Fitting a function to time-dependent ensemble averaged data

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science. A crucial objective when interpreting such data is to fit these averages (for instance, squared displacements) with a function and extract parameters (such as diffusion constants). A commonly overlooked challenge in such function fitting procedures is that fluctuations around mean values, by construction, exhibit temporal correlations. We show that the only available general purpose function fitting methods, correlated chi-square method and the weighted least squares method (which neglects correlation), fail at either robust parameter estimation or accurate error estimation. We remedy this by deriving a new closed-form error estimation formula for weighted least square fitting. The new formula uses the full covariance matrix, i.e., rigorously includes temporal correlations, but is free of the robustness issues, inherent to the correlated chi-square method. We demonstrate its accuracy in four examples of importance in many fields: Brownian motion, damped harmonic oscillation, fractional Brownian motion and continuous time random walks. We also successfully apply our method, weighted least squares including correlation in error estimation (WLS-ICE), to particle tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and we provide a publically available WLS-ICE software.Comment: 47 pages (main text: 15 pages, supplementary: 32 pages

Three-helix-bundle Protein in a Ramachandran Model

We study the thermodynamic behavior of a model protein with 54 amino acids that forms a three-helix bundle in its native state. The model contains three types of amino acids and five to six atoms per amino acid and has the Ramachandran torsional angles $\phi_i$, $\psi_i$ as its degrees of freedom. The force field is based on hydrogen bonds and effective hydrophobicity forces. For a suitable choice of the relative strength of these interactions, we find that the three-helix-bundle protein undergoes an abrupt folding transition from an expanded state to the native state. Also shown is that the corresponding one- and two-helix segments are less stable than the three-helix sequence.Comment: 15 pages, 7 figure

Folding in two-dimenensional off-lattice models of proteins

Model off-lattice sequences in two dimensions are constructed so that their native states are close to an on-lattice target. The Hamiltonian involves the Lennard-Jones and harmonic interactions. The native states of these sequences are determined with a high degree of certainty through Monte Carlo processes. The sequences are characterized thermodynamically and kinetically. It is shown that the rank-ordering-based scheme of the assignment of contact energies typically fails in off-lattice models even though it generates high stability of on-lattice sequences. Similar to the on-lattice case, Go-like modeling, in which the interaction potentials are restricted to the native contacts in a target shape, gives rise to good folding properties. Involving other contacts deteriorates these properties.Comment: REVTeX, 9 pages, 8 EPS figure

The Crumpling Transition Revisited

The ``crumpling" transition, between rigid and crumpled surfaces, has been object of much discussion over the past years. The common lore is that such transition should be of second order. However, some lattice versions of the rigidity term on fixed connectivity surfaces seem to suggest that the transition is of higher order instead. While some models exhibit what appear to be lattice artifacts, others are really indistiguishable from models where second order transitions have been reported and yet appear to have third order transitions.Comment: Contribution to Lattice 92. 4 pages. espcrc2.sty file included. 6 figures upon request. UB-ECM-92/30 and UAB-FT-29

Use of the Metropolis algorithm to simulate the dynamics of protein chains

The Metropolis implementation of the Monte Carlo algorithm has been developed to study the equilibrium thermodynamics of many-body systems. Choosing small trial moves, the trajectories obtained applying this algorithm agree with those obtained by Langevin's dynamics. Applying this procedure to a simplified protein model, it is possible to show that setting a threshold of 1 degree on the movement of the dihedrals of the protein backbone in a single Monte Carlo step, the mean quantities associated with the off-equilibrium dynamics (e.g., energy, RMSD, etc.) are well reproduced, while the good description of higher moments requires smaller moves. An important result is that the time duration of a Monte Carlo step depends linearly on the temperature, something which should be accounted for when doing simulations at different temperatures.Comment: corrections to the text and to the figure

Disordered Heteropolymers with Crosslinks - Phase Diagram and Conformational Transitions

