Dynamics of Living Polymers

We study theoretically the dynamics of living polymers which can add and subtract monomer units at their live chain ends. The classic example is ionic living polymerization. In equilibrium, a delicate balance is maintained in which each initiated chain has a very small negative average growth rate (``velocity'') just sufficient to negate the effect of growth rate fluctuations. This leads to an exponential molecular weight distribution (MWD) with mean Nbar. After a small perturbation of relative amplitude epsilon, e.g. a small temperature jump, this balance is destroyed: the velocity acquires a boost greatly exceeding its tiny equilibrium value. For epsilon > epsilon_c = 1/Nbar^{1/2} the response has 3 stages: (1) Coherent chain growth or shrinkage, leaving a highly non-linear hole or peak in the MWD at small chain lengths. During this episode, lasting time taufast ~ Nbar, the MWD's first moment and monomer concentration m relax very close to equilibrium. (2) Hole-filling (or peak decay) after taufill ~ epsilon^2 Nbar^2. The absence or surfeit of small chains is erased. (3) Global MWD shape relaxation after tauslow ~ Nbar^2. By this time second and higher MWD moments have relaxed. During episodes (2) and (3) the fast variables (Nbar,m) are enslaved to the slowly varying number of free initiators (chains of zero length). Thus fast variables are quasi-statically fine-tuned to equilibrium. The outstanding feature of these dynamics is their ultrasensitivity: despite the perturbation's linearity, the response is non-linear until the late episode (3). For very small perturbations, epsilon < epsilon_c, response remains non-linear but with a less dramatic peak or hole during episode (1). Our predictions are in agreement with viscosity measurements on the most widely studied system, alpha-methylstyrene.Comment: 16 pages, submitted to Eur. Phys. J.

Reaction Kinetics in Polymer Melts

We study the reaction kinetics of end-functionalized polymer chains dispersed in an unreactive polymer melt. Starting from an infinite hierarchy of coupled equations for many-chain correlation functions, a closed equation is derived for the 2nd order rate constant $k$ after postulating simple physical bounds. Our results generalize previous 2-chain treatments (valid in dilute reactants limit) by Doi, de Gennes, and Friedman and O'Shaughnessy, to arbitrary initial reactive group density $n_0$ and local chemical reactivity $Q$. Simple mean field (MF) kinetics apply at short times, $k \sim Q$. For high $Q$, a transition occurs to diffusion-controlled (DC) kinetics with $k \approx x_t^3/t$ (where $x_t$ is rms monomer displacement in time $t$) leading to a density decay $n_t \approx n_0 - n_0^2 x_t^3$. If $n_0$ exceeds the chain overlap threshold, this behavior is followed by a regime where $n_t \approx 1/x_t^3$ during which $k$ has the same power law dependence in time, $k \approx x_t^3/t$, but possibly different numerical coefficient. For unentangled melts this gives $n_t \sim t^{-3/4}$ while for entangled cases one or more of the successive regimes $n_t \sim t^{-3/4}$, $t^{-3/8}$ and $t^{-3/4}$ may be realized depending on the magnitudes of $Q$ and $n_0$. Kinetics at times longer than the longest polymer relaxation time $\tau$ are always MF. If a DC regime has developed before $\tau$ then the long time rate constant is $k \approx R^3/\tau$ where $R$ is the coil radius. We propose measuring the above kinetics in a model experiment where radical end groups are generated by photolysis.Comment: 24 pages, 5 figures, uses bulk.sty, submitted to Eur.Phys.J.B discussion section expande

