Dynamic instability of a growing adsorbed polymorphic filament

Abstract

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.

Figure 1

Filament conformation—statics. (a) Schematic representation of the system consisting of an elastic filament with a nonzero natural curvature adhered to a substrate via a series of springs. Here the filament is akin to a protofilament whereas the substrate represents the “bath” of the other filaments. (b) Static phase diagram showing the unbinding threshold in terms of the average local bending angle as a function of the intrinsic angle ϕ for different values of the stiffness S. (Inset) Critical value of the dimensionless parameter α, that characterizes the ratio of the filament and substrate energy, as a function of the ratio between the healing length l_h and the discretization step a. (Solid line) Theoretical result, in excellent agreement with the simulations. (Dashed line) Asymptotic result as a → 0. (c) The characteristic shape of the attached filament in the neighborhood of the free edge shows the healing length characterizing the balance between filament and substrate deformation. The simulation results are compared with our linearized continuum theory.

Figure 2

Filament dynamics for different values of the intrinsic curvature κ and spring stiffness S, corresponding to the unbound phase α > α_c. The filament unbinds dynamically. There are two possible mechanisms that determine the kinetics of the process. (a) Diffusive unbinding occurs when the filament detaches partially but remains intact. In this case, the figure shows that the detached length increases as l/a = V^∗(ηt)^1/2. (Inset) Coefficient V^∗, for different values of S, follows a single linear function when plotted as a function of (κ² – κ²_c)^1/2. (b) Stokesian unbinding occurs when the subunits break off from the filament once they are detached. The figure shows that the detached length follows the form l/a = V(ηt)/a. (Inset) Coefficient V scales as is a linear function of (κ² – κ²_c) as expected for small/ moderate values of the parameters.

Figure 3

Filament dynamics in the presence of quenched disorder and thermal fluctuations. (a) When quenched disorder is introduced in the spring stiffness, the unbinding threshold in Fig. 1 is broadened due to the statistics of extremes as characterized in terms of the coefficient of variation C_ν = σ/μ, the ratio of the standard deviation to the mean of the spring stiffness S. (Inset) Critical value of the dimensionless parameter $\alpha \equiv \sqrt{B}\kappa /\sqrt{\mu }{r}_c$ as a function of the coefficient of variation of the disorder. The error bar represents the variance of α_c. rather than the error on the mean. Both the mean and the variance of α_c are seen to increase with increasing C_v as predicted by extreme value statistics (line). (b) Due to thermal fluctuations, the filament can unbind also for α < α_c. Here, we show the decrease of the filament length for α = 0.5α_c. (Inset) Velocity in the initial stage satisfies the Arrhenius law.

Figure 4

Filament growth and collapse kinetics. (a) Filament length as a function of time for a range of α₀, the mechanical control parameter. (b) Filament length as a function of time for a range of β = τ_κ/τ_G, the ratio of the time constant for the intrinsic curvature to equilibrate relative to the time constant for subunit addition. (c) Filament length when the hydrolysis is random also shows the transition between growth and catastrophe, but with one important difference from the case when the toughness parameter α is random; shrinkage goes all the way to collapse in the absence of any structural mechanical inhomogeneities.

Figure 5

Phase diagram in α–β space summarizing the regimes of growth, collapse, and dynamic instability—a window that appears in the presence of quenched disorder either in the structural toughness of the filament-substrate bond or in the hydrolysis. (Dashed line) Theoretical prediction corresponding to Eq. 15.