Nonequilibrium self-assembly of a filament coupled to ATP/GTP hydrolysis

Enzymatic Dissociation of DNA-Histone Condensates in an Electrophoretic Setting: Modulating DNA Patterning and Hydrogel Viscoelasticity

Development of an energy-driven self-assembly process is a matter of interest for understanding and mimicking diverse ranges of biological and environmental patterns in a synthetic system. In this article, first we demonstrate transient and temporally controlled self-assembly of a DNA-histone condensate where trypsin (already present in the system) hydrolyzes histone, resulting in disassembly. Upon performing this dynamic self-assembly process in a gel matrix under an electric field, we observe diverse kinds of DNA patterning across the gel matrix depending on the amount of trypsin, incubation time of the reaction mixture, and gel porosity. Notably, here, the micrometer-sized DNA-histone condensate does not move through the gel and only free DNA can pass; therefore, transport and accumulation of DNA at different zones depend on the release rate of DNA by trypsin. Furthermore, we show that the viscoelasticity of the native gel increases in the presence of DNA and a pattern over gel viscoelasticity at different zones can be achieved by tuning the amount of enzyme, i.e., the dissociation rate of the DNA-histone condensate. We believe enabling spatiotemporally controlled DNA patterning by applying an electric field will be potentially important in designing different kinds of spatiotemporally distinct dynamic materials.

The effect of motor-induced shaft dynamics on microtubule stability and length

Control of microtubule abundance, stability, and length is crucial to regulate intracellular transport as well as cell polarity and division. How microtubule stability depends on tubulin addition or removal at the dynamic ends is well studied. However, microtubule rescue, the event when a microtubule switches from shrinking to growing, occurs at tubulin exchange sites along the shaft. Molecular motors have recently been shown to promote such exchanges. Using a stochastic theoretical description, we study how microtubule stability and length depend on motor-induced tubulin exchange and thus rescue. Our theoretical description matches our in vitro experiments on microtubule dynamics in the presence of kinesin-1 molecular motors. Although the overall dynamics of a population of microtubules can be captured by an effective rescue rate, by assigning rescue to exchange sites, we reveal that the dynamics of individual microtubules within the population differ dramatically. Furthermore, we study in detail a transition from bounded to unbounded microtubule growth. Our results provide novel insights into how molecular motors imprint information of microtubule stability on the microtubule network.

Effects of random hydrolysis on biofilament length distributions in a shared subunit pool

The sizes of filamentous structures in a cell are often regulated for many physiological processes. A key question in cell biology is how such size control is achieved. Here, we theoretically study the length distributions of multiple filaments, growing by stochastic assembly and disassembly of subunits from a limiting subunit pool. Importantly, we consider a chemical switching of subunits (hydrolysis) prevalent in many biofilaments like microtubules (MTs). We show by simulations of different models that hydrolysis leads to a skewed unimodal length distribution for a single MT. In contrast, hydrolysis can lead to bimodal distributions of individual lengths for two MTs, where individual filaments toggle stochastically between bigger and smaller sizes. For more than two MTs, length distributions are also bimodal, although the bimodality becomes less prominent. We further show that this collective phenomenon is connected with the nonequilibrium nature of hydrolysis, and the bimodality disappears for reversible dynamics. Consistent with earlier theoretical studies, a homogeneous subunit pool, without hydrolysis, cannot control filament lengths. We thus elucidate the role of hydrolysis as a control mechanism on MT length diversity.

Atomistic molecular dynamics simulations of tubulin heterodimers explain the motion of a microtubule

Microtubules are essential parts of the cytoskeleton that are built by polymerization of tubulin heterodimers into a hollow tube. Regardless that their structures and functions have been comprehensively investigated in a modern soft matter, it is unclear how properties of tubulin heterodimer influence and promote the self-assembly. A detailed knowledge of such structural mechanisms would be helpful in drug design against neurodegenerative diseases, cancer, diabetes etc. In this work atomistic molecular dynamics simulations were used to investigate the fundamental dynamics of tubulin heterodimers in a sheet and a short microtubule utilizing well-equilibrated structures. The breathing motions of the tubulin heterodimers during assembly show that the movement at the lateral interface between heterodimers (wobbling) dominates in the lattice. The simulations of the protofilament curvature agrees well with recently published experimental data, showing curved protofilaments at polymerization of the microtubule plus end. The tubulin heterodimers exposed at the microtubule minus end were less curved and displayed altered interactions at the site of sheet closure around the outmost heterodimers, which may slow heterodimer binding and polymerization, providing a potential explanation for the limited dynamics observed at the minus end.

