Use of virtual cell in studies of cellular dynamics

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.

Computational investigation of the dynamic control of cAMP signaling by PDE4 isoform types

Cyclic adenosine monophosphate (cAMP) is a generic signaling molecule that, through precise control of its signaling dynamics, exerts distinct cellular effects. Consequently, aberrant cAMP signaling can have detrimental effects. Phosphodiesterase 4 (PDE4) enzymes profoundly control cAMP signaling and comprise different isoform types of which the enzymatic activity is modulated by differential feedback mechanisms. Because these feedback dynamics are non-linear and occur coincidentally, their effects are difficult to examine experimentally, but can be well simulated computationally. Through understanding the role of PDE4 isoform types in regulating cAMP signaling, PDE4-targeted therapeutic strategies can be better specified. Here, we established a computational model to study how feedback mechanisms on different PDE4 isoform types lead to dynamic, isoform-specific control of cAMP signaling. Ordinary differential equations describing cAMP dynamics were implemented in the VirtualCell (VCell) environment. Simulations indicated that long PDE4 isoforms exert the most profound control on oscillatory cAMP signaling, as opposed to the PDE4-mediated control of single cAMP input pulses. Moreover, elevating cAMP levels or decreasing PDE4 levels revealed different effects on downstream signaling. Together these results underline that cAMP signaling is distinctly regulated by different PDE4 isoform types and that this isoform-specificity should be considered in both computational and experimental follow-up studies to better define PDE4 enzymes as therapeutic targets in diseases in which cAMP signaling is aberrant.

Exploring the influence of cytosolic and membrane FAK activation on YAP/TAZ nuclear translocation

Membrane binding and unbinding dynamics play a crucial role in the biological activity of several nonintegral membrane proteins, which have to be recruited to the membrane to perform their functions. By localizing to the membrane, these proteins are able to induce downstream signal amplification in their respective signaling pathways. Here, we present a 3D computational approach using reaction-diffusion equations to investigate the relation between membrane localization of focal adhesion kinase (FAK), Ras homolog family member A (RhoA), and signal amplification of the YAP/TAZ signaling pathway. Our results show that the theoretical scenarios in which FAK is membrane bound yield robust and amplified YAP/TAZ nuclear translocation signals. Moreover, we predict that the amount of YAP/TAZ nuclear translocation increases with cell spreading, confirming the experimental findings in the literature. In summary, our in silico predictions show that when the cell membrane interaction area with the underlying substrate increases, for example, through cell spreading, this leads to more encounters between membrane-bound signaling partners and downstream signal amplification. Because membrane activation is a motif common to many signaling pathways, this study has important implications for understanding the design principles of signaling networks.

The Influence of OAT1 Density and Functionality on Indoxyl Sulfate Transport in the Human Proximal Tubule: An Integrated Computational and In Vitro Study

Research has shown that traditional dialysis is an insufficient long-term therapy for patients suffering from end-stage kidney disease due to the high retention of uremic toxins in the blood as a result of the absence of the active transport functionality of the proximal tubule (PT). The PT’s function is defined by the epithelial membrane transporters, which have an integral role in toxin clearance. However, the intricate PT transporter–toxin interactions are not fully explored, and it is challenging to decouple their effects in toxin removal in vitro. Computational models are necessary to unravel and quantify the toxin–transporter interactions and develop an alternative therapy to dialysis. This includes the bioartificial kidney, where the hollow dialysis fibers are covered with kidney epithelial cells. In this integrated experimental–computational study, we developed a PT computational model that focuses on indoxyl sulfate (IS) transport by organic anionic transporter 1 (OAT1), capturing the transporter density in detail along the basolateral cell membrane as well as the activity of the transporter and the inward boundary flux. The unknown parameter values of the OAT1 density (1.15×107 transporters µm−2), IS uptake (1.75×10−5 µM−1 s−1), and dissociation (4.18×10−4 s−1) were fitted and validated with experimental LC-MS/MS time-series data of the IS concentration. The computational model was expanded to incorporate albumin conformational changes present in uremic patients. The results suggest that IS removal in the physiological model was influenced mainly by transporter density and IS dissociation rate from OAT1 and not by the initial albumin concentration. While in uremic conditions considering albumin conformational changes, the rate-limiting factors were the transporter density and IS uptake rate, which were followed closely by the albumin-binding rate and IS dissociation rate. In summary, the results of this study provide an exciting avenue to help understand the toxin–transporter complexities in the PT and make better-informed decisions on bioartificial kidney designs and the underlining transporter-related issues in uremic patients.

