Dynamics of living polymers

Enhanced Glass Transition Temperature of Thin Polystyrene Films Having an Underneath Cross-Linked Layer

Due to the importance of the interface in the segmental dynamics of supported macromolecule ultrathin films, the glass transition temperature (T_g) of polystyrene (PS) ultrathin films upon solid substrates modified with a cross-linked PS (CLPS) layer has been investigated. The results showed that the T_g of the thin PS films on a silica surface with a ∼5 nm cross-linked layer increased with reducing film thickness. Meanwhile, the increase in T_g of the thin PS films became more pronounced with increasing the cross-linking density of the layer. For example, a 20 nm thick PS film supported on CLPS with 1.8 kDa of cross-linking degree exhibited a ∼35 and ∼50 K increase in T_g compared to its bulk and that on neat SiO₂ substrate, respectively. Such a large T_g elevation for the ultrathin PS films was attributed to the interfacial aggregation states in which chains diffused through nanolevel voids formed in the cross-linked layer to the SiO₂-Si surface. In such a situation, the chains were topologically constrained in the cross-linked layer with less mobility. These results offer us the opportunity to tailor interfacial effects by changing the degree of cross-linking, which has great potential application in many polymer nanocomposites.

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.

Dynamic Landau theory for supramolecular self-assembly

Although pathway-specific kinetic theories are fundamentally important to describe and understand reversible polymerisation kinetics, they come in principle at a cost of having a large number of system-specific parameters. Here, we construct a dynamical Landau theory to describe the kinetics of activated linear supramolecular self-assembly, which drastically reduces the number of parameters and still describes most of the interesting and generic behavior of the system in hand. This phenomenological approach hinges on the fact that if nucleated, the polymerisation transition resembles a phase transition. We are able to describe hysteresis, overshooting, undershooting and the existence of a lag time before polymerisation takes off, and pinpoint the conditions required for observing these types of phenomenon in the assembly and disassembly kinetics. We argue that the phenomenological kinetic parameter in our theory is a pathway controller, i.e., it controls the relative weights of the molecular pathways through which self-assembly takes place.

End-growth/evaporation living polymerization kinetics revisited

End-growth/evaporation kinetics in living polymer systems with "association-ready" free unimers (no initiator) is considered theoretically. The study is focused on the systems with long chains (typical aggregation number N ≫ 1) at long times. A closed system of continuous equations is derived and is applied to study the kinetics of the chain length distribution (CLD) following a jump of a parameter (T-jump) inducing a change of the equilibrium mean chain length from N(0) to N. The continuous approach is asymptotically exact for t ≫ t(1), where t(1) is the dimer dissociation time. It yields a number of essentially new analytical results concerning the CLD kinetics in some representative regimes. In particular, we obtained the asymptotically exact CLD response (for N ≫ 1) to a weak T-jump (ε = N(0)∕N - 1 ≪ 1). For arbitrary T-jumps we found that the longest relaxation time t(max ) = 1∕γ is always quadratic in N (γ is the relaxation rate of the slowest normal mode). More precisely t(max )∝4N(2) for N(0) < 2N and t(max )∝NN(0)∕(1 - N∕N(0)) for N(0) > 2N. The mean chain length N(n) is shown to change significantly during the intermediate slow relaxation stage t(1) ≪ t ≪ t(max ). We predict that N(n)(t)-N(n)(0)∝√t in the intermediate regime for weak (or moderate) T-jumps. For a deep T-quench inducing strong increase of the equilibrium N(n) (N ≫ N(0) ≫ 1), the mean chain length follows a similar law, N(n)(t)∝√t, while an opposite T-jump (inducing chain shortening, N(0) ≫ N ≫ 1) leads to a power-law decrease of N(n): N(n)(t)∝t(-1∕3). It is also shown that a living polymer system gets strongly polydisperse in the latter regime, the maximum polydispersity index r = N(w)∕N(n) being r∗ ≈ 0.77N(0)∕N ≫ 1. The concentration of free unimers relaxes mainly during the fast process with the characteristic time t(f) ∼ t(1)N(0)∕N(2). A nonexponential CLD dominated by short chains develops as a result of the fast stage in the case of N(0) = 1 and N ≫ 1. The obtained analytical results are supported, in part, by comparison with numerical results found both previously and in the present paper.

Non-linear scission/recombination kinetics of living polymerization

Living polymers are formed by reversible association of primary units (unimers). Generally the chain statistical weight involves a factor sigma < 1 suppressing short chains in comparison with free unimers. Living polymerization is a sharp thermodynamic transition for sigma << 1 which is typically the case. We show that this sharpness has an important effect on the kinetics of living polymerization (one-dimensional association). The kinetic model involves i) the unimer activation step (a transition to an assembly-competent state); ii) the scission/recombination processes providing growth of polymer chains and relaxation of their length distribution. Analyzing the polymerization with no chains but unimers at t = 0, with initial concentration of unimers M greater or approximately M(*) (M(*) is the critical polymerization concentration)), we determine the time evolution of the chain length distribution and find that: 1) for M(*) << M << M(*) /sigma the kinetics is characterized by 5 distinct time stages demarcated by 4 characteristic times t(1), t(2), t(3) and t(*); 2) there are transient regimes (t(1) less or approximately t less or approximately t(3)) when the molecular-weight distribution is strongly non-exponential; 3) the chain scissions are negligible at times shorter than t(2). The chain growth is auto-accelerated for t(1) less or approximately t less or approximately t(2) : the cut-off chain length (= polymerization degree ((n)w) N(1) proportional, variant t(2) in this regime. 4) For t(2) < t < t(3) the length distribution is characterized by essentially 2 non-linear modes; the shorter cut-off length N(1) is decreasing with time in this regime, while the length scale N(2) of the second mode is increasing. (5) The terminal relaxation time of the polymer length distribution, t(*), shows a sharp maximum in the vicinity of M(*); the effective exponent (partial partial differential ln 1/t(*)) divided by (partial partial differential ln M) is as high as approximately sigma(-1/3) just above M(*).

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.

Actin polymerization kinetics, cap structure, and fluctuations

Polymerization of actin proteins into dynamic structures is essential to eukaryotic cell life, motivating many in vitro experiments measuring polymerization kinetics of individual filaments. Here, we model these kinetics, accounting for all relevant steps revealed by experiment: polymerization, depolymerization, random ATP hydrolysis, and release of phosphate (P(i)). We relate filament growth rates to the dynamics of ATP-actin and ADP-P(i)-actin caps that develop at filament ends. At the critical concentration of the barbed end, c(crit), we find a short ATP cap and a long fluctuation-stabilized ADP-P(i) cap. We show that growth rates and the critical concentration at the barbed end are intimately related to cap structure and dynamics. Fluctuations in filament lengths are described by the length diffusion coefficient, D. Recently Fujiwara et al. [Fujiwara, I., Takahashi, S., Takaduma, H., Funatsu, T. & Ishiwata, S. (2002) Nat. Cell Biol. 4, 666-673] and Kuhn and Pollard [Kuhn, J. & Pollard, T. D. (2005) Biophys. J. 88, 1387-1402] observed large length fluctuations slightly above c(crit), provoking speculation that growth may proceed by oligomeric rather than monomeric on-off events. For the single-monomer growth process, we find that D exhibits a pronounced peak below c(crit), due to filaments alternating between capped and uncapped states, a mild version of the dynamic instability of microtubules. Fluctuations just above c(crit) are enhanced but much smaller than those reported experimentally. Future measurements of D as a function of concentration can help identify the origin of the observed fluctuations.