Reversible membrane deformations by straight DNA origami filaments

Membrane-active cytoskeletal elements, such as FtsZ, septin or actin, form filamentous polymers able to induce and stabilize curvature on cellular membranes. In order to emulate the characteristic dynamic self-assembly properties of cytoskeletal subunits in vitro, biomimetic synthetic scaffolds were here developed using DNA origami. In contrast to our earlier work with pre-curved scaffolds, we specifically assessed the potential of origami mimicking straight filaments, such as actin and microtubules, by origami presenting cholesteryl anchors for membrane binding and additional blunt end stacking interactions for controllable polymerization into linear filaments. By assessing the interaction of our DNA nanostructures with model membranes using fluorescence microscopy, we show that filaments can be formed, upon increasing MgCl2 in solution, for structures displaying blunt ends; and can subsequently depolymerize, by decreasing the concentration of MgCl2. Distinctive spike-like membrane protrusions were generated on giant unilamellar vesicles at high membrane-bound filament densities, and the presence of such deformations was reversible and shown to correlate with the MgCl2-triggered polymerization of DNA origami subunits into filamentous aggregates. In the end, our approach reveals the formation of membrane-bound filaments as a minimal requirement for membrane shaping by straight cytoskeletal-like objects.

Microtubule catastrophe under force: mathematical and computational results from a Brownian ratchet model

In the intracellular environment, the intrinsic dynamics of microtubule filaments is often hindered by the presence of barriers of various kind, such as kinetochore complexes and cell cortex, which impact their polymerisation force and dynamical properties such as catastrophe frequency. We present a theoretical study of the effect of a forced barrier, also subjected to thermal noise, on the statistics of catastrophe events in a single microtubule as well as a 'bundle' of two parallel microtubules. For microtubule dynamics, which includes growth, detachment, hydrolysis and the consequent dynamic instability, we employ a one-dimensional discrete stochastic model. The dynamics of the barrier is captured by over-damped Langevin equation, while its interaction with a growing filament is assumed to be hard-core repulsion. A unified treatment of the continuum dynamics of the barrier and the discrete dynamics of the filament is realized using a hybrid Fokker-Planck equation. An explicit mathematical formula for the force-dependent catastrophe frequency of a single microtubule is obtained by solving the above equation, under some assumptions. The prediction agrees well with results of numerical simulations in the appropriate parameter regime. More general situations are studied via numerical simulations. To investigate the extent of 'load-sharing' in a microtubule bundle, and its impact on the frequency of catastrophes, the dynamics of a two-filament bundle is also studied. Here, two parallel, non-interacting microtubules interact with a common, forced barrier. The equations for the two-filament model, when solved using a mean-field assumption, predicts equal sharing of load between the filaments. However, numerical results indicate the existence of a wide spectrum of load-sharing behaviour, which is characterized using a dimensionless parameter.

Dynamic metastable long-living droplets formed by sticker-spacer proteins

Multivalent biopolymers phase separate into membrane-less organelles (MLOs) which exhibit liquid-like behavior. Here, we explore formation of prototypical MOs from multivalent proteins on various time and length scales and show that the kinetically arrested metastable multi-droplet state is a dynamic outcome of the interplay between two competing processes: a diffusion-limited encounter between proteins, and the exhaustion of available valencies within smaller clusters. Clusters with satisfied valencies cannot coalesce readily, resulting in metastable, long-living droplets. In the regime of dense clusters akin to phase-separation, we observe co-existing assemblies, in contrast to the single, large equilibrium-like cluster. A system-spanning network encompassing all multivalent proteins was only observed at high concentrations and large interaction valencies. In the regime favoring large clusters, we observe a slow-down in the dynamics of the condensed phase, potentially resulting in loss of function. Therefore, metastability could be a hallmark of dynamic functional droplets formed by sticker-spacer proteins.

