Design of active transport must be highly intricate: a possible role of myosin and Ena/VASP for G-actin transport in filopodia

Multi-Grid Reaction-Diffusion Master Equation: Applications to Morphogen Gradient Modelling

The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem.

Modelling intracellular transport in crowded environments: effects of motor association to cargos

In intracellular transports, motor proteins transport macromolecules as cargos to desired locations by moving on biopolymers such as microtubules. Recent experiments suggest that, while moving in crowded environments, cargos that can associate motor proteins during their translocation have larger run-length and association time compared to free motors. Here, we model the dynamics of a cargo that can associate at the most m free motors present on the microtubule track as obstacles to its motion. The proposed models display competing effects of association and crowding, leading to a peak in the run-length with the free-motor density. For m = 2 and 3, we show that this feature is governed by the largest eigenvalue of the transition matrix describing the cargo dynamics. In all the above cases, free motors are assumed to be present on the microtubule as stalled obstacles. We finally compare simulation results for the run-length for general scenarios where the free motors undergo processive motion in addition to binding and unbinding to or from the microtubule.

IGF1R undergoes active and directed centripetal transport on filopodia upon receptor activation

Filopodia are thin, actin-based membrane protrusions with roles in sensing external mechanical and chemical cues, such as growth factor gradients in tissues. It was proposed that the chemical sensing role of filopodia is achieved through clearance of activated signaling receptors from filopodia. Type I insulin-like growth factor receptor (IGF1R) is a key regulator of normal development and growth, as well as tumor development and progression. Its biological roles depend on its activation upon IGF1 binding at the cell membrane. IGF1R behavior at the cell membrane and in particular in filopodia, has not been established. We found that IGF1 activation led to a gradual reduction in IGF1R puncta in filopodia, and that this clearance depended on actin, non-muscle myosin II, and IGF1R kinase activity. Using single particle tracking of filopodial IGF1R, we established that ligand-free IGF1R undergoes non-directional unidimensional diffusion along the filopodium. Moreover, after initial diffusion, the ligand-bound IGF1R is actively transported along the filopodium towards the filopodium base, and consequently cleared from the filopodium. Our results show that IGF1R can move directionally on the plasma membrane protrusions, supporting a sensory role for filopodia in interpreting local IGF1 gradients.

Turnover versus treadmilling in actin network assembly and remodeling

Actin networks are highly dynamic cytoskeletal structures that continuously undergo structural remodeling. One prominent way to probe these processes is via Fluorescence Recovery After Photobleaching (FRAP), which can be used to estimate the rate of turnover for filamentous actin monomers. It is thought that head-to-tail treadmilling and de novo filament nucleation constitute two primary mechanisms underlying turnover kinetics. More generally, these self-assembly activities are responsible for many important cellular functions such as force generation, cellular shape dynamics, and cellular motility. In what relative proportions filament treadmilling and de novo filament nucleation contribute to actin network turnover is still not fully understood. We used an advanced stochastic reaction-diffusion model in three dimensions, MEDYAN, to study turnover dynamics of actin networks containing Arp2/3, formin and capping protein at experimentally meaningful length- and time-scales. Our results reveal that, most commonly, treadmilling of older filaments is the main contributor to actin network turnover. On the other hand, although turnover and treadmilling are often used interchangeably, we show clear instances where this assumption would not be justified, for example, finding that rapid turnover is accompanied by slow treadmilling in highly dendritic Arp2/3 networks.

Remarkable structural transformations of actin bundles are driven by their initial polarity, motor activity, crosslinking, and filament treadmilling

