Providing positional information with active transport on dynamic microtubules

eGFRD in all dimensions

Biochemical reactions often occur at low copy numbers but at once in crowded and diverse environments. Space and stochasticity therefore play an essential role in biochemical networks. Spatial-stochastic simulations have become a prominent tool for understanding how stochasticity at the microscopic level influences the macroscopic behavior of such systems. While particle-based models guarantee the level of detail necessary to accurately describe the microscopic dynamics at very low copy numbers, the algorithms used to simulate them typically imply trade-offs between computational efficiency and biochemical accuracy. eGFRD (enhanced Green's Function Reaction Dynamics) is an exact algorithm that evades such trade-offs by partitioning the N-particle system into M ≤ N analytically tractable one- and two-particle systems; the analytical solutions (Green's functions) then are used to implement an event-driven particle-based scheme that allows particles to make large jumps in time and space while retaining access to their state variables at arbitrary simulation times. Here we present "eGFRD2," a new eGFRD version that implements the principle of eGFRD in all dimensions, thus enabling efficient particle-based simulation of biochemical reaction-diffusion processes in the 3D cytoplasm, on 2D planes representing membranes, and on 1D elongated cylinders representative of, e.g., cytoskeletal tracks or DNA; in 1D, it also incorporates convective motion used to model active transport. We find that, for low particle densities, eGFRD2 is up to 6 orders of magnitude faster than conventional Brownian dynamics. We exemplify the capabilities of eGFRD2 by simulating an idealized model of Pom1 gradient formation, which involves 3D diffusion, active transport on microtubules, and autophosphorylation on the membrane, confirming recent experimental and theoretical results on this system to hold under genuinely stochastic conditions.

Determinants of Polar versus Nematic Organization in Networks of Dynamic Microtubules and Mitotic Motors

During cell division, mitotic motors organize microtubules in the bipolar spindle into either polar arrays at the spindle poles or a "nematic" network of aligned microtubules at the spindle center. The reasons for the distinct self-organizing capacities of dynamic microtubules and different motors are not understood. Using in vitro reconstitution experiments and computer simulations, we show that the human mitotic motors kinesin-5 KIF11 and kinesin-14 HSET, despite opposite directionalities, can both organize dynamic microtubules into either polar or nematic networks. We show that in addition to the motor properties the natural asymmetry between microtubule plus- and minus-end growth critically contributes to the organizational potential of the motors. We identify two control parameters that capture system composition and kinetic properties and predict the outcome of microtubule network organization. These results elucidate a fundamental design principle of spindle bipolarity and establish general rules for active filament network organization.

Metaphase kinetochore movements are regulated by kinesin-8 motors and microtubule dynamic instability

During metaphase, sister chromatids are connected to microtubules extending from the opposite spindle poles via kinetochores to protein complexes on the chromosome. Kinetochores congress to the equatorial plane of the spindle and oscillate around it, with kinesin-8 motors restricting these movements. Yet, the physical mechanism underlying kinetochore movements is unclear. We show that kinetochore movements in the fission yeast Schizosaccharomyces pombe are regulated by kinesin-8-promoted microtubule catastrophe, force-induced rescue, and microtubule dynamic instability. A candidate screen showed that among the selected motors only kinesin-8 motors Klp5/Klp6 are required for kinetochore centering. Kinesin-8 accumulates at the end of microtubules, where it promotes catastrophe. Laser ablation of the spindle resulted in kinetochore movement toward the intact spindle pole in wild-type and klp5Δ cells, suggesting that kinetochore movement is driven by pulling forces. Our theoretical model with Langevin description of microtubule dynamic instability shows that kinesin-8 motors are required for kinetochore centering, whereas sensitivity of rescue to force is necessary for the generation of oscillations. We found that irregular kinetochore movements occur for a broader range of parameters than regular oscillations. Thus, our work provides an explanation for how regulation of microtubule dynamic instability contributes to kinetochore congression and the accompanying movements around the spindle center.