We study the phase behavior of random heteropolymers (RHPs) with quenched cross-links, a novel polymer class of technological and biological relevance, and show the possible occurrence of freezing with few chain conformations sampled. The sensitivity of the frozen phase microstructure to the disorder components is elucidated at positive solubility parameter values; at low T's segregated microphases form, while at a finite T, a first order conformational transition occurs, and is attributed to statistical matching of large microphases bounded by cross-links. The end of the symmetry broken regime stabilization by cross-links occurs at a higher T by a second order conformational transition. \\icrophases form, while at a finite T, a first order conformational transition occurs, and is attributed to statistical matching of large microphases bounded by cross-links. The end of the symmetry broken regime stabilization by cross-links occurs at a higher T by a second order conformational transition.Comment: 5 pages, 2 ps. figures. submitted to Chem. Phys. Let

The Relevant Scale Parameter in the High Temperature Phase of QCD

We introduce the running coupling constant of QCD in the high temperature phase, $\tilde{g}^2(T)$, through a renormalization scheme where the dimensional reduction is optimal at the one-loop level. We then calculate the relevant scale parameter, $\Lambda_T$, which characterizes the running of $\tilde{g}^2(T)$ with $T$, using the background field method in the static sector. It is found that $\Lambda_T/\Lambda_{\overline{\text{MS}}} =e^{(\gamma_E+1/22)}/(4\pi)\approx 0.148$. We further verify that the coupling $\tilde{g}^2(T)$ is also optimal for lattice perturbative calculations. Our result naturally explains why the high temperature limit of QCD sets in at temperatures as low as a few times the critical temperature. In addition, our $\Lambda_T$ agrees remarkably well with the scale parameter determined from the lattice measurement of the spatial string tension of the SU(2) gauge theory at high $T$.Comment: 13 pages, RevTeX 3.0, no figures. Full postscript version available via anonymous ftp (192.84.132.4) at ftp://risc0.ca.infn.it/pub/private/lissia/infnca-th-94-24.p

Parity breaking at high temperature and density

We investigate the question of parity breaking in three-dimensional Euclidean SU(2) gauge-Higgs theory by Monte Carlo simulations. We observe no sign of spontaneous parity breaking in the behaviour of both local and non-local gauge invariant operators. However, the presence of parity odd terms in the action can induce a phase transition to a parity odd ground state which is characterized by a Chern-Simons like condensate. The implications for various proposed scenarios of fermion number non-conservation is discussed.Comment: 20 pages, 13 figures not included, sorr

Sequence Dependence of Self-Interacting Random Chains

We study the thermodynamic behavior of the random chain model proposed by Iori, Marinari and Parisi, and how this depends on the actual sequence of interactions along the chain. The properties of randomly chosen sequences are compared to those of designed ones, obtained through a simulated annealing procedure in sequence space. We show that the transition to the folded phase takes place at a smaller strength of the quenched disorder for designed sequences. As a result, folding can be relatively fast for these sequences.Comment: 14 pages, uuencoded compressed postscript fil

Refolding upon force quench and pathways of mechanical and thermal unfolding of ubiquitin

The refolding from stretched initial conformations of ubiquitin (PDB ID: 1ubq) under the quenched force is studied using the Go model and the Langevin dynamics. It is shown that the refolding decouples the collapse and folding kinetics. The force quench refolding times scale as tau_F ~ exp(f_q*x_F/k_B*T), where f_q is the quench force and x_F = 0.96 nm is the location of the average transition state along the reaction coordinate given by the end-to-end distance. This value is close to x_F = 0.8 nm obtained from the force-clamp experiments. The mechanical and thermal unfolding pathways are studied and compared with the experimental and all-atom simulation results in detail. The sequencing of thermal unfolding was found to be markedly different from the mechanical one. It is found that fixing the N-terminus of ubiquitin changes its mechanical unfolding pathways much more drastically compared to the case when the C-end is anchored. We obtained the distance between the native state and the transition state x_UF=0.24 nm which is in reasonable agreement with the experimental data.Comment: 35 pages, 15 figures, 1 tabl