Structural effects of cap, crack, and intrinsic curvature on the microtubule catastrophe kinetics

Microtubules (MTs) experience an effect called "catastrophe," which is the transition from the MT growth to a sudden dramatic shrinkage in length. The straight guanosine triphosphate (GTP)-tubulin cap at the filament tip and the intrinsic curvature of guanosine diphosphate (GDP)-tubulins are known to be the key thermodynamic factors that determine MT catastrophe, while the hydrolysis of this GTP-cap acts as the kinetic control of the process. Although several theoretical models have been developed, assuming the catastrophe occurs when the GTP-cap shrinks to a minimal stabilizing size, the structural effect of the GTP-cap and GDP-curvature is not explicitly included; thus, their influence on catastrophe kinetics remains less understood. To investigate this structural effect, we apply a single-protofilament model with one GTP-cap while assuming a random hydrolysis mechanism and take the occurrence of a crack in the lateral bonds between neighboring protofilaments as the onset of the catastrophe. Therein, we find the effective potential of the tip along the peel-off direction and formulate the catastrophe kinetics as a mean first-passage time problem, subject to thermal fluctuations. We consider cases with and without a compressive force on the MT tip, both of which give a quadratic effective potential, making MT catastrophe an Ornstein-Uhlenbeck process in our formalism. In the free-standing case, the mean catastrophe time has a sensitive tubulin-concentration dependence, similar to a double-exponential function, and agrees well with the experiment. For a compressed MT, we find a modified exponential function of force that shortens the catastrophe time.

Sufficient conditions for the additivity of stall forces generated by multiple filaments or motors

Molecular motors and cytoskeletal filaments work collectively most of the time under opposing forces. This opposing force may be due to cargo carried by motors or resistance coming from the cell membrane pressing against the cytoskeletal filaments. Some recent studies have shown that the collective maximum force (stall force) generated by multiple cytoskeletal filaments or molecular motors may not always be just a simple sum of the stall forces of the individual filaments or motors. To understand this excess or deficit in the collective force, we study a broad class of models of both cytoskeletal filaments and molecular motors. We argue that the stall force generated by a group of filaments or motors is additive, that is, the stall force of N number of filaments (motors) is N times the stall force of one filament (motor), when the system is reversible at stall. Conversely, we show that this additive property typically does not hold true when the system is irreversible at stall. We thus present a novel and unified understanding of the existing models exhibiting such non-addivity, and generalise our arguments by developing new models that demonstrate this phenomena. We also propose a quantity similar to thermodynamic efficiency to easily predict this deviation from stall-force additivity for filament and motor collectives.

Profilin Interaction with Actin Filament Barbed End Controls Dynamic Instability, Capping, Branching, and Motility

Cell motility and actin homeostasis depend on the control of polarized growth of actin filaments. Profilin, an abundant regulator of actin dynamics, supports filament assembly at barbed ends by binding G-actin. Here, we demonstrate how, by binding and destabilizing filament barbed ends at physiological concentrations, profilin also controls motility, cell migration, and actin homeostasis. Profilin enhances filament length fluctuations. Profilin competes with Capping Protein at barbed ends, which generates a lower amount of profilin-actin than expected if barbed ends were tightly capped. Profilin competes with barbed end polymerases, such as formins and VopF, and inhibits filament branching by WASP-Arp2/3 complex by competition for filament barbed ends, accounting for its as-yet-unknown effects on motility and metastatic cell migration observed in this concentration range. In conclusion, profilin is a major coordinator of polarized growth of actin filaments, controlled by competition between barbed end cappers, trackers, destabilizers, and filament branching machineries.

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The binding of profilin to barbed ends accounts for its effects on cell migration

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Profilin enhances length fluctuations of actin filaments by destabilizing barbed ends

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Profilin competes with capping protein at filament barbed ends

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Profilin competes with polymerases and filament branching machineries at barbed ends

A minimal conformational switching-dependent model for amyloid self-assembly

Amyloid formation is associated with various pathophysiological conditions like Alzheimer’s and Parkinson’s diseases as well as many useful functions. The hallmark of amyloid assemblies is a conformational transition of the constituent proteins into a β - sheet rich filament. Accounting for this conformational transition in amyloidogenic proteins, we develop an analytically solvable model that can probe the dynamics of an ensemble of single filaments. Using the theory and Monte Carlo simulations, we show the presence of two kinetic regimes for the growth of a self-assembling filament – switching-dependent and –independent growth regimes. We observe a saturation in fibril elongation velocities at higher concentrations in the first regime, providing a novel explanation to the concentration-independence of growth velocities observed experimentally. We also compute the length fluctuation of the filaments to characterize aggregate heterogeneity. From the early velocities and length fluctuation, we propose a novel way of estimating the conformational switching rate. Our theory predicts a kinetic phase diagram that has three distinct phases – short oligomers/monomers, disordered aggregates and β -rich filaments. The model also predicts the force generation potential and the intermittent growth of amyloid fibrils evident from single molecular experiments. Our model could contribute significantly to the physical understanding of amyloid aggregation.