Ten steps to investigate a cellular system with mathematical modeling

Cellular and intracellular processes are inherently complex due to the large number of components and interactions, which are often nonlinear and occur at different spatiotemporal scales. Because of this complexity, mathematical modeling is increasingly used to simulate such systems and perform experiments in silico, many orders of magnitude faster than real experiments and often at a higher spatiotemporal resolution. In this article, we will focus on the generic modeling process and illustrate it with an example model of membrane lipid turnover.

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.

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.

Simultaneous quantification of actin monomer and filament dynamics with modeling-assisted analysis of photoactivation

Photoactivation allows one to pulse-label molecules and obtain quantitative data about their behavior. We have devised a new modeling-based analysis for photoactivatable actin experiments that simultaneously measures properties of monomeric and filamentous actin in a three-dimensional cellular environment. We use this method to determine differences in the dynamic behavior of β- and γ-actin isoforms, showing that both inhabit filaments that depolymerize at equal rates but that β-actin exists in a higher monomer-to-filament ratio. We also demonstrate that cofilin (cofilin 1) equally accelerates depolymerization of filaments made from both isoforms, but is only required to maintain the β-actin monomer pool. Finally, we used modeling-based analysis to assess actin dynamics in axon-like projections of differentiating neuroblastoma cells, showing that the actin monomer concentration is significantly depleted as the axon develops. Importantly, these results would not have been obtained using traditional half-time analysis. Given that parameters of the publicly available modeling platform can be adjusted to suit the experimental system of the user, this method can easily be used to quantify actin dynamics in many different cell types and subcellular compartments.

Computational neurobiology is a useful tool in translational neurology: the example of ataxia

Hereditary ataxia, or motor incoordination, affects approximately 150,000 Americans and hundreds of thousands of individuals worldwide with onset from as early as mid-childhood. Affected individuals exhibit dysarthria, dysmetria, action tremor, and diadochokinesia. In this review, we consider an array of computational studies derived from experimental observations relevant to human neuropathology. A survey of related studies illustrates the impact of integrating clinical evidence with data from mouse models and computational simulations. Results from these studies may help explain findings in mice, and after extensive laboratory study, may ultimately be translated to ataxic individuals. This inquiry lays a foundation for using computation to understand neurobiochemical and electrophysiological pathophysiology of spinocerebellar ataxias and may contribute to development of therapeutics. The interdisciplinary analysis suggests that computational neurobiology can be an important tool for translational neurology.

Dynamic metabolic control of an ion channel

G-protein-coupled receptors mediate responses to external stimuli in various cell types. We are interested in the modulation of KCNQ2/3 potassium channels by the Gq-coupled M1 muscarinic (acetylcholine) receptor (M1R). Here, we describe development of a mathematical model that incorporates all known steps along the M1R signaling cascade and accurately reproduces the macroscopic behavior we observe when KCNQ2/3 currents are inhibited following M1R activation. Gq protein-coupled receptors of the plasma membrane activate phospholipase C (PLC) which cleaves the minor plasma membrane lipid phosphatidylinositol 4,5-bisphosphate (PI(4,5)P2) into the second messengers diacylgycerol and inositol 1,4,5-trisphosphate, leading to calcium release, protein kinase C (PKC) activation, and PI(4,5)P2 depletion. Combining optical and electrical techniques with knowledge of relative abundance of each signaling component has allowed us to develop a kinetic model and determine that (i) M1R activation and M1R/Gβ interaction are fast; (ii) Gαq/Gβ separation and Gαq/PLC interaction have intermediate time constants; (iii) the amount of activated PLC limits the rate of KCNQ2/3 suppression; (iv) weak PLC activation can elicit robust calcium signals without net PI(4,5)P2 depletion or KCNQ2/3 channel inhibition; and (v) depletion of PI(4,5)P2, and not calcium/CaM or PKC-mediated phosphorylation, closes KCNQ2/3 potassium channels, thereby increasing neuronal excitability.

