Nonequilibrium self-assembly of linear fibers: microscopic treatment of growth, decay, catastrophe and rescue

Universal catastrophe time distributions of dynamically unstable polymers

Dynamic instability-the growth, catastrophe, and shrinkage of quasi-one-dimensional filaments-has been observed in multiple biopolymers. Scientists have long understood the catastrophic cessation of growth and subsequent depolymerization as arising from the interplay of hydrolysis and polymerization at the tip of the polymer. Here we show that for a broad class of catastrophe models, the expected catastrophe time distribution is exponential. We show that the distribution shape is insensitive to noise, but that depletion of monomers from a finite pool can dramatically change the distribution shape by reducing the polymerization rate. We derive a form for this finite-pool catastrophe time distribution and show that finite-pool effects can be important even when the depletion of monomers does not greatly alter the polymerization rate.

Dynamic Instability from Non-equilibrium Structural Transitions on the Energy Landscape of Microtubule

Microtubules are the backbone of the cytoskeleton and vital to numerous cellular processes. The central dogma of microtubules is that all their functions are driven by dynamic instability, but its mechanism has remained unresolved for over 30 years because of conceptual difficulties inherent in the dominant GTP-cap framework. We present a physically rigorous structural mechanochemical model: dynamic instability is driven by non-equilibrium transitions between the bent (B), straight (S), and curved (C) forms of tubulin monomers and longitudinal interfaces in the two-dimensional lattice of microtubule. All the different phenomena (growth, shortening, catastrophe, rescue, and pausing) are controlled by the kinetic pathways for B↔S↔C transitions and corresponding energy landscapes. Different kinetics at minus end are due to different B↔S↔C pathways imposed by the polarity of microtubule lattice. This model enables us to reproduce all the observed phenomena of dynamic instability of purified tubulins in kinetic simulations.

Circuit-Host Coupling Induces Multifaceted Behavioral Modulations of a Gene Switch

Quantitative modeling of gene circuits is fundamentally important to synthetic biology, as it offers the potential to transform circuit engineering from trial-and-error construction to rational design and, hence, facilitates the advance of the field. Currently, typical models regard gene circuits as isolated entities and focus only on the biochemical processes within the circuits. However, such a standard paradigm is getting challenged by increasing experimental evidence suggesting that circuits and their host are intimately connected, and their interactions can potentially impact circuit behaviors. Here we systematically examined the roles of circuit-host coupling in shaping circuit dynamics by using a self-activating gene switch as a model circuit. Through a combination of deterministic modeling, stochastic simulation, and Fokker-Planck equation formalism, we found that circuit-host coupling alters switch behaviors across multiple scales. At the single-cell level, it slows the switch dynamics in the high protein production regime and enlarges the difference between stable steady-state values. At the population level, it favors cells with low protein production through differential growth amplification. Together, the two-level coupling effects induce both quantitative and qualitative modulations of the switch, with the primary component of the effects determined by the circuit's architectural parameters. This study illustrates the complexity and importance of circuit-host coupling in modulating circuit behaviors, demonstrating the need for a new paradigm-integrated modeling of the circuit-host system-for quantitative understanding of engineered gene networks.

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.

A minimal transcriptional controlling network of regulatory T cell development

Regulatory T cells (Treg) are a subpopulation of T cells that are central to immune homeostasis and develop under the control of a complex regulatory network consisting of FoxP3 and its partner factors. A central question about this network is how does it enable T cells to robustly specify and stably maintain their states despite intrinsic and environmental fluctuations. Inspired by recent experimental advances, we propose here a minimal transcriptional controlling network and use it to illustrate the robustness and dynamic features of Treg development. Our study shows that the controlling network may exhibit distinct dynamics depending on its parameter regimes and that the maintenance of multistability requires the orchestration of both its positive and negative feedback loops. In addition, system volume contributes monotonically to the increase in the network's robustness. We further show that the dynamics of our model varies with the alteration of FoxP3-DNA binding affinity, consistent with recent experimental findings. This minimal model thereby offers new insights into the dynamics and robustness of Treg development and may serve as a platform for future exploration toward a more quantitative and systematic understanding of the immune system.