Force-induced dynamical properties of multiple cytoskeletal filaments are distinct from that of single filaments

How cytoskeletal filaments collectively undergo growth and shrinkage is an intriguing question. Collective properties of multiple bio-filaments (actin or microtubules) undergoing hydrolysis have not been studied extensively earlier within simple theoretical frameworks. In this paper, we study the collective dynamical properties of multiple filaments under force, and demonstrate the distinct properties of a multi-filament system in comparison to a single filament. Comparing stochastic simulation results with recent experimental data, we show that multi-filament collective catastrophes are slower than catastrophes of single filaments. Our study also shows further distinctions as follows: (i) force-dependence of the cap-size distribution of multiple filaments are quantitatively different from that of single filaments, (ii) the diffusion constant associated with the system length fluctuations is distinct for multiple filaments, and (iii) switching dynamics of multiple filaments between capped and uncapped states and the fluctuations therein are also distinct. We build a unified picture by establishing interconnections among all these collective phenomena. Additionally, we show that the collapse times during catastrophes can be sharp indicators of collective stall forces exceeding the additive contributions of single filaments.

Branching influences force-velocity curves and length fluctuations in actin networks

We investigate collective dynamics of branched actin networks growing against a rigid movable wall constrained by a resistive force. Computing the force velocity relations, we show that the stall force of such networks depends not only on the average number of filaments touching the wall, but also on the amount of fluctuation of the leading edge of the network. These differences arise due to differences in the network architecture, namely, distance between two adjacent branching points and the initial distance of the starting filament from the wall, with their relative magnitudes influencing the nature of the force velocity curves (convex versus concave). We also show that the introduction of branching results in nonmonotonic diffusion constant, a quantity that measures the growth in length fluctuation of the leading edge of the network, as a function of externally applied force. Together our results demonstrate how the collective dynamics of a branched network differs from that of a parallel filament network.

Effects of microtubule mechanics on hydrolysis and catastrophes

We introduce a model for microtubule (MT) mechanics containing lateral bonds between dimers in neighboring protofilaments, bending rigidity of dimers, and repulsive interactions between protofilaments modeling steric constraints to investigate the influence of mechanical forces on hydrolysis and catastrophes. We use the allosteric dimer model, where tubulin dimers are characterized by an equilibrium bending angle, which changes from 0° to 22° by hydrolysis of a dimer. This also affects the lateral interaction and bending energies and, thus, the mechanical equilibrium state of the MT. As hydrolysis gives rise to conformational changes in dimers, mechanical forces also influence the hydrolysis rates by mechanical energy changes modulating the hydrolysis rate. The interaction via the MT mechanics then gives rise to correlation effects in the hydrolysis dynamics, which have not been taken into account before. Assuming a dominant influence of mechanical energies on hydrolysis rates, we investigate the most probable hydrolysis pathways both for vectorial and random hydrolysis. Investigating the stability with respect to lateral bond rupture, we identify initiation configurations for catastrophes along the hydrolysis pathways and values for a lateral bond rupture force. If we allow for rupturing of lateral bonds between dimers in neighboring protofilaments above this threshold force, our model exhibits avalanche-like catastrophe events.

Treadmilling and length distributions of active polar filaments

The cytoskeleton is a network of filamentous proteins, notably, actin filaments and microtubules. These filaments are active as their assembly is driven by the hydrolysis of nucleotides bound to the constituting protomers. In addition, the assembly kinetics differs at the two respective ends, making them active polar filaments. Experimental evidence suggests, that, in vivo, actin filaments and microtubules can grow at one and shrink at the other end at the same rate, a state that is known as treadmilling. In this work, we use a generic discrete two-state model for active polar filaments to analyze the conditions leading to treadmilling. We find that a single filament can self-organize into the treadmilling state for a broad range of monomer concentrations. In this regime the corresponding length distribution has a pronounced maximum at a finite value. We then extend our description to consider specifically the dynamics of actin filaments. We show that actin treadmilling should be observable in vitro in the presence of appropriate depolymerization promoting factors.

