Universal behavior in reacting polymer systems

Loop formation and translational diffusion of intrinsically disordered proteins

The conformational flexibility and dynamics of unfolded peptide chains is of major interest in the context of protein folding and protein functioning. The rate with which amino acids at different positions along the peptide chain meet sets an upper speed limit for protein folding. By using single-molecule photo-induced energy transfer spectroscopy, we have systematically measured end-to-end and end-to-internal site contact formation rates for several intrinsically disordered protein fragments (IDPs) (11 to 41 amino acids) and have also determined their hydrodynamic radius using dual-focus fluorescence correlation spectroscopy. For interpreting the measured values, we have developed a Brownian dynamics model (based on bead-rod chain dynamics in a thermal bath including hydrodynamic interactions) which quantitatively reproduces all measured data surprisingly well while requiring only two fit parameters. The model provides a complete picture of the peptides' dynamics and allows us to translate the experimental rates and radii into molecular properties of the peptides: We find a persistence length of l_{P}=5.2±1.9Å, a hydrodynamic radius of a=3.5±0.7Å per amino acid, and that excluded volume effects play an important role in the dynamics of IDPs.

Renormalization group analysis of polymer cyclization with non-equilibrium initial conditions

We develop a renormalization group approach for cyclizing polymers for the case when chain ends are initially close together (ring initial conditions). We analyze the behavior at times much shorter than the longest polymer relaxation time. In agreement with our previous work (Europhys. Lett. 73, 621 (2006)) we find that the leading time dependence of the reaction rate k(t) for ring initial conditions and equilibrium initial conditions are related, namely k (ring)(t) proportional, variant t (-delta) and k (eq)(t) proportional, variant t (1-delta) for times less than the longest polymer relaxation time. Here delta is an effective exponent which approaches delta = 5/4 for very long Rouse chains. Our present analysis also suggests a "sub-leading" term proportional to (ln t)/t which should be particularly significant for smaller values of the renormalized reaction rate and early times. For Zimm dynamics, our RG analysis indicates that the leading time dependence for the reaction rate is k(t) approximately 1/t for very long chains. The leading term is again consistent with the expected relation between ring and equilibrium initial conditions. We also find a logarithmic correction term which we "exponentiate" to a logarithmic form with a Landau pole. The presence of the logarithm is particularly important for smaller chains and, in the Zimm case, large values of the reaction rate.

Diffusion-controlled first contact of the ends of a polymer: crossover between two scaling regimes

We report on Monte Carlo simulations of loop formation of an ideal flexible polymer consisting of N bonds with two reactive ends. We determine the first-passage time associated with chain looping that yields a conformation in which the end monomers are separated by a distance a--the reaction radius. In particular, our numerical results demonstrate how this time scale crosses over from tau(first) approximately N(3/2)/a to the a-independent tau(first) approximately N2 as N is increased. The existence and characteristics, of the two scaling regimes and the crossover between the two, are further illuminated by a scaling argument.

Temperature-Dependent Tail−Tail Dynamics of Pyrene-Labeled Poly(dimethylsiloxane) Oligomers Dissolved in Ethyl Acetate

We have determined the tail-tail dynamics and energetics of linear poly(dimethylsiloxane) (PDMS) polymers (M(n) = 3100; M(w)/M(n) = 1.07) that have been terminally labeled with pyrene (Py-PDMS-Py) in ethyl acetate between 255 and 323 K. The upper critical solution temperature (theta(u)) for the PDMS/ethyl acetate system is 278 K. The Py-PDMS-Py photophysics follow the Birks model at all of the temperatures studied. However, there is evidence for an increase in the local composition of PDMS surrounding the Py- residues in Py-PDMS-Py below the theta temperature. The recovered activation energy for Py-PDMS-Py tail-tail cyclization is the same as the activation energy for diffusion in ethyl acetate. The activation energy for cyclization is about fourfold larger in comparison to the barrier for PDMS internal backbone rotations. The estimated internal activation barrier for Py-PDMS-Py in ethyl acetate is in line with the internal barrier rotation for PDMS backbone rotation. The intramolecular excimer binding energy for Py-PDMS-Py in ethyl acetate is similar to the intermolecular pyrene excimer. In the ground state, the mean distance between the Py-PDMS-Py tail segments does not change over the temperature range studied. A comparison of these temperature-dependent Py-PDMS-Py/ethyl acetate results with previous results on Py-PDMS-Py in pure supercritical CO2 and CO2-dilated liquids shows that the addition of CO2 to the liquid phase offers the way to tune to Py-PDMS-Py tail-tail dynamics over the greatest range.

Cyclization of Rouse chains at long- and short-time scales

We have investigated cyclization of a Rouse chain at long and short times by a Langevin dynamics simulation method. We measure St, the fraction of nonreacted chains, for polymerizations ranging from Z=5 to Z=800 and capture distances ranging from a=0.1b to a=8b where b is the bond length. Comparison is made with two theoretical approaches. The first is a decoupling approximation used by Wilemski and Fixman to close the relevant master equation [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)]. The second approach is the renormalization group arguments of Friedman and O'Shaughnessy [Phys. Rev. Lett 60, 64 (1988); J. Phys. II 1, 471 (1991)]. We find that at long times St decays as a single exponential with rate k(infinity). The scaled decay rate K=k(infinity)tauR appears to approach a constant value independent of the capture distance for very large chains consistent with the predictions of both the renormalization group (RG) and Wilemski-Fixman closure approximation. We extract K*, the long chain limit of K, from the fixed point a=a* where K is independent of Z. K* is larger than both the RG and closure predictions but much closer to the RG result. More convincing evidence for the RG analysis is obtained by comparing the short-time decay of St to long-time results. The RG analysis predicts that dSdt should decay as a power law at early times and that the exponent in the power law is related to K by a simple expression with no free parameters. Our simulations find remarkable agreement with this parameter-free prediction even for relatively short chains. We discuss possible experimental consequences of our result.