Bundled actin structures play a key role in maintaining cellular shape, in aiding force transmission to and from extracellular substrates, and in affecting cellular motility. Recent studies have also brought to light new details on stress generation, force transmission and contractility of actin bundles. In this work, we are primarily interested in the question of what determines the stability of actin bundles and what network geometries do unstable bundles eventually transition to. To address this problem, we used the MEDYAN mechano-chemical force field, modeling several micron-long actin bundles in 3D, while accounting for a comprehensive set of chemical, mechanical and transport processes. We developed a hierarchical clustering algorithm for classification of the different long time scale morphologies in our study. Our main finding is that initially unipolar bundles are significantly more stable compared with an apolar initial configuration. Filaments within the latter bundles slide easily with respect to each other due to myosin activity, producing a loose network that can be subsequently severely distorted. At high myosin concentrations, a morphological transition to aster-like geometries was observed. We also investigated how actin treadmilling rates influence bundle dynamics, and found that enhanced treadmilling leads to network fragmentation and disintegration, while this process is opposed by myosin and crosslinking activities. Interestingly, treadmilling bundles with an initial apolar geometry eventually evolve to a whole gamut of network morphologies based on relative positions of filament ends, such as sarcomere-like organization. We found that apolar bundles show a remarkable sensitivity to environmental conditions, which may be important in enabling rapid cytoskeletal structural reorganization and adaptation in response to intracellular and extracellular cues.

Myosin X is recruited to nascent focal adhesions at the leading edge and induces multi-cycle filopodial elongation

Filopodia protrude from the leading edge of cells and play important roles in cell motility. Here we report the mechanism of myosin X (encoded by Myo10)-induced multi-cycle filopodia extension. We found that actin, Arp2/3, vinculin and integrin-β first accumulated at the cell's leading edge. Myosin X was then gathered at these sites, gradually clustered by lateral movement, and subsequently initiated filopodia formation. During filopodia extension, we found the translocation of Arp2/3 and integrin-β along filopodia. Arp2/3 and integrin-β then became localized at the tip of filopodia, from where myosin X initiated the second extension of filopodia with a change in extension direction, thus producing long filopodia. Elimination of integrin-β, Arp2/3 and vinculin by siRNA significantly attenuated the myosin-X-induced long filopodia formation. We propose the following mechanism. Myosin X accumulates at nascent focal adhesions at the cell's leading edge, where myosin X promotes actin convergence to create the base of filopodia. Then myosin X moves to the filopodia tip and attracts integrin-β and Arp2/3 for further actin nucleation. The tip-located myosin X then initiates the second cycle of filopodia elongation to produce the long filopodia.

Physical model for the geometry of actin-based cellular protrusions

Actin-based cellular protrusions are a ubiquitous feature of cell morphology, e.g., filopodia and microvilli, serving a huge variety of functions. Despite this, there is still no comprehensive model for the mechanisms that determine the geometry of these protrusions. We present here a detailed computational model that addresses a combination of multiple biochemical and physical processes involved in the dynamic regulation of the shape of these protrusions. We specifically explore the role of actin polymerization in determining both the height and width of the protrusions. Furthermore, we show that our generalized model can explain multiple morphological features of these systems, and account for the effects of specific proteins and mutations.

Modelling the effect of myosin X motors on filopodia growth

We present a numerical simulation study of the dynamics of filopodial growth in the presence of active transport by myosin X motors. We employ both a microscopic agent-based model, which captures the stochasticity of the growth process, and a continuum mean-field theory which neglects fluctuations. We show that in the absence of motors, filopodia growth is overestimated by the continuum mean-field theory. Thus fluctuations slow down the growth, especially when the protrusions are driven by a small number (10 or less) of F-actin fibres, and when the force opposing growth (coming from membrane elasticity) is large enough. We also show that, with typical parameter values for eukaryotic cells, motors are unlikely to provide an actin transport mechanism which enhances filopodial size significantly, unless the G-actin concentration within the filopodium greatly exceeds that of the cytosol bulk. We explain these observations in terms of order-of-magnitude estimates of diffusion-induced and advection-induced growth of a bundle of Brownian ratchets.

Linking actin networks and cell membrane via a reaction-diffusion-elastic description of nonlinear filopodia initiation

Reaction-diffusion models have been used to describe pattern formation on the cellular scale, and traditionally do not include feedback between cellular shape changes and biochemical reactions. We introduce here a distinct reaction-diffusion-elasticity approach: The reaction-diffusion part describes bistability between two actin orientations, coupled to the elastic energy of the cell membrane deformations. This coupling supports spatially localized patterns, even when such solutions do not exist in the uncoupled self-inhibited reaction-diffusion system. We apply this concept to describe the nonlinear (threshold driven) initiation mechanism of actin-based cellular protrusions and provide support by several experimental observations.