Interfacial Reaction Kinetics

We study irreversible A-B reaction kinetics at a fixed interface separating two immiscible bulk phases, A and B. We consider general dynamical exponent $z$, where $x_t\sim t^{1/z}$ is the rms diffusion distance after time $t$. At short times the number of reactions per unit area, $R_t$, is {\em 2nd order} in the far-field reactant densities $n_A^{\infty},n_B^{\infty}$. For spatial dimensions $d$ above a critical value $d_c=z-1$, simple mean field (MF) kinetics pertain, $R_t\sim Q_b t n_A^{\infty} n_B^{\infty}$ where $Q_b$ is the local reactivity. For low dimensions $d<d_c$, this MF regime is followed by 2nd order diffusion controlled (DC) kinetics, $R_t \approx x_t^{d+1} n_A^{\infty} n_B^{\infty}$, provided $Q_b > Q_b^* \sim (n_B^{\infty})^{[z-(d+1)]/d}$. Logarithmic corrections arise in marginal cases. At long times, a cross-over to {\em 1st order} DC kinetics occurs: $R_t \approx x_t n_A^{\infty}$. A density depletion hole grows on the more dilute A side. In the symmetric case ($n_A^{\infty}=n_B^{\infty}$), when $d<d_c$ the long time decay of the interfacial reactant density, $n_A^s$, is determined by fluctuations in the initial reactant distribution, giving $n_A^s \sim t^{-d/(2z)}$. Correspondingly, A-rich and B-rich regions develop at the interface analogously to the segregation effects established by other authors for the bulk reaction $A+B\to\emptyset$. For $d>d_c$ fluctuations are unimportant: local mean field theory applies at the interface (joint density distribution approximating the product of A and B densities) and $n_A^s \sim t^{(1-z)/(2z)}$. We apply our results to simple molecules (Fickian diffusion, $z=2$) and to several models of short-time polymer diffusion ($z>2$).Comment: 39 pages, 7 figures, uses fund2.sty, submitted to Eur. Phys. J. B, 1 figure added, for short version see cond-mat/980409

Steps of actin filament branch formation by Arp2/3 complex investigated with coarse-grained molecular dynamics

The nucleation of actin filament branches by the Arp2/3 complex involves activation through nucleation promotion factors (NPFs), recruitment of actin monomers, and binding of the complex to the side of actin filaments. Because of the large system size and processes that involve flexible regions and diffuse components, simulations of branch formation using all-atom molecular dynamics are challenging. We applied a coarse-grained model that retains amino-acid level information and allows molecular dynamics simulations in implicit solvent, with globular domains represented as rigid bodies and flexible regions allowed to fluctuate. We used recent electron microscopy structures of the inactive Arp2/3 complex bound to NPF domains and to mother actin filament for the activated Arp2/3 complex. We studied interactions of Arp2/3 complex with the activating VCA domain of the NPF Wiskott-Aldrich syndrome protein, actin monomers, and actin filament. We found stable configurations with one or two actin monomers bound along the branch filament direction and with CA domain of VCA associated to the strong and weak binding sites of the Arp2/3 complex, supporting prior structural studies and validating our approach. We reproduced delivery of actin monomers and CA to the Arp2/3 complex under different conditions, providing insight into mechanisms proposed in previous studies. Simulations of active Arp2/3 complex bound to a mother actin filament indicate the contribution of each subunit to the binding. Addition of the C-terminal tail of Arp2/3 complex subunit ArpC2, which is missing in the cryo-EM structure, increased binding affinity, indicating a possible stabilizing role of this tail

Cellular Hokey Pokey: A Coarse-Grained Model of Lamellipodia Protrusion Dynamics Driven by Fluctuations in Actin Polymerization

Animal cells that spread onto a surface often rely on actin-rich cell extensions called lamellipodia to execute cell protrusion. XTC cells on a two-dimensional substrate exhibit regular protrusion and retraction of their lamellipodium, even though the cell is not translating. Travelling waves of protrusion have also been observed, similar to those observed in crawling cells. These periodic fluctuations in leading edge position have been linked to excitable actin dynamics near the cell edge using a one dimensional model of actin dynamics, as a function of arc-length along the cell. In this work we extend this earlier model of actin dynamics into two-dimensions (along the arc-length and radial directions of the cell) and include a model membrane that protrudes and retracts in response to the changing number of free barbed ends of actin filaments near the membrane. We show that if the polymerization rate of these barbed ends changes depending on their local concentration at the leading edge and the opposing force from the cell membrane, the model can reproduce the patterns of membrane protrusion and retraction seen in experiment. We investigate both Brownian ratchet and switch-like force-velocity relationships between the membrane load forces and actin polymerization rate