Microtubule dynamic instability: the role of cracks between protofilaments

Microtubules (MTs) are cytoplasmic protein polymers that are essential for fundamental cellular processes including the maintenance of cell shape, organelle transport and formation of the mitotic spindle. Microtubule dynamic instability is critical for these processes, but it remains poorly understood, in part because the relationship between the structure of the MT tip and the growth/depolymerization transitions is enigmatic. In previous work, we used computational models of dynamic instability to provide evidence that cracks (laterally unbonded regions) between protofilaments play a key role in the regulation of dynamic instability. Here we use computational models to investigate the connection between cracks and dynamic instability in more detail. Our work indicates that while cracks contribute to dynamic instability in a fundamental way, it is not the depth of the cracks per se that governs MT dynamic instability. Instead, what matters more is whether the cracks terminate in GTP-rich or GDP-rich regions of the MT. Based on these observations, we suggest that a functional "GTP cap" (i.e., one capable of promoting MT growth) is one where the cracks terminate in pairs of GTP-bound subunits, and that the likelihood of catastrophe rises significantly with the fraction of crack-terminating subunits that contain GDP. In addition to helping clarify the mechanism of dynamic instability, this idea could also explain how MT stabilizers work: proteins that introduce lateral cross-links between protofilaments would produce islands of GDP-bound tubulin that mimic GTP-rich regions in having strong lateral bonds, thus reducing crack propagation, suppressing catastrophe and promoting rescue.

Theoretical analysis of microtubules dynamics using a physical-chemical description of hydrolysis

Microtubules are cytoskeleton multifilament proteins that support many fundamental biological processes such as cell division, cellular transport, and motility. They can be viewed as dynamic polymers that function in nonequilibrium conditions stimulated by hydrolysis of GTP (guanosine triphosphate) molecules bound to their monomers. We present a theoretical description of microtubule dynamics based on discrete-state stochastic models that explicitly takes into account all relevant biochemical transitions. In contrast to previous theoretical analysis, a more realistic physical-chemical description of GTP hydrolysis is presented, in which the hydrolysis rate at a given monomer depends on the chemical composition of the neighboring monomers. This dependence naturally leads to a cooperativity in the hydrolysis. It is found that this cooperativity significantly influences all dynamic properties of microtubules. It is suggested that the dynamic instability in cytoskeleton proteins might be observed only for weak cooperativity, while the strong cooperativity in hydrolysis suppresses the dynamic instability. The presented microscopic analysis is compared with existing phenomenological descriptions of hydrolysis processes. Our analytical calculations, supported by computer Monte Carlo simulations, are also compared with available experimental observations.

Synergy between XMAP215 and EB1 increases microtubule growth rates to physiological levels

In cells, a complex network of proteins regulates the dynamic growth of microtubules that is essential for division and migration. In vitro approaches with purified components have so far been unable to reconstitute fast microtubule growth observed in vivo . Here we show that two well-studied plus-end-binding proteins-end-tracking protein EB1 and microtubule polymerase XMAP215-act together to strongly promote microtubule growth to cellular rates. Unexpectedly, the combined effects of XMAP215 and EB1 are highly synergistic, with acceleration of growth well beyond the product of the individual effects of either protein. The synergistic growth promotion does not rely on any of the canonical EB1 interactions, suggesting an allosteric interaction through the microtubule end. This hypothesis is supported by the finding that taxol and XMAP215, which have non-overlapping binding sites on tubulin, also act synergistically on growth. The increase in growth rates is accompanied by a strong enhancement of microtubule catastrophe by EB1, thereby rendering the fast and dynamic microtubule behaviour typically observed in cells.

Microtubule dynamic instability: a new model with coupled GTP hydrolysis and multistep catastrophe

A key question in understanding microtubule dynamics is how GTP hydrolysis leads to catastrophe, the switch from slow growth to rapid shrinkage. We first provide a review of the experimental and modeling literature, and then present a new model of microtubule dynamics. We demonstrate that vectorial, random, and coupled hydrolysis mechanisms are not consistent with the dependence of catastrophe on tubulin concentration and show that, although single-protofilament models can explain many features of dynamics, they do not describe catastrophe as a multistep process. Finally, we present a new combined (coupled plus random hydrolysis) multiple-protofilament model that is a simple, analytically solvable generalization of a single-protofilament model. This model accounts for the observed lifetimes of growing microtubules, the delay to catastrophe following dilution and describes catastrophe as a multistep process.

Sensor potency of the moonlighting enzyme-decorated cytoskeleton: the cytoskeleton as a metabolic sensor

Background

There is extensive evidence for the interaction of metabolic enzymes with the eukaryotic cytoskeleton. The significance of these interactions is far from clear.

Presentation of the hypothesis

In the cytoskeletal integrative sensor hypothesis presented here, the cytoskeleton senses and integrates the general metabolic activity of the cell. This activity depends on the binding to the cytoskeleton of enzymes and, depending on the nature of the enzyme, this binding may occur if the enzyme is either active or inactive but not both. This enzyme-binding is further proposed to stabilize microtubules and microfilaments and to alter rates of GTP and ATP hydrolysis and their levels.

