Growth of branched actin networks against obstacles

A method for simulating the growth of branched actin networks against obstacles has been developed. The method is based on simple stochastic events, including addition or removal of monomers at filament ends, capping of filament ends, nucleation of branches from existing filaments, and detachment of branches; the network structure for several different models of the branching process has also been studied. The models differ with regard to their inclusion of effects such as preferred branch orientations, filament uncapping at the obstacle, and preferential branching at filament ends. The actin ultrastructure near the membrane in lamellipodia is reasonably well produced if preferential branching in the direction of the obstacle or barbed-end uncapping effects are included. Uncapping effects cause the structures to have a few very long filaments that are similar to those seen in pathogen-induced "actin tails." The dependence of the growth velocity, branch spacing, and network density on the rate parameters for the various processes is quite different among the branching models. An analytic theory of the growth velocity and branch spacing of the network is described. Experiments are suggested that could distinguish among some of the branching models.     

A biomimetic motility assay provides insight into the mechanism of actin-based motility

Abiomimetic motility assay is used to analyze the mechanism of force production by site-directed polymerization of actin. Polystyrene microspheres, functionalized in a controlled fashion by the N-WASP protein, the ubiquitous activator of Arp2/3 complex, undergo actin-based propulsion in a medium that consists of five pure proteins. We have analyzed the dependence of velocity on N-WASP surface density, on the concentration of capping protein, and on external force. Movement was not slowed down by increasing the diameter of the beads (0.2 to 3 microm) nor by increasing the viscosity of the medium by 10(5)-fold. This important result shows that forces due to actin polymerization are balanced by internal forces due to transient attachment of filament ends at the surface. These forces are greater than the viscous drag. Using Alexa488-labeled Arp2/3, we show that Arp2/3 is incorporated in the actin tail like G-actin by barbed end branching of filaments at the bead surface, not by side branching, and that filaments are more densely branched upon increasing gelsolin concentration. These data support models in which the rates of filament branching and capping control velocity, and autocatalytic branching of filament ends, rather than filament nucleation, occurs at the particle surface.     

The effect of branching on the critical concentration and average filament length of actin

The dependences of the steady-state critical concentration and average filament length of actin solutions, on the filament branching and capping rates, are calculated using a rate methodology based on the total number of actin filaments. The methodology generalizes calculations of the "treadmilling" actin concentration at which an average filament has net zero growth rate. The predictions of the rate methodology are validated by comparison with stochastic-growth simulations that track the positions of all filament subunits over time. For side branching, the critical concentration drops proportionally to the square root of the branching rate; for end branching the drop is linear. The polymerization response to branching has a maximum as a function of the capping-protein concentration. The average filament length drops with increasing branching, because the critical concentration drops. Even small rates of filament uncapping have a large impact on the average filament length in vitro. The potential significance of these phenomena for cell behavior is evaluated.     

Actin filament branching and protrusion velocity in a simple 1D model of a motile cell

We formulate and analyse a 1D model for the spatial distribution of actin density at the leading edge of a motile cell. The model incorporates nucleation, capping, growth and decay of actin filaments, as well as retrograde flow of the actin meshwork and known parameter values based on the literature. Using a simplified geometry, and reasonable assumptions about the biochemical processes, we derive PDEs for the density of actin filaments and their tips. Analytic travelling wave solutions are used to predict how the speed of the cell depends on rates of nucleation, capping, polymerization and membrane resistance. Analysis and simulations agree with experimental profiles for measured actin distributions. Extended versions of the model are studied numerically. We find that our model produces stable travelling wave solutions with reasonable cell speeds. Increasing the rate of nucleation of filaments (by the actin related protein Arp2/3) or the rate of actin polymerization leads to faster cell speed, whereas increasing the rate of capping or the membrane resistance reduces cell speed. We consider several variants of nucleation (spontaneous, tip, and side branching) and find best agreement with experimentally measured spatial profiles of filament and tip density in the side branching case.     

Cellular motility driven by assembly and disassembly of actin filaments

Motile cells extend a leading edge by assembling a branched network of actin filaments that produces physical force as the polymers grow beneath the plasma membrane. A core set of proteins including actin, Arp2/3 complex, profilin, capping protein, and ADF/cofilin can reconstitute the process in vitro, and mathematical models of the constituent reactions predict the rate of motion. Signaling pathways converging on WASp/Scar proteins regulate the activity of Arp2/3 complex, which mediates the initiation of new filaments as branches on preexisting filaments. After a brief spurt of growth, capping protein terminates the elongation of the filaments. After filaments have aged by hydrolysis of their bound ATP and dissociation of the gamma phosphate, ADF/cofilin proteins promote debranching and depolymerization. Profilin catalyzes the exchange of ADP for ATP, refilling the pool of ATP-actin monomers bound to profilin, ready for elongation.     