Polymerisation force of a rigid filament bundle: diffusive interaction leads to sublinear force-number scaling

Polymerising filaments generate force against an obstacle, as in, e.g., microtubule-kinetochore interactions in the eukaryotic cell. Earlier studies of this problem have not included explicit three-dimensional monomer diffusion, and consequently, missed out on two important aspects: (i) the barrier, even when it is far from the polymers, affects free diffusion of monomers and reduces their adsorption at the tips, while (ii) parallel filaments could interact through the monomer density field ("diffusive coupling"), leading to negative interference between them. In our study, both these effects are included and their consequences investigated in detail. A mathematical treatment based on a set of continuum Fokker-Planck equations for combined filament-wall dynamics suggests that the barrier-induced monomer depletion reduces the growth velocity and also the stall force, while the total force produced by many filaments remains additive. However, Brownian dynamics simulations show that the linear force-number scaling holds only when the filaments are far apart; when they are arranged close together, forming a bundle, sublinear scaling of force with number appears, which could be attributed to diffusive interaction between the growing polymer tips.

Motor Protein Accumulation on Antiparallel Microtubule Overlaps

Biopolymers serve as one-dimensional tracks on which motor proteins move to perform their biological roles. Motor protein phenomena have inspired theoretical models of one-dimensional transport, crowding, and jamming. Experiments studying the motion of Xklp1 motors on reconstituted antiparallel microtubule overlaps demonstrated that motors recruited to the overlap walk toward the plus end of individual microtubules and frequently switch between filaments. We study a model of this system that couples the totally asymmetric simple exclusion process for motor motion with switches between antiparallel filaments and binding kinetics. We determine steady-state motor density profiles for fixed-length overlaps using exact and approximate solutions of the continuum differential equations and compare to kinetic Monte Carlo simulations. Overlap motor density profiles and motor trajectories resemble experimental measurements. The phase diagram of the model is similar to the single-filament case for low switching rate, while for high switching rate we find a new (to our knowledge) low density-high density-low density-high density phase. The overlap center region, far from the overlap ends, has a constant motor density as one would naïvely expect. However, rather than following a simple binding equilibrium, the center motor density depends on total overlap length, motor speed, and motor switching rate. The size of the crowded boundary layer near the overlap ends is also dependent on the overlap length and switching rate in addition to the motor speed and bulk concentration. The antiparallel microtubule overlap geometry may offer a previously unrecognized mechanism for biological regulation of protein concentration and consequent activity.

Phase-plane analysis of the totally asymmetric simple exclusion process with binding kinetics and switching between antiparallel lanes

Motor protein motion on biopolymers can be described by models related to the totally asymmetric simple exclusion process (TASEP). Inspired by experiments on the motion of kinesin-4 motors on antiparallel microtubule overlaps, we analyze a model incorporating the TASEP on two antiparallel lanes with binding kinetics and lane switching. We determine the steady-state motor density profiles using phase-plane analysis of the steady-state mean field equations and kinetic Monte Carlo simulations. We focus on the density-density phase plane, where we find an analytic solution to the mean field model. By studying the phase-space flows, we determine the model's fixed points and their changes with parameters. Phases previously identified for the single-lane model occur for low switching rate between lanes. We predict a multiple coexistence phase due to additional fixed points that appear as the switching rate increases: switching moves motors from the higher-density to the lower-density lane, causing local jamming and creating multiple domain walls. We determine the phase diagram of the model for both symmetric and general boundary conditions.

