in the authors' laboratory, Virtual Cell. The theories behind both stochastic and deterministic simulations are discussed. Examples of respective applications to cell biological problems in RNA trafficking and neuronal calcium dynamics are provided to illustrate these ideas.

Systems analysis of Ran transport

The separate components of nucleocytoplasmic transport have been well characterized, including the key regulatory role of Ran, a guanine nucleotide triphosphatase. However, the overall system behavior in intact cells is difficult to analyze because the dynamics of these components are interdependent. We used a combined experimental and computational approach to study Ran transport in vivo. The resulting model provides the first quantitative picture of Ran flux between the nuclear and cytoplasmic compartments in eukaryotic cells. The model predicts that the Ran exchange factor RCC1, and not the flux capacity of the nuclear pore complex (NPC), is the crucial regulator of steady-state flux across the NPC. Moreover, it provides the first estimate of the total in vivo flux (520 molecules per NPC per second and predicts that the transport system is robust.

Modeling of transcellular Ca transport in rat duodenum points to coexistence of two mechanisms of apical entry

Employing realistic parameters, we have demonstrated that a relatively simple mathematical model can reproduce key features of steady-state Ca2+ transport with the assumption of two mechanisms of Ca2+ entry: a channel-like flux and a carrier-mediated transport. At low luminal [Ca2+] (1-5 mM), facilitated entry dominates and saturates with Km = 0.4 mM. At luminal [Ca2+] of tens of millimolar, apical permeability is dominated by the channel flux that in turn is regulated by cytosolic Ca2+. The model reproduces the linear relationship between maximum Ca2+ transport rate and intestinal calbindin D9K (CaBP) content. At luminal [Ca2+] > 50 mM, local sensitivity analysis shows transcellular transport to be most sensitive to variations in CaBP. At low luminal [Ca2+], transport becomes sensitive to apical entry regulation. The simulations have been run within the Virtual Cell modeling environment, yielding the time course of external Ca2+ and spatiotemporal distributions of both intracellular Ca2+ and CaBP. Coexistence of two apical entry mechanisms accords with the properties of the duodenal Ca2+ transport protein CaT1 and the epithelial Ca2+ channel ECaC.

Analysis of nonlinear dynamics on arbitrary geometries with the Virtual Cell

The Virtual Cell is a modeling tool that allows biologists and theorists alike to specify and simulate cell-biophysical models on arbitrarily complex geometries. The framework combines an intuitive, front-end graphical user interface that runs in a web browser, sophisticated server-side numerical algorithms, a database for storage of models and simulation results, and flexible visualization capabilities. In this paper, we present an overview of the capabilities of the Virtual Cell, and, for the first time, the detailed mathematical formulation used as the basis for spatial computations. We also present summaries of two rather typical modeling projects, in order to illustrate the principal capabilities of the Virtual Cell. (c) 2001 American Institute of Physics.

Physiological modeling with virtual cell framework

This article describes a computational framework for cell biological modeling and simulation that is based on the mapping of experimental biochemical and electrophysiological data onto experimental images. The framework is designed to enable the construction of complex general models that encompass the general class of problems coupling reaction and diffusion.