Igor L. Novak

Diffusion on a curved surface coupled to diffusion in the volume

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.

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.

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. WIREs Syst Biol Med 2012, 4:129–140. doi: 10.1002/wsbm.165

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.

Nonaffine response of skeletal muscles on the ‘descending limb’

Tetanized muscle myofibrils are often modeled as one-dimensional chains where springs represent half-sarcomeres (HS). The force–length relation for individual HSs (isometric tetanus) is known to have a ‘descending limb’, a segment with an apparently negative stiffness. Despite the potential mechanical instability on the descending limb, the isometric tetanus is usually interpreted as describing an affine deformation. At the same time, active stretching during tetanus around the descending limb is known to produce non-affine sarcomere patterns. In view of this paradox, the question whether the mechanical behavior of a myofibril can be interpreted as a response of a single contractile unit has been a subject of considerable controversy over the last 50 years. In this paper we question the claim that the isometric tetanus describes homogeneous configurations of the HS chain. To distinguish between the multitudes of non-affine equilibrium states available to this mechanical system, we propose to use the concept of a stored mechanical energy. While the notion of energy is natural from a mechanical point of view, physiologists have resisted it so far on the grounds that the contractile elements are active. We discuss how this objection can be overcome and show that the appropriately defined stored energy of a tetanized myofibril with N contractile units has exponentially many local minima. We then argue that the ruggedness of the ensuing energy landscape is responsible for the experimentally observed history dependence and hysteresis in the mechanical response of a tetanized muscle near the descending limb. A nonlocal extension of the chain model, accounting for surrounding tissues, shows that both the ground states and the marginally stable states are fine mixtures of short and long HSs. These mixtures are homogeneous at the macro-scale and inhomogeneous at the micro-scale and we show that the negative overall slope of the step-wise tetanus can coexist with a positive instantaneous stiffness. A salient feature of the nonlocal model is that the variation of the degree of non-uniformity with elongation follows a complete devil’s staircase.

A minimal actomyosin-based model predicts the dynamics of filopodia on neuronal dendrites

A combination of computational and experimental approaches is used to show that the complex dynamics of dendritic filopodia, which is essential for synaptogenesis, is explained by a conceptually simple interplay among actin retrograde flow, myosin contractility, and substrate adhesion.

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 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.

Actomyosin Dynamics Determine the Extension and Retraction of Filopodia on Neuronal Dendrites

Impact Statement In this study, using a combination of computational and experimental approaches we show that a complex dynamic behavior of dendritic filopodia that is essential for synaptogenesis is explained by an interplay among forces generated by actin retrograde flow, myosin contractility, and substrate adhesion. Abstract Dendritic filopodia are actin-filled dynamic subcellular structures that sprout on neuronal dendrites during neurogenesis. The exploratory motion of the filopodia is crucial for synaptogenesis but the underlying mechanisms are poorly understood. To study the filopodial motility, we collected and analyzed image data on filopodia in cultured rat hippocampal neurons. We hypothesized that mechanical feedback among the actin retrograde flow, myosin activity and substrate adhesion gives rise to various filopodial behaviors. We have formulated a minimal one-dimensional partial differential equation model that reproduced the range of observed motility. To validate our model, we systematically manipulated experimental correlates of parameters in the model: substrate adhesion strength, actin polymerization rate, myosin contractility and the integrity of the putative microtubule-based barrier at the filopodium base. The model predicts the response of the system to each of these experimental perturbations, supporting the hypothesis that our actomyosin-driven mechanism controls dendritic filopodia dynamics.