Testing the hypothesis

Evidence consistent with the cytoskeletal integrative sensor hypothesis is presented in the case of glycolysis. Several testable predictions are made. There should be a relationship between post-translational modifications of tubulin and of actin and their interaction with metabolic enzymes. Different conditions of cytoskeletal dynamics and enzyme-cytoskeleton binding should reveal significant differences in local and perhaps global levels and ratios of ATP and GTP. The different functions of moonlighting enzymes should depend on cytoskeletal binding.

Implications of the hypothesis

The physical and chemical effects arising from metabolic sensing by the cytoskeleton would have major consequences on cell shape, dynamics and cell cycle progression. The hypothesis provides a framework that helps the significance of the enzyme-decorated cytoskeleton be determined.

Hybrid molecular dynamics simulations of living filaments

We propose a hybrid molecular dynamics/multi-particle collision dynamics model to simulate a set of self-assembled semiflexible filaments and free monomers. Further, we introduce a Monte Carlo scheme to deal with single monomer addition (polymerization) or removal (depolymerization), satisfying the detailed balance condition within a proper statistical mechanical framework. This model of filaments, based on the wormlike chain, aims to represent equilibrium polymers with distinct reaction rates at both ends, such as self-assembled adenosine diphosphate-actin filaments in the absence of adenosine triphosphate (ATP) hydrolysis and other proteins. We report the distribution of filament lengths and the corresponding dynamical fluctuations on an equilibrium trajectory. Potential generalizations of this method to include irreversible steps like ATP-actin hydrolysis are discussed.

History-dependent depolymerization of actin filaments

Depolymerizing cytoskeletal filaments are involved in cell division, cell motility, and other cellular functions. Understanding the dynamics of depolymerization is as important as understanding the dynamics of polymerization. We study nonequilibrium depolymerization of actin filaments using a simple two-state model. We show that the polymerization history influences the dynamics of depolymerization as well as the length fluctuations during depolymerization. We also simulate depolymerization under different experimentally feasible conditions. Under conditions of constant concentration, we show that the depolymerization happens in two regimes. Under the conditions of mass conservation, the depolymerization can have three regimes.

Random hydrolysis controls the dynamic instability of microtubules

Uncovering mechanisms that control the dynamics of microtubules is fundamental for our understanding of multiple cellular processes such as chromosome separation and cell motility. Building on previous theoretical work on the dynamic instability of microtubules, we propose here a stochastic model that includes all relevant biochemical processes that affect the dynamics of microtubule plus-end, namely, the binding of GTP-bound monomers, unbinding of GTP- and GDP-bound monomers, and hydrolysis of GTP monomers. The inclusion of dissociation processes, present in our approach but absent from many previous studies, is essential to guarantee the thermodynamic consistency of the model. Our theoretical method allows us to compute all dynamic properties of microtubules explicitly. Using experimentally determined rates, it is found that the cap size is ∼3.6 layers, an estimate that is compatible with several experimental observations. In the end, our model provides a comprehensive description of the dynamic instability of microtubules that includes not only the statistics of catastrophes but also the statistics of rescues.

Polymerization of actin filaments coupled with adenosine triphosphate hydrolysis: Brownian dynamics and theoretical analysis

Polymerization dynamics of single actin filaments coupled with adenosine triphosphate (ATP) hydrolysis is investigated via both theoretical analysis and Brownian dynamics simulations. Brownian dynamics simulations have been applied recently to study the growth behaviors of long filaments as a function of the free actin monomer concentrations, C(T), which is found to be in agreement with the associated experiments. In the present study, both ATP cap length and length diffusivity are studied as a function of the free ATP-actin monomer concentrations, C(T). The exact analytical expressions are found to be in perfect consistency with Brownian dynamics simulations. Likewise, we find that the length diffusion coefficient is peaked near the critical concentration, C(T,cr). It is, therefore, expected that the dependence of length diffusivity on ATP-actin monomer concentrations is utilized to analyze the surprising experiments on the length fluctuations of individual actin filaments.

The mechanisms of microtubule catastrophe and rescue: implications from analysis of a dimer-scale computational model

ETOC: The behavior of a dimer-scale computational model predicts that short interprotofilament “cracks” (laterally unbonded regions between protofilaments) exist even at the tips of growing MTs and that rapid fluctuations in the depths of these cracks govern both catastrophe and rescue.