Feedback mechanism for microtubule length regulation by stathmin gradients

We formulate and analyze a theoretical model for the regulation of microtubule (MT) polymerization dynamics by the signaling proteins Rac1 and stathmin. In cells, the MT growth rate is inhibited by cytosolic stathmin, which, in turn, is inactivated by Rac1. Growing MTs activate Rac1 at the cell edge, which closes a positive feedback loop. We investigate both tubulin sequestering and catastrophe promotion as mechanisms for MT growth inhibition by stathmin. For a homogeneous stathmin concentration in the absence of Rac1, we find a switchlike regulation of the MT mean length by stathmin. For constitutively active Rac1 at the cell edge, stathmin is deactivated locally, which establishes a spatial gradient of active stathmin. In this gradient, we find a stationary bimodal MT-length distribution for both mechanisms of MT growth inhibition by stathmin. One subpopulation of the bimodal length distribution can be identified with fast-growing and long pioneering MTs in the region near the cell edge, which have been observed experimentally. The feedback loop is closed through Rac1 activation by MTs. For tubulin sequestering by stathmin, this establishes a bistable switch with two stable states: one stable state corresponds to upregulated MT mean length and bimodal MT length distributions, i.e., pioneering MTs; the other stable state corresponds to an interrupted feedback with short MTs. Stochastic effects as well as external perturbations can trigger switching events. For catastrophe-promoting stathmin, we do not find bistability.

Physical and Mathematical Modeling in Experimental Papers

An increasing number of publications include modeling. Often, such studies help us to gain a deeper insight into the phenomena studied and break down barriers between experimental and theoretical communities. However, combining experimental and theoretical work is challenging for authors, reviewers, and readers. To help maximize the usefulness and impact of combined theoretical and experimental research, this Primer describes the purpose, usefulness, and different types of models and addresses the practical aspect of integrated publications by outlining characteristics of good modeling, presentation, and fruitful collaborations.

Delayed feedback model of axonal length sensing

A fundamental question in cell biology is how the sizes of cells and organelles are regulated at various stages of development. Size homeostasis is particularly challenging for neurons, whose axons can extend from hundreds of microns to meters (in humans). Recently, a molecular-motor-based mechanism for axonal length sensing has been proposed, in which axonal length is encoded by the frequency of an oscillating retrograde signal. In this article, we develop a mathematical model of this length-sensing mechanism in which advection-diffusion equations for bidirectional motor transport are coupled to a chemical signaling network. We show that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. Knockdown of either kinesin or dynein causes an increase in the oscillation frequency, suggesting that the length-sensing mechanism would produce longer axons, which is consistent with experimental findings. One major prediction of the model is that fluctuations in the transport of molecular motors lead to a reduction in the reliability of the frequency-encoding mechanism for long axons.

Effects of aging in catastrophe on the steady state and dynamics of a microtubule population

Several independent observations have suggested that the catastrophe transition in microtubules is not a first-order process, as is usually assumed. Recent in vitro observations by Gardner et al. [M. K. Gardner et al., Cell 147, 1092 (2011)] showed that microtubule catastrophe takes place via multiple steps and the frequency increases with the age of the filament. Here we investigate, via numerical simulations and mathematical calculations, some of the consequences of the age dependence of catastrophe on the dynamics of microtubules as a function of the aging rate, for two different models of aging: exponential growth, but saturating asymptotically, and purely linear growth. The boundary demarcating the steady-state and non-steady-state regimes in the dynamics is derived analytically in both cases. Numerical simulations, supported by analytical calculations in the linear model, show that aging leads to nonexponential length distributions in steady state. More importantly, oscillations ensue in microtubule length and velocity. The regularity of oscillations, as characterized by the negative dip in the autocorrelation function, is reduced by increasing the frequency of rescue events. Our study shows that the age dependence of catastrophe could function as an intrinsic mechanism to generate oscillatory dynamics in a microtubule population, distinct from hitherto identified ones.

Kinesin-8 motors improve nuclear centering by promoting microtubule catastrophe

In fission yeast, microtubules push against the cell edge, thereby positioning the nucleus in the cell center. Kinesin-8 motors regulate microtubule catastrophe; however, their role in nuclear positioning is not known. Here we develop a physical model that describes how kinesin-8 motors affect nuclear centering by promoting a microtubule catastrophe. Our model predicts the improved centering of the nucleus in the presence of motors, which we confirmed experimentally in living cells. The model also predicts a characteristic time for the recentering of a displaced nucleus, which is supported by our experiments where we displaced the nucleus using optical tweezers.

