Non-Gaussian dynamics from a simulation of a short peptide: Loop closure rates and effective diffusion coefficients

Abstract

Intrachain contact formation rates, fundamental to the dynamics of biopolymer self-organization such as protein folding, can be monitored in the laboratory through fluorescence quenching measurements. The common approximations for the intrachain contact rate given by the theory of Szabo, Schulten, and Schulten (SSS) [J. Chem. Phys. 72, 4350 (1980)] and Wilemski-Fixman (WF) [J. Chem. Phys. 60,878 (1973)] are shown to be complementary variational bounds: the SSS and WF approximations are lower and upper bounds, respectively, on the mean first contact times. As reported in the literature, the SSS approximation requires an effective diffusion coefficient 10 to 100 times smaller than expected to fit experimentally measured quenching rates. An all atom molecular dynamics simulation of an eleven residue peptide sequence in explicit water is analyzed to investigate the source of this surprising parameter value. The simulated diffusion limited contact time is approximately 6 ns for a reaction radius of 4Å for solvent viscosity corresponding to that of water at 298 K and 1 atm (η = 1.0 cP). In analytical work, the polymer is typically modeled by a Gaussian chain of effective monomers. Compared to Gaussian dynamics, the simulated end-to-end distance autocorrelation has a much slower relaxation. The long time behavior of the distance autocorrelation function can be approximated by a Gaussian model in which the monomer diffusion coefficient D0 is reduced to D0/6. This value of the diffusion coefficient brings the mean end-to-end contact time from analytical approximations and simulation into agreement in the sense that the SSS and WF approximations bracket the simulated mean first contact time. Attention to the non-Gaussian nature of the dynamics has direct implications for the development of improved analytical models.

Dynamics of Unfolded Polypeptide Chains in Crowded Environment Studied by Fluorescence Correlation Spectroscopy

“…37 Various simulations suggest that these theories represent complementary bounds of end-to-end contact formation rates in a Gaussian chain. 38,39 However, for real polymer chains the proposed scaling laws deviate for short chains due to chain stiffness influencing intrachain diffusion over short peptide segments. Flory introduced the characteristic ratio C N as a measure for the dimensions of a stiff chain compared to a freely jointed chain (FJC): 40…”

Diffusion-limited loop formation of semiflexible polymers: Kramers theory and the intertwined time scales of chain relaxation and closing

“…Note that the boundaries between Regions I and II are not sharp but are crossovers. Loop-formation kinetics in the crossover area will likely combine aspects of both regimes, as indicated in recent simulations [10] and by results that show that τ SSS and τ Doi are respectively lower and upper bounds for τ c [11]. Similarly, based on their BD simulation results, Podtelezhnikov et al [27] suggested that τ c ≃ τ R /α near the boundaries.…”

“…The discrepancy between the two continues to spur debate [10,11]. For the important case of stiff chains [12,13], where the polymer length L is not too much longer than the persistence length ℓ p , only limited numerical results are known [see, for example, [14,15] and references therein].…”

“…Even for the simplest case of an ideally flexible polymer with no hydrodynamic effects (or simply a Rouse chain), there are two rival theoretical approaches that lead to contradictory results: Szabo, Schulten, and Schulten (SSS) conclude that the time for a loop to form ("closing time" τ c ) should scale for moderately large polymer lengths L as τ c ∼ L 3/2 [7], while Doi, applying Wilemski-Fixmann (WF) theory [8], finds τ Doi ∼ L 2 [9]. The discrepancy between the two continues to spur debate [10,11]. For the important case of stiff chains [12,13], where the polymer length L is not too much longer than the persistence length ℓ p , only limited numerical results are known [see, for example, [14,15] and references therein].…”

The Rate of Intramolecular Loop Formation in DNA and Polypeptides: The Absence of the Diffusion-Controlled Limit and Fractional Power-Law Viscosity Dependence

“…The two most commonly used approximations, the Szabo–Schulten–Schulten (SSS) theory27 and the Wilemski–Fixman (WF) approximation,15 utilize two different local equilibrium approximations for the polymer dynamics and cannot be systematically improved in any straightforward way. Computer simulations2,3,6–9,12 have provided further insights and revealed the limitations inherent in these approximations. A recent paper by Thirumalai and co-workers12 presents a comprehensive comparison of various theories with the results of such simulations.…”