Microtubule (MT) dynamic instability is fundamental to many cell functions, but its mechanism remains poorly understood, in part because it is difficult to gain information about the dimer-scale events at the MT tip. To address this issue, we used a dimer-scale computational model of MT assembly that is consistent with tubulin structure and biochemistry, displays dynamic instability, and covers experimentally relevant spans of time. It allows us to correlate macroscopic behaviors (dynamic instability parameters) with microscopic structures (tip conformations) and examine protofilament structure as the tip spontaneously progresses through both catastrophe and rescue. The model's behavior suggests that several commonly held assumptions about MT dynamics should be reconsidered. Moreover, it predicts that short, interprotofilament “cracks” (laterally unbonded regions between protofilaments) exist even at the tips of growing MTs and that rapid fluctuations in the depths of these cracks influence both catastrophe and rescue. We conclude that experimentally observed microtubule behavior can best be explained by a “stochastic cap” model in which tubulin subunits hydrolyze GTP according to a first-order reaction after they are incorporated into the lattice; catastrophe and rescue result from stochastic fluctuations in the size, shape, and extent of lateral bonding of the cap.

Active biopolymers confer fast reorganization kinetics

Many cytoskeletal biopolymers are "active," consuming energy in large quantities. In this Letter, we identify a fundamental difference between active polymers and passive, equilibrium polymers: for equal mean lengths, active polymers can reorganize faster than equilibrium polymers. We show that equilibrium polymers are intrinsically limited to linear scaling between mean lifetime (or mean first-passage time, or MFPT) and mean length, MFPT∼<L>, by analogy to 1D Potts models. By contrast, we present a simple active-polymer model that improves upon this scaling, such that MFPT∼<L>(1/2). Since, to be biologically useful, structural biopolymers must typically be many monomers long yet respond dynamically to the needs of the cell, the difference in reorganization kinetics may help to justify the active polymers' greater energy cost.

Dynamic instability of a growing adsorbed polymorphic filament

The intermittent transition between slow growth and rapid shrinkage in polymeric assemblies is termed "dynamic instability", a feature observed in a variety of biochemically distinct assemblies including microtubules, actin, and their bacterial analogs. The existence of this labile phase of a polymer has many functional consequences in cytoskeletal dynamics, and its repeated appearance suggests that it is relatively easy to evolve. Here, we consider the minimal ingredients for the existence of dynamic instability by considering a single polymorphic filament that grows by binding to a substrate, undergoes a conformation change, and may unbind as a consequence of the residual strains induced by this change. We identify two parameters that control the phase space of possibilities for the filament: a structural mechanical parameter that characterizes the ratio of the bond strengths along the filament to those with the substrate (or equivalently the ratio of longitudinal to lateral interactions in an assembly), and a kinetic parameter that characterizes the ratio of timescales for growth and conformation change. In the deterministic limit, these parameters serve to demarcate a region of uninterrupted growth from that of collapse. However, in the presence of disorder in either the structural or the kinetic parameter the growth and collapse phases can coexist where the filament can grow slowly, shrink rapidly, and transition between these phases, thus exhibiting dynamic instability. We exhibit the window for the existence of dynamic instability in a phase diagram that allows us to quantify the evolvability of this labile phase.

ENGINEERING AND THERAPEUTIC APPLICATIONS OF MICROTUBULES

Living organisms are fascinating systems. The macromolecules that make up a living cell possess equally astounding structural and functional characteristics. By taking simple cues from how these biopolymers organize and work inside the cell, one can draw inspiration to utilize them outside their natural environment for several purposes. Microtubules are example of biopolymers that demonstrate extraordinary properties of hierarchical self-organization, dynamic remodeling and mechanical rigidity. Mimicking the principles and properties of microtubules and improving them have opened novel engineering avenues. In addition, due to the functions that microtubules perform during cell division, they are excellent therapeutic drug targets for anticancer agents. In this work, we describe the biological properties and functions of microtubules, and discuss their engineering and therapeutic applications.

Mean-field study of the role of lateral cracks in microtubule dynamics

A link between dimer-scale processes and microtubule (MT) dynamics at macroscale is studied by comparing simulations obtained using computational dimer-scale model with its mean-field approximation. The novelty of the mean-field model (MFM) is in its explicit representation of inter-protofilament cracks, as well as in the direct incorporation of the dimer-level kinetics. Due to inclusion of both longitudinal and lateral dimer interactions, the MFM is two dimensional, in contrast to previous theoretical models of MTs. It is the first analytical model that predicts and quantifies crucial features of MT dynamics such as (i) existence of a minimal soluble tubulin concentration needed for the polymerization (with concentration represented as a function of model parameters), (ii) existence of steady-state growth and shortening phases (given with their respective velocities), and (iii) existence of an unstable pause state near zero velocity. In addition, the size of the GTP cap of a growing MT is estimated. Theoretical predictions are shown to be in good agreement with the numerical simulations.