Actin-based motility as a self-organized system: mechanism and reconstitution in vitro

Site-directed actin polymerisation in response to signalling is responsible for the formation of cell protrusions. These elementary 'actin-based motility processes' are involved in cell locomotion, cell metastasis, organ morphogenesis and microbial pathogenesis. We have reconstituted actin-based propulsive movement of particles of various sizes and geometries (rods, microspheres) in a minimum motility medium containing five pure proteins. The ATP-supported treadmilling of actin filaments, regulated by Actin Depolymerizing Factor (ADF/cofilin), profilin and capping proteins provides the thermodynamic basis for sustained actin-based movement. Local activation of Arp2/3 complex at the surface of the particle promotes autocatalytic barbed end branching of filaments, generating a polarized arborescent array. Barbed end growth of branched filaments against the surface generates a propulsive force and is eventually arrested by capping proteins. Understanding the mechanism of actin-based movement requires elucidation of the biochemical properties and mode of action of Arp2/3 complex in filament branching, in particular the role of ATP binding and hydrolysis in Arp2/3, and a physical analysis of the movement of functionalised particles. Because the functionalisation of the particle by an activator of Arp2/3 complex (N-WASP or the Listeria protein ActA) and the concentrations of effectors in the medium are controlled, the reconstituted motility assay allows an analysis of the mechanism of force production at the mesoscopic and molecular levels.     

On the edge: modeling protrusion

Actin-based protrusion is the first step in cell crawling. In the last two decades, the studies of actin networks in the lamellipodium and Listeria's comet tail advanced so far that the last goal of the reductionist agenda - reconstitution of protrusion from purified components in vitro and in silico - became viable. Earlier models dealt with growth of and force generation by a single actin filament. Modern models of tethered ratchet, autocatalytic branching, end-tracking motor action and elastic- and nano- propulsion have recently helped to elucidate dynamics and forces in complex actin networks. By considering these models, their limitations and their relationships to recent biophysical data, progress is being made toward a unified model of protrusion.     

End versus side branching by Arp2/3 complex

We investigate the issue of end versus side branching of actin filaments by Arp2/3 complex, using a combination of analytic theory, polymerization assays, and quantitative modeling. The analytic theory shows that the effect of capping protein on the initial stages of actin polymerization in the presence of Arp2/3 complex depends strongly on whether new Arp2/3 complex-induced branches grow from the sides or ends of existing filaments. Motivated by these results, we measure and quantitatively model the kinetics of actin polymerization in the presence of activated Arp2/3 complex, for a range of concentrations of capping protein. Our model includes the most important types of events involving actin and actin-binding proteins, and can be adjusted to include end branching, side branching, or both. The side-branching model gives a better fit to the experimental data than the end-branching model. An end-plus-side model including both types of branching gives a moderate improvement in the quality of the fit. Another side-branching model, based on aging of subunits' capacity for branch formation, gives a significantly better fit than the end-plus-side model. We discuss implications for actin polymerization in cells.     

Branching is coordinated with mitosis in growing hyphae of Aspergillus nidulans

Filamentous fungi like Aspergillus nidulans can effectively colonize their surroundings by the formation of new branches along the existing hyphae. While growth conditions, chemical perturbations, and mutations affecting branch formation have received great attention during the last decades, the mechanisms that regulates branching is still poorly understood. In this study, a possible relation between cell cycle progression and branching was studied by testing the effect of a nuclei distribution mutation, cell cycle inhibitors, and conditional cell cycle mutations in combination with tip-growth inhibitors and varying substrate concentrations on branch initiation. Formation of branches was blocked after inhibition of nuclear division, which was not caused by a reduced growth rate. In hyphae of a nuclei distribution mutant branching was severely reduced in anucleated hyphae whereas the number of branches per hyphal length was linearly correlated to the concentration of nuclei, in the nucleated hyphae. In wild type cells, branching intensity was increased when the tip extension was reduced, and reduced when growing on poor substrates. In these situations, the hyphal concentration of nuclei was maintained and it is suggested that branching is correlated to cell cycle progression in order to maintain a minimum required cytoplasmic volume per nucleus and to avoid the formation of anucleated hyphae in the absence of nuclear divisions. The presented results further suggest the hyphal diameter as a key point through which the hyphal element regulates its branching intensity in response to the surrounding substrate concentrations.     