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.

Computational analysis of Rho GTPase cycling

The Rho family of GTPases control actin organization during diverse cellular responses (migration, cytokinesis and endocytosis). Although the primary members of this family (RhoA, Rac and Cdc42) have different downstream effects on actin remodeling, the basic mechanism involves targeting to the plasma membrane and activation by GTP binding. Our hypothesis is that the details of GTPase cycling between membrane and cytosol are key to the differential upstream regulation of these biochemical switches. Accordingly, we developed a modeling framework to analyze experimental data for these systems. This analysis can reveal details of GDI-mediated cycling and help distinguish between GDI-dependent and -independent mechanisms, including vesicle trafficking and direct association-dissociation of GTPase with membrane molecules. Analysis of experimental data for Rac membrane cycling reveals that the lower apparent affinity of GDI for RacGTP compared to RacGDP can be fully explained by the faster dissociation of the latter from the membrane. Non-dimensional steady-state solutions for membrane fraction of GTPase are presented in multidimensional charts. This methodology is then used to analyze glucose stimulated Rac cycling in pancreatic β-cells. The charts are used to illustrate the effects of GEFs/GAPs and regulated affinities between GTPases and membrane and/or GDI on the amount of membrane bound GTPase. In a similar fashion, the charts can be used as a guide in assessing how targeted modifications may compensate for altered GTPase-GDI balance in disease scenarios.

Among the functions of the small GTPases Rac, RhoA and Cdc42 are the regulation of protein traffic, insulin secretion, cell shape, survival and motility. The last two are important steps for tumor growth and metastasis. The function of these proteins relies on their expression levels, proper membrane localization and activation. In addition, all three proteins compete for the same protein ‘GDI’, which modulates their cycling. These proteins are ubiquitous in mammalian cells, but also studied in simpler in vitro systems and cultured yeast. Here we show, using a series of computational analyses, that for each of these experimental systems the dominant pathway for membrane cycling of GTPases seems to differ. This means that the researcher interested in the physiological function of any of those proteins must make sure that the experimental system is appropriate. We present a methodology to identify the dominant pathways by measuring the apparent membrane dissociation rate of the protein as a function of GDI concentration. We provide charts generated from parametric scans. This analysis is then applied to the Rac-dependent insulin secretion pathway in pancreatic ß-cells, revealing that direct signaling between Rac and the membrane is an essential mechanism that emerges from the data.

Tunable oscillations and chaotic dynamics in systems with localized synthesis

Biological systems contain biochemical control networks that reside within a remarkable spatial structure. We present a theoretical study of a biological system in which two chemically coupled species, an activating species and an inhibiting species forming a negative feedback, are synthesized at unique sites and interact with each other through diffusion. The dynamical behaviors in these systems depend on the spatial locations of these synthetic sites. In a negative feedback system with two sites, we find two dynamical modes: fixed point and stable oscillations whose frequency can be tuned by varying the distance between the sites. When there are multiple synthetic sites, we find more diverse dynamics, including chaos, quasiperiodicity, and bistability. Based on this theoretical analysis, it should be possible to create in the laboratory synthetic circuits displaying these dynamics. This study illustrates the concept of "spatial switching," in which bifurcations in the dynamics occur as a function of the geometry of the system.

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.