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.

THE BROWNIAN RATCHET REVISITED: DIFFUSION FORMALISM, POLYMER-BARRIER ATTRACTIONS, AND MULTIPLE FILAMENTOUS BUNDLE GROWTH

Actin polymerization driven stochastic movement of the bacteria Listeria monocytogenes is often measured using single-particle tracking (SPT) methodology and analyzed in terms of statistics. Experimental results suggested a dynamic association between the growing actin filaments and the propelled bacteria. Based on an alternative mathematical formalism for a Brownian ratchet (BR), we introduce such an attractive interaction into the one-dimensional BR model and show that its effect is equivalent to an external resistant force on the bacterium. Such a force significantly reduces the Brownian motion of a driven bacterium, and accentuates the stepping due to polymerization. We then consider the growth, with and without a barrier, of a filamentous bundle consisting of N identical filaments. It is shown that the bundle grows with a similar rate as a single filament in the absence of a load, but can oppose N times the external force under the stalling condition. A set of relationships describing the velocity of the bacterium movement (Vz) and its apparent diffusivity (Dz) as functions of the resistant force (F) and the number of filaments in a bundle (N) are obtained. The theoretical study suggests methods for data analysis in future experiments with applied external resistant force.

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.

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.

The stability of cellulose: a statistical perspective from a coarse-grained model of hydrogen-bond networks

A critical roadblock to the production of biofuels from lignocellulosic biomass is the efficient degradation of crystalline microfibrils of cellulose to glucose. A microscopic understanding of how different physical conditions affect the overall stability of the crystalline structure of microfibrils could facilitate the design of more effective protocols for their degradation. One of the essential physical interactions that stabilizes microfibrils is a network of hydrogen (H) bonds: both intrachain H-bonds between neighboring monomers of a single cellulose polymer chain and interchain H-bonds between adjacent chains. We construct a statistical mechanical model of cellulose assembly at the resolution of explicit hydrogen-bond networks. Using the transfer matrix method, the partition function and the subsequent statistical properties are evaluated. With the help of this lattice-based model, we capture the plasticity of the H-bond network in cellulose due to frustration and redundancy in the placement of H-bonds. This plasticity is responsible for the stability of cellulose over a wide range of temperatures. Stable intrachain and interchain H-bonds are identified as a function of temperature that could possibly be manipulated toward rational destruction of crystalline cellulose.

Molecular noise of capping protein binding induces macroscopic instability in filopodial dynamics

Capping proteins are among the most important regulatory proteins involved in controlling complicated stochastic dynamics of filopodia, which are dynamic finger-like protrusions used by eukaryotic motile cells to probe their environment and help guide cell motility. They attach to the barbed end of a filament and prevent polymerization, leading to effective filament retraction due to retrograde flow. When we simulated filopodial growth in the presence of capping proteins, qualitatively different dynamics emerged, compared with actin-only system. We discovered that molecular noise due to capping protein binding and unbinding leads to macroscopic filopodial length fluctuations, compared with minuscule fluctuations in the actin-only system. Thus, our work shows that molecular noise of signaling proteins may induce micrometer-scale growth-retraction cycles in filopodia. When capped, some filaments eventually retract all the way down to the filopodial base and disappear. This process endows filopodium with a finite lifetime. Additionally, the filopodia transiently grow several times longer than in actin-only system, since less actin transport is required because of bundle thinning. We have also developed an accurate mean-field model that provides qualitative explanations of our numerical simulation results. Our results are broadly consistent with experiments, in terms of predicting filopodial growth retraction cycles and the average filopodial lifetimes.

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

We study the stochastic dynamics of growth and shrinkage of single actin filaments or microtubules taking into account insertion, removal, and ATP/GTP hydrolysis of subunits. The resulting phase diagram contains three different phases: two phases of unbounded growth: a rapidly growing phase and an intermediate phase, and one bounded growth phase. We analyze all these phases, with an emphasis on the bounded growth phase. We also discuss how hydrolysis affects force-velocity curves. The bounded growth phase shows features of dynamic instability, which we characterize in terms of the time needed for the ATP/GTP cap to disappear as well as the time needed for the filament to reach a length of zero (i.e., to collapse) for the first time. We obtain exact expressions for all these quantities, which we test using Monte Carlo simulations.