Distribution of Cell Adhesion Molecules (CAM) along the Branching Neurite Surface: Model-Inherited Effects of the Branch Diameters and Mode of CAM Transport

In many cases, an increase in the surface density of cell adhesion molecules (CAM) in the distal parts of a growing neurite is favorable for the neurite elongation. This increase is attained by exocytotic insertion of CAM-containing vesicles into the growth cones with subsequent redistribution of CAM along the cell surface due to lateral diffusion and endocytosis. Using a mathematical model describing these processes, we quantitatively describe conditions providing two qualitatively different profiles in a branching neurite: (i) the CAM surface density increases along both daughter branches, which would be in favor of further outgrowth of both branches, i.e., successful branching, or (ii) the CAM surface density increases along one daughter branch and decreases along another branch, which could lead to the retraction of the latter. The geometric factors and mechanisms underlying the intracellular CAM transport to the daughter growth cones were proved to determine the profile of CAM surface density. A similarity in the diameters of daughter branches, their short lengths, a high value of the lateral transfer constant, and partitioning of CAM transport at the branching point proportionally to the surface areas of daughter branches are in favor of an increase in the CAM surface density along both daughter branches. Asymmetric branching can lead to a decrease in the CAM surface density along the thinner or thicker daughter branch, if CAM trafficking was equally partitioned or was proportional to the branch cross-sectional areas, respectively. The proposed model helps to understand possible relationships between the intracellular CAM trafficking, CAM surface distribution, and geometry of branching of the neurites.     

Computer simulation of growth of anastomosing microvascular networks

Stochastic growth of polygonal microvascular networks was simulated on computer by dichotomous terminal branching and bridging (anastomosing with an existing segment). The model was applied to describe microvascular growth into a rectangular plane from the sides when vessels bifurcate in a probabilistic manner. The angle of bifurcation was drawn from a normal distribution, the mean of which was varied between 40 degrees and 80 degrees. The resulting networks contained an average of 88-104 nodes of which 30-38% were due to bridging. Number of nodes, number of branches, number of vascular polygons and a fractal dimension representing the density of nodes were calculated for each simulated network. Capillary density increased when mean angle of bifurcation was increased between 40 degrees and 80 degrees. Distributions of normalized vessel lengths and polygon shapes were compared with those of a mesenteric vascular network. The distributions were not found to be significantly different (p less than 0.05) for most values of the mean angle of bifurcation, matching best for the mean bifurcation angle of 50 degrees. Vascular polygons had an average shape between pentagonal and hexagonal for the mesenteric network as well as for all values of the mean bifurcation angle used in this study.     

Analysis of Branching in Spring-sown White Lupins (Lupinus albusL.): The Significance of the Number of Axillary Buds

Plant density and sowing date were shown to affect branchingin spring-sown white lupin (Lupinus albusL.), but the responsevaried among environments. The patterns of primary and secondarybranching in the cv. Lublanc were studied as a function of boththe number of axillary buds and the plant growth rate. Fieldexperiments that used a wide range of sowing dates and plantdensities to alter plant architecture were conducted over 5years, and these were supplemented with data from additionalglasshouse and growth cabinet experiments. The number of axillary buds on the main stem or primary branches,which determined the potential number of branches, increasedlinearly with the number of nodes. In situations where all axillarybuds did not produce branches, it was found that the numberof primary and secondary branches produced was related to theplant growth rate at the beginning of branch elongation. Knowledgeof the number of axillary buds improved the analysis of theinteraction between cultural practices and environmental conditionson plant architecture. The variability of branching potentialamong genotypes was discussed. Lupinus albus; branches; axillary buds; growth; sowing date; plant density     

Simulated distributions of the density of cell adhesion molecules over branching processes with different geometry and intracellular trafficking

Adhesive cell-cell and cell-substrate interactions mediated by different types of cell adhesion molecules (CAM) are important for growth and migration processes. In simulation study we investigated the impact of geometry of branching cellular processes on the lateral distribution of CAM due to retrograde lateral diffusion from a growing part, where they were delivered by different assumed types of trafficking. The model incorporates trafficking of CAM to and installation in the growing active part of the cell, their lateral diffusive redistribution, formation and dissociation of CAM/ligand complexes, and CAM internalization by endocytosis. Since the rate of growth is two and one order(s) of magnitude slower than the rate of trafficking and lateral diffusion, respectively, steady state distributions of CAM were considered. Three possible types of intracellular CAM partitioning between sister branches were considered: equal, proportional to the branch cross-section area, and proportional to the branch surface area. Asymmetry of branching led to various inhomogeneous distributions of the CAM surface density along the branches, and these distributions depended on the type of intracellular trafficking, which might provide a basis for different modes of growth. One can speculate that, depending on these modes, initially asymmetrical branching can be either reinforced or symmetrized during further development.     