Redundant mechanisms for stable cell locomotion revealed by minimal models

Crawling of eukaryotic cells on flat surfaces is underlain by the protrusion of the actin network, the contractile activity of myosin II motors, and graded adhesion to the substrate regulated by complex biochemical networks. Some crawling cells, such as fish keratocytes, maintain a roughly constant shape and velocity. Here we use moving-boundary simulations to explore four different minimal mechanisms for cell locomotion: 1), a biophysical model for myosin contraction-driven motility; 2), a G-actin transport-limited motility model; 3), a simple model for Rac/Rho-regulated motility; and 4), a model that assumes that microtubule-based transport of vesicles to the leading edge limits the rate of protrusion. We show that all of these models, alone or in combination, are sufficient to produce half-moon steady shapes and movements that are characteristic of keratocytes, suggesting that these mechanisms may serve redundant and complementary roles in driving cell motility. Moving-boundary simulations demonstrate local and global stability of the motile cell shapes and make testable predictions regarding the dependence of shape and speed on mechanical and biochemical parameters. The models shed light on the roles of membrane-mediated area conservation and the coupling of mechanical and biochemical mechanisms in stabilizing motile cells.

Modeling cellular processes in 3D

Recent advances in photonic imaging and fluorescent protein technology offer unprecedented views of molecular space-time dynamics in living cells. At the same time, advances in computing hardware and software enable modeling of ever more complex systems, from global climate to cell division. As modeling and experiment become more closely integrated we must address the issue of modeling cellular processes in 3D. Here, we highlight recent advances related to 3D modeling in cell biology. While some processes require full 3D analysis, we suggest that others are more naturally described in 2D or 1D. Keeping the dimensionality as low as possible reduces computational time and makes models more intuitively comprehensible; however, the ability to test full 3D models will build greater confidence in models generally and remains an important emerging area of cell biological modeling.

The emergence of sarcomeric, graded-polarity and spindle-like patterns in bundles of short cytoskeletal polymers and two opposite molecular motors

We use linear stability analysis and numerical solutions of partial differential equations to investigate pattern formation in the one-dimensional system of short dynamic polymers and one (plus-end directed) or two (one is plus-end, another minus-end directed) molecular motors. If polymer sliding and motor gliding rates are slow and/or the polymer turnover rate is fast, then the polymer-motor bundle has mixed polarity and homogeneous motor distribution. However, if motor gliding is fast, a sarcomeric pattern with periodic bands of alternating polymer polarity separated by motor aggregates evolves. On the other hand, if polymer sliding is fast, a graded-polarity bundle with motors at the center emerges. In the presence of the second, minus-end directed motor, the sarcomeric pattern is more ubiquitous, while the graded-polarity pattern is destabilized. However, if the minus-end motor is weaker than the plus-end directed one, and/or polymer nucleation is autocatalytic, and/or long polymers are present in the bundle, then a spindle-like architecture with a sorted-out polarity emerges with the plus-end motors at the center and minus-end motors at the edges. We discuss modeling implications for actin-myosin fibers and in vitro and meiotic spindles.

Developing models in virtual cell

This Teaching Resource provides lecture notes, slides, and a student assignment for a two-part lecture on mathematical modeling using the Virtual Cell environment. The lectures discuss the steps involved in developing and running simulations using Virtual Cell, with particular focus on spatial partial differential equation models. We discuss how to construct both ordinary differential equation models, in which the cytoplasm is considered a well-mixed cellular compartment, and partial differential equation models, which calculate how chemical species change as a function of both time and location. The Virtual Cell environment is especially well suited for models that explore spatial specificity of cellular reactions. Partial differential equation models in Virtual Cell can give rise to simulations using predefined cellular geometries, which enable direct comparison with imaging data. These models address questions regarding the regulatory capability arising from spatial organization of the cell. Examples are provided of studies that have successfully exploited the Virtual Cell software to address the spatial contribution to signaling.

Modeling of spatially-restricted intracellular signaling

Understanding the signaling capabilities of a cell presents a major challenge, not only due to the number of molecules involved, but also because of the complex network connectivity of intracellular signaling. Recently, the proliferation of quantitative imaging techniques has led to the discovery of the vast spatial organization of intracellular signaling. Computational modeling has emerged as a powerful tool for understanding how inhomogeneous signaling originates and is maintained. This article covers the current imaging techniques used to obtain quantitative spatial data and the mathematical approaches used to model spatial cell biology. Modeling-derived hypotheses have been experimentally tested and the integration of modeling and imaging approaches has led to non-intuitive mechanistic insights.

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.