Theory of active transport in filopodia and stereocilia

The biological processes in elongated organelles of living cells are often regulated by molecular motor transport. We determined spatial distributions of motors in such organelles, corresponding to a basic scenario when motors only walk along the substrate, bind, unbind, and diffuse. We developed a mean-field model, which quantitatively reproduces elaborate stochastic simulation results as well as provides a physical interpretation of experimentally observed distributions of Myosin IIIa in stereocilia and filopodia. The mean-field model showed that the jamming of the walking motors is conspicuous, and therefore damps the active motor flux. However, when the motor distributions are coupled to the delivery of actin monomers toward the tip, even the concentration bump of G actin that they create before they jam is enough to speed up the diffusion to allow for severalfold longer filopodia. We found that the concentration profile of G actin along the filopodium is rather nontrivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this nonmonotonous shape is expected to occur under a broad set of conditions. We also find that the stationary motor distribution is universal for the given set of model parameters regardless of the organelle length, which follows from the form of the kinetic equations and the boundary conditions.

Myosin-X: a MyTH-FERM myosin at the tips of filopodia

Myosin-X (Myo10) is an unconventional myosin with MyTH4-FERM domains that is best known for its striking localization to the tips of filopodia and its ability to induce filopodia. Although the head domain of Myo10 enables it to function as an actin-based motor, its tail contains binding sites for several molecules with central roles in cell biology, including phosphatidylinositol (3,4,5)-trisphosphate, microtubules and integrins. Myo10 also undergoes fascinating long-range movements within filopodia, which appear to represent a newly recognized system of transport. Myo10 is also unusual in that it is a myosin with important roles in the spindle, a microtubule-based structure. Exciting new studies have begun to reveal the structure and single-molecule properties of this intriguing myosin, as well as its mechanisms of regulation and induction of filopodia. At the cellular and organismal level, growing evidence demonstrates that Myo10 has crucial functions in numerous processes ranging from invadopodia formation to cell migration.

Protein fluxes along the filopodium as a framework for understanding the growth-retraction dynamics: the interplay between diffusion and active transport

We present a picture of filopodial growth and retraction from physics perspective, where we emphasize the significance of the role played by protein fluxes due to spatially extended nature of the filopodium. We review a series of works, which used stochastic simulations and mean field analytical modeling to find the concentration profile of G-actin inside a filopodium, which, in turn, determines the stationary filopodial length. In addition to extensively reviewing the prior works, we also report some new results on the role of active transport in regulating the length of filopodia. We model a filopodium where delivery of actin monomers towards the tip can occur both through passive diffusion and active transport by myosin motors. We found that the concentration profile of G-actin along the filopodium is rather non-trivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this non-monotonous shape is expected to occur under a broad set of conditions. We also raise the issue of slow approach to the stationary length and the possibility of multiple steady state solutions.

How does the antagonism between capping and anti-capping proteins affect actin network dynamics?

Actin-based cell motility is essential to many biological processes. We built a simplified, three-dimensional computational model and subsequently performed stochastic simulations to study the growth dynamics of lamellipodia-like branched networks. In this work, we shed light on the antagonism between capping and anti-capping proteins in regulating actin dynamics in the filamentous network. We discuss detailed mechanisms by which capping and anti-capping proteins affect the protrusion speed of the actin network and the rate of nucleation of filaments. We computed a phase diagram showing the regimes of motility enhancement and inhibition by these proteins. Our work shows that the effects of capping and anti-capping proteins are mainly transmitted by modulation of the filamentous network density and local availability of monomeric actin. We discovered that the combination of the capping/anti-capping regulatory network with nucleation-promoting proteins introduces robustness and redundancy in cell motility machinery, allowing the cell to easily achieve maximal protrusion speeds under a broader set of conditions. Finally, we discuss distributions of filament lengths under various conditions and speculate on their potential implication for the emergence of filopodia from the lamellipodial network.