Branching and self-organization in marine modular colonial organisms: a model

Despite the universality of branching patterns in marine modular colonial organisms, there is neither a clear explanation about the growth of their branching forms nor an understanding of how these organisms conserve their shape during development. This study develops a model of branching and colony growth using parameters and variables related to actual modular structures (e.g., branches) in Caribbean gorgonian corals (Cnidaria). Gorgonians exhibiting treelike networks branch subapically, creating hierarchical mother-daughter relationships among branches. We modeled both the intrinsic subapical branching along with an ecological-physiological limit to growth or maximum number of mother branches (k). Shape is preserved by maintaining a constant ratio (c) between the total number of branches and the mother branches. The size frequency distribution of mother branches follows a scaling power law suggesting self-organized criticality. Differences in branching among species with the same k values are determined by r (branching rate) and c. Species with rr/2 or c>r>0). Ecological/physiological constraints limit growth without altering colony form or the interaction between r and c. The model described the branching dynamics giving the form to colonies and how colony growth declines over time without altering the branching pattern. This model provides a theoretical basis to study branching as a simple function of the number of branches independently of ordering- and bifurcation-based schemes.     

The architecture of shrubs after defoliation and the subsequent feeding behavior of browsers

We examined regrowth architecture of 4 species of savanna shrubs following 4 levels of defoliation. Defoliation had little effect on the regrowth architecture of honey mesquite ( Prosopis glandulosa ), which is rarely browsed by mammalian herbivores. The 3 acacia species ( Acacia berlandieri , A. greggii , A. schaffneri ) responded to defoliation by increasing leaf and spine density on the regrowth branches, but spine length and branching architecture remained unchanged. Only A. greggii , which is a preferred food plant of many browsers, exhibited an increase in the number and length of current annual growth branches in response to defoliation. The changes in plant architecture due to defoliation had little effect on the subsequent feeding behavior of captive white-tailed deer ( Odocoileus virginianus ). Food intake rate of the deer was most strongly related to internode distance, a parameter not significantly altered by defoliation. This suggests that the architectural responses of these shrubs to defoliation may not provide increased defense against browsing by co-evolved mammals.     

Tests of a cellular model for constant branch distribution in the filamentous fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Tests of a Cellular Model for Constant Branch Distribution in the Filamentous Fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Growth cone behavior and production of traction force

The growth cone must push its substrate rearward via some traction force in order to propel itself forward. To determine which growth cone behaviors produce traction force, we observed chick sensory growth cones under conditions in which force production was accommodated by movement of obstacles in the environment, namely, neurites of other sensory neurons or glass fibers. The movements of these obstacles occurred via three, different, stereotyped growth cone behaviors: (a) filopodial contractions, (b) smooth rearward movement on the dorsal surface of the growth cone, and (c) interactions with ruffling lamellipodia. More than 70% of the obstacle movements were caused by filopodial contractions in which the obstacle attached at the extreme distal end of a filopodium and moved only as the filopodium changed its extension. Filopodial contractions were characterized by frequent changes of obstacle velocity and direction. Contraction of a single filopodium is estimated to exert 50-90 microdyn of force, which can account for the pull exerted by chick sensory growth cones. Importantly, all five cases of growth cones growing over the top of obstacle neurites (i.e., geometry that mimics the usual growth cone/substrate interaction), were of the filopodial contraction type. Some 25% of obstacle movements occurred by a smooth backward movement along the top surface of growth cones. Both the appearance and rate of movements were similar to that reported for retrograde flow of cortical actin near the dorsal growth cone surface. Although these retrograde flow movements also exerted enough force to account for growth cone pulling, we did not observe such movements on ventral growth cone surfaces. Occasionally obstacles were moved by interaction with ruffling lamellipodia. However, we obtained no evidence for attachment of the obstacles to ruffling lamellipodia or for directed obstacle movements by this mechanism. These data suggest that chick sensory growth cones move forward by contractile activity of filopodia, i.e., isometric contraction on a rigid substrate. Our data argue against retrograde flow of actin producing traction force.