A theoretical analysis of filament length fluctuations in actin and other polymers

Control of the structure and dynamics of the actin cytoskeleton is essential for cell motility and for maintaining the structural integrity of cells. Central to understanding the control of these features is an understanding of the dynamics of actin filaments, first as isolated filaments, then as integrated networks, and finally as networks containing higher-order structures such as bundles, stress fibers and acto-myosin complexes. It is known experimentally that single filaments can exhibit large fluctuations, but a detailed understanding of the transient dynamics involved is still lacking. Here we first study stochastic models of a general system involving two-monomer types that can be analyzed completely, and then we report stochastic simulations on the complete actin model with three monomer types. We systematically examine the transient behavior of filament length dynamics so as to gain a better understanding of the time scales involved in reaching a steady state. We predict the lifetime of a cap of one monomer type and obtain the mean and variance of the survival time of a cap at the filament end, which together determine the filament length fluctuations.

Control of actin filament treadmilling in cell motility

Recent advances in structural, biochemical, biophysical, and live cell imaging approaches have furthered our understanding of the molecular mechanisms by which regulated assembly dynamics of actin filaments drive motile processes. Attention is focused on lamellipodium protrusion, powered by the turnover of a branched filament array. ATP hydrolysis on actin is the key reaction that allows filament treadmilling. It regulates barbed-end dynamics and length fluctuations at steady state and specifies the functional interaction of actin with essential regulatory proteins such as profilin and ADF/cofilin. ATP hydrolysis on actin and Arp2/3 acts as a timer, regulating the assembly and disassembly of the branched array to generate tropomyosin-mediated heterogeneity in the structure and dynamics of the lamellipodial network. The detailed molecular mechanisms of ATP hydrolysis/Pi release on F-actin remain elusive, as well as the mechanism of filament branching with Arp2/3 complex or that of the formin-driven processive actin assembly. Novel biophysical methods involving single-molecule measurements should foster progress in these crucial issues.

Role of ATP-hydrolysis in the dynamics of a single actin filament

We study the stochastic dynamics of growth and shrinkage of single actin filaments taking into account insertion, removal, and ATP hydrolysis of subunits either according to the vectorial mechanism or to the random mechanism. In a previous work, we developed a model for a single actin or microtubule filament where hydrolysis occurred according to the vectorial mechanism: the filament could grow only from one end, and was in contact with a reservoir of monomers. Here we extend this approach in two ways--by including the dynamics of both ends and by comparing two possible mechanisms of ATP hydrolysis. Our emphasis is mainly on two possible limiting models for the mechanism of hydrolysis within a single filament, namely the vectorial or the random model. We propose a set of experiments to test the nature of the precise mechanism of hydrolysis within actin filaments.

A theory of microtubule catastrophes and their regulation

Dynamic instability, in which abrupt transitions occur between growing and shrinking states, is an intrinsic property of microtubules that is regulated by both mechanics and specialized proteins. We discuss a model of dynamic instability based on the popular idea that growth is maintained by a cap at the tip of the fiber. The loss of this cap is thought to trigger the transition from growth to shrinkage, called a catastrophe. The model includes longitudinal interactions between the terminal tubulins of each protofilament and "gating rescues" between neighboring protofilaments. These interactions allow individual protofilaments to transiently shorten during a phase of overall microtubule growth. The model reproduces the reported dependency of the catastrophe rate on tubulin concentration, the time between tubulin dilution and catastrophe, and the induction of microtubule catastrophes by walking depolymerases. The model also reproduces the comet tail distribution that is characteristic of proteins that bind to the tips of growing microtubules.

Continuous model for microtubule dynamics with catastrophe, rescue, and nucleation processes

Microtubules are a major component of the cytoskeleton distinguished by highly dynamic behavior both in vitro and in vivo referred to as dynamic instability. We propose a general mathematical model that accounts for the growth, catastrophe, rescue, and nucleation processes in the polymerization of microtubules from tubulin dimers. Our model is an extension of various mathematical models developed earlier formulated in order to capture and unify the various aspects of tubulin polymerization. While attempting to use a minimal number of adjustable parameters, the proposed model covers a broad range of behaviors and has predictive features discussed in the paper. We have analyzed the range of resultant dynamical behavior of the microtubules by changing each of the parameter values at a time and observing the emergence of various dynamical regimes that agree well with the previously reported experimental data and behavior.