A model for the regulatory network controlling the dynamics of kinetochore microtubule plus-ends and poleward flux in metaphase

Tight regulation of kinetochore microtubule dynamics is required to generate the appropriate position and movement of chromosomes on the mitotic spindle. A widely studied but mysterious aspect of this regulation occurs during metaphase when polymerization of kinetochore microtubule plus-ends is balanced by depolymerization at their minus-ends. Thus, kinetochore microtubules maintain a constant net length, allowing chromosomes to persist at the spindle equator, but consist of tubulin subunits that continually flux toward spindle poles. Here, we construct a feasible network of regulatory proteins for controlling kinetochore microtubule plus-end dynamics, which was combined with a Monte Carlo algorithm to simulate metaphase tubulin flux. We also test the network model by combining it with a force-balancing model explicitly taking force generators into account. Our data reveal how relatively simple interrelationships among proteins that stimulate microtubule plus-end polymerization, depolymerization, and dynamicity can induce robust flux while accurately predicting apparently contradictory results of knockdown experiments. The model also provides a simple and robust physical mechanism through which the regulatory networks at kinetochore microtubule plus- and minus-ends could communicate.

Dynamics of an idealized model of microtubule growth and catastrophe

We investigate a simple dynamical model of a microtubule that evolves by attachment of guanosine triphosphate (GTP) tubulin to its end, irreversible conversion of GTP to guanosine diphosphate (GDP) tubulin by hydrolysis, and detachment of GDP at the end of a microtubule. As a function of rates of these processes, the microtubule can grow steadily or its length can fluctuate wildly. In the regime where detachment can be neglected, we find exact expressions for the tubule and GTP cap length distributions, as well as power-law length distributions of GTP and GDP islands. In the opposite limit of instantaneous detachment, we find the time between catastrophes, where the microtubule shrinks to zero length, and determine the size distribution of avalanches (sequence of consecutive GDP detachment events). We obtain the phase diagram for general rates and verify our predictions by numerical simulations.

Dynamics of Microtubule Instabilities

We investigate an idealized model of microtubule dynamics that involves: (i) attachment of guanosine triphosphate (GTP) at rate λ, (ii) conversion of GTP to guanosine diphosphate (GDP) at rate 1, and (iii) detachment of GDP at rate μ. As a function of these rates, a microtubule can grow steadily or its length can fluctuate wildly. For μ = 0, we find the exact tubule and GTP cap length distributions, and power-law length distributions of GTP and GDP islands. For μ = ∞, we argue that the time between catastrophes, where the microtubule shrinks to zero length, scales as e(λ). We also discuss the nature of the phase boundary between a growing and shrinking microtubule.

Analysis of a mesoscopic stochastic model of microtubule dynamic instability

A theoretical model of dynamic instability of a system of linear one-dimensional microtubules (MTs) in a bounded domain is introduced for studying the role of a cell edge in vivo and analyzing the effect of competition for a limited amount of tubulin. The model differs from earlier models in that the evolution of MTs is based on the rates of single-mesoscopic-unit (e.g., a heterodimer per protofilament) transformations, in contrast to postulating effective rates and frequencies of larger-scale macroscopic changes, extracted, e.g., from the length history plots of MTs. Spontaneous GTP hydrolysis with finite rate after polymerization is assumed, and theoretical estimates of an effective catastrophe frequency as well as other parameters characterizing MT length distributions and cap size are derived. We implement a simple cap model which does not include vectorial hydrolysis. We demonstrate that our theoretical predictions, such as steady-state concentration of free tubulin and parameters of MT length distributions, are in agreement with the numerical simulations. The present model establishes a quantitative link between mesoscopic parameters governing the dynamics of MTs and macroscopic characteristics of MTs in a closed system. Last, we provide an explanation for nonexponential MT length distributions observed in experiments. In particular, we show that the appearance of such nonexponential distributions in the experiments can occur because a true steady state has not been reached and/or due to the presence of a cell edge.