Biophysics of filament length regulation by molecular motors

Regulating physical size is an essential problem that biological organisms must solve from the subcellular to the organismal scales, but it is not well understood what physical principles and mechanisms organisms use to sense and regulate their size. Any biophysical size-regulation scheme operates in a noisy environment and must be robust to other cellular dynamics and fluctuations. This work develops theory of filament length regulation inspired by recent experiments on kinesin-8 motor proteins, which move with directional bias on microtubule filaments and alter microtubule dynamics. Purified kinesin-8 motors can depolymerize chemically-stabilized microtubules. In the length-dependent depolymerization model, the rate of depolymerization tends to increase with filament length, because long filaments accumulate more motors at their tips and therefore shorten more quickly. When balanced with a constant filament growth rate, this mechanism can lead to a fixed polymer length. However, the mechanism by which kinesin-8 motors affect the length of dynamic microtubules in cells is less clear. We study the more biologically realistic problem of microtubule dynamic instability modulated by a motor-dependent increase in the filament catastrophe frequency. This leads to a significant decrease in the mean filament length and a narrowing of the filament length distribution. The results improve our understanding of the biophysics of length regulation in cells.

Dynamics and length distribution of microtubules under force and confinement

We investigate the microtubule polymerization dynamics with catastrophe and rescue events for three different confinement scenarios, which mimic typical cellular environments: (i) The microtubule is confined by rigid and fixed walls, (ii) it grows under constant force, and (iii) it grows against an elastic obstacle with a linearly increasing force. We use realistic catastrophe models and analyze the microtubule dynamics, the resulting microtubule length distributions, and force generation by stochastic and mean field calculations; in addition, we perform stochastic simulations. Freely growing microtubules exhibit a phase of bounded growth with finite microtubule length and a phase of unbounded growth. The main results for the three confinement scenarios are as follows: (i) In confinement by fixed rigid walls, we find exponentially decreasing or increasing stationary microtubule length distributions instead of bounded or unbounded phases, respectively. We introduce a realistic model for wall-induced catastrophes and investigate the behavior of the average length as a function of microtubule growth parameters. (ii) Under a constant force, the boundary between bounded and unbounded growth is shifted to higher tubulin concentrations and rescue rates. The critical force f(c) for the transition from unbounded to bounded growth increases logarithmically with tubulin concentration and the rescue rate, and it is smaller than the stall force. (iii) For microtubule growth against an elastic obstacle, the microtubule length and polymerization force can be regulated by microtubule growth parameters. For zero rescue rate, we find that the average polymerization force depends logarithmically on the tubulin concentration and is always smaller than the stall force in the absence of catastrophes and rescues. For a nonzero rescue rate, we find a sharply peaked steady-state length distribution, which is tightly controlled by microtubule growth parameters. The corresponding average microtubule length self-organizes such that the average polymerization force equals the critical force f(c) for the transition from unbounded to bounded growth. We also investigate the force dynamics if growth parameters are perturbed in dilution experiments. Finally, we show the robustness of our results against changes of catastrophe models and load distribution factors.

Microtubule length regulation by molecular motors

Length regulation of microtubules (MTs) is essential for many cellular processes. Molecular motors like kinesin-8, which move along MTs and also act as depolymerases, are known as key players in MT dynamics. However, the regulatory mechanisms of length control remain elusive. Here, we investigate a stochastic model accounting for the interplay between polymerization kinetics and motor-induced depolymerization. We determine the dependence of MT length and variance on rate constants and motor concentration. Moreover, our analyses reveal how collective phenomena lead to a well-defined MT length.

Length-dependent dynamics of microtubules

Certain regulatory proteins influence the polymerization dynamics of microtubules by inducing catastrophe with a rate that depends on the microtubule length. Using a discrete formulation, here we show that, for a catastrophe rate proportional to the microtubule length, the steady-state probability distributions of length decay much faster with length than an exponential decay as seen in the absence of these proteins.