Life at the leading edge

Cell migration requires sustained forward movement of the plasma membrane at the cell's front or "leading edge." To date, researchers have uncovered four distinct ways of extending the membrane at the leading edge. In lamellipodia and filopodia, actin polymerization directly pushes the plasma membrane forward, whereas in invadopodia, actin polymerization couples with the extracellular delivery of matrix-degrading metalloproteases to clear a path for cells through the extracellular matrix. Membrane blebs drive the plasma membrane forward using a combination of actomyosin-based contractility and reversible detachment of the membrane from the cortical actin cytoskeleton. Each protrusion type requires the coordination of a wide spectrum of signaling molecules and regulators of cytoskeletal dynamics. In addition, these different protrusion methods likely act in concert to move cells through complex environments in vivo.

The Eps8/IRSp53/VASP network differentially controls actin capping and bundling in filopodia formation

There is a body of literature that describes the geometry and the physics of filopodia using either stochastic models or partial differential equations and elasticity and coarse-grained theory. Comparatively, there is a paucity of models focusing on the regulation of the network of proteins that control the formation of different actin structures. Using a combination of in-vivo and in-vitro experiments together with a system of ordinary differential equations, we focused on a small number of well-characterized, interacting molecules involved in actin-dependent filopodia formation: the actin remodeler Eps8, whose capping and bundling activities are a function of its ligands, Abi-1 and IRSp53, respectively; VASP and Capping Protein (CP), which exert antagonistic functions in controlling filament elongation. The model emphasizes the essential role of complexes that contain the membrane deforming protein IRSp53, in the process of filopodia initiation. This model accurately accounted for all observations, including a seemingly paradoxical result whereby genetic removal of Eps8 reduced filopodia in HeLa, but increased them in hippocampal neurons, and generated quantitative predictions, which were experimentally verified. The model further permitted us to explain how filopodia are generated in different cellular contexts, depending on the dynamic interaction established by Eps8, IRSp53 and VASP with actin filaments, thus revealing an unexpected plasticity of the signaling network that governs the multifunctional activities of its components in the formation of filopodia.

Cells move and interact with the environment by forming migratory structures composed of self organized polymers of actin. These protrusions can be flat and short surfaces, the lamellipodia, or adopt an elongated, finger-like shape called filopodia. In this article, we analyze the ‘computation’ performed by cells when they opt to form filopodia. We focus our attention on some initiators of filopodia that play an essential role due to their interaction with the cell membrane. We analyze the formation of these filopodia initiators in different genotypes, thus providing a way to rationalize the behaviors of different cells in terms of tendency to form filopodia. Our results, based on the combination of experimental and computational approaches, suggest that cells have developed molecular networks that are extremely flexible in their capability to follow the path leading to filopodia formation. In this sense the role of an element of the network, Eps8, is paradigmatic, as this protein can both induce or inhibit the formation of filopodia depending on the cellular context.

Robust patterns in the stochastic organization of filopodia

Background

Filopodia are actin-based cellular projections that have a critical role in initiating and sustaining directional migration in vertebrate cells. Filopodia are highly dynamic structures that show a rich diversity in appearance and behavior. While there are several mathematical models of filopodia initiation and growth, testing the capacity of these theoretical models in predicting empirical behavior has been hampered by a surprising shortage of quantitative data related to filopodia. Neither is it clear how quantitatively robust the cellular filopodial network is and how perturbations alter it.

Results

We have measured the length and interfilopodial separation distances of several thousand filopodia in the rodent cell line Rat2 and measured these parameters in response to genetic, chemical and physical perturbation. Our work shows that length and separation distance have a lognormal pattern distribution over their entire detection range (0.4 μm to 50 μm).

Conclusions

We find that the lognormal distribution of length and separation is robust and highly resistant to perturbation. We also find that length and separation are independent variables. Most importantly, our empirical data is not entirely in agreement with predictions made based on existing theoretical models and that filopodial size and separation are an order of magnitude larger than what existing models suggest.