Thermal stability of TC5b.

<p>A. Fraction of formed α-helix L²YIQWLK⁸ (dashed black), β-turn S¹⁴ (solid green), and polyprolyl helix P¹⁸ (dotted blue), as defined using self-consistent clustering and enumeration of their backbone dihedral angles. Note that P¹⁸ remains unchanged in its backbone conformation due to its definition in CHARMM. Individual α-helical residues have varying thermal stability, with the more N-terminal ones being less stable, consistent with the existence of α-helical fraying. B. Fraction of formed α-helical salt/secondary bridge Q⁵∶K⁸ (solid red), α-helical hydrogen bond Y³∶L⁷ (dotted red), β-turn/tertiary salt bridge D⁹∶R¹⁶ (solid blue), β-turn hydrogen bond D⁹∶S¹⁴ (dashed green), tertiary hydrophobic core W⁶∶P¹⁹ and Y³∶P¹⁹ (solid and dashed black), and secondary hydrophobic core Y³∶W⁶ (dashed red), as defined by using self-consistent clustering and enumeration of their distances. Note that the α-helical salt/secondary bridge is only partially formed at low temperature, even though the rest of the structure is nearly fully folded by other measures. Similarly, the secondary hydrophobic core Y³∶W⁶ persists even at high temperature, where the rest of the protein is largely unfolded by other measures. Importantly, substantial amount of residual native structure persists at high temperature. C. Fraction of formed mean α-helical structure (dashed black), mean β-turn structure (solid green), mean tertiary structure (solid black) in the REMS calculated ensembles, and native fraction measured experimentally using chemical shift dispersion (squares), as adapted from the first study of TC5b <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0000446#pone.0000446-Neidigh1" target="_blank">[28]</a>.</p

Equilibration and calibration of REMS simulations of TC5b in explicit water.

<p>A. Instantaneous potential energy (<i>U</i>) as a function of MD time during evolution of the 273 K replica in the canonical <i>NVT</i> ensemble prior to initiating REMS, demonstrating its equilibration, as reflected in the energetic stability during the last 50 ps. B. Instantaneous potential energy (<i>U</i>) as a function of MD time upon replica exchange from 276 to 273 K, demonstrating thermalization in less than 2 ps. C. Average potential energy 〈<i>U</i>〉 of 273 K replica as a function of energy smoothing time (<i>ts</i>). As <i>ts</i> approaches 2000 fs, the standard deviation of <<i>U></i> approaches the fluctuation of the energy distribution in that time domain. At <i>ts</i>  =  200 fs, energy-smoothed 〈<i>U</i>〉 of REMS is statistically indistinguishable from the instantaneous <i>U</i> used during conventional REM; double-sided <i>p</i> = 0.73. Bars represent ±1σ.</p

Manifold of unfolded mesostates.

<p>Mapping of the unfolded state ensemble, as calculated using the 363 K replica, onto the two top coordinates of its locally linear embedding space (open black circles), and the two top coordinates of its principal component projection (solid green circles). Principle component analysis fails to discern mesostate structure of the unfolded state ensemble, with the entire ensemble located near the origin of the PCA projection. On the other hand, displacement along the manifold from the origin of the LLE map coincides with the formation of native-like mesostates, containing: 1) α-helical/secondary salt bridge (red), 2) β-turn/tertiary salt bridge (blue), 3) α-helix and α-helical hydrophobic core, and 4) nearly native configurations with both the α-helix and the tertiary hydrophobic core.</p

Sampling and efficiency of REMS simulations of TC5b in explicit water.

<p>A. Mean probabilities 〈<i>P</i>〉 of MC exchange between adjoining replicas <i>x_n</i> and <i>x_m</i> as a function of temperature, demonstrating that usage of REMS leads to efficient replica exchange. B. Exchanges of replicas in the temperature space, tracking the initial lowest (red dashed) and highest (blue solid) containing the predominantly native and unfolded states, respectively, as they diffuse in temperature space in the course of the simulation. C. Divergence of the normalized difference (<i>Δ</i>) of fraction of formed hydrophobic core W⁶∶P¹⁹ (closed squares), hydrophobic core Y³∶W⁶ (open circles), salt bridge D⁹∶R¹⁶ (closed stars), α-helical Y3∶L7 (solid circles) and the β-turn D9∶S14 (open squares) hydrogen bonds between initial and final structures as a function of replica exchange for the 363 K replica. These measure were chosen because their non-local nature should be most sensitive to initial configuration memory effects. The total length of REMS simulation exceeds the apparent computational time constant of self-diffusion by nearly three orders of magnitude.</p

REMS calculation of approximately canonical ensembles of pure water.

<p>A. Histograms of potential energies (<i>U</i>) of different temperature replicas, demonstrating energy overlaps between adjoining temperature REMS replicas, as required for efficient MC exchange. B. Comparison of histograms of potential energies (<i>U</i>) of water ensembles at 273 K calculated using canonical MD (<i>NVT</i>) and REMS. Usage of REMS yields statistically indistinguishable mean energies and slightly increased energy fluctuations, as compared to those of canonical MD simulations, as shown by their normal fits (solid curves). C. Heat capacities at constant volume (<i>C_V</i>) of pure water at different temperatures, as obtained experimentally (solid squares), and calculated using canonical MD (dotted diamonds) and REMS (dashed circles). Usage of REMS with extremely long smoothing time of 600 fs (REMS^*, solid stars) leads to a significant underestimation of the heat capacity of water at low temperature. Sizes of symbols represent ±1σ.</p

Folding reaction manifold.

<p>Mapping of TC5b's folding ensemble at the midpoint of its thermal transition, as calculated using the 310 K replica, onto the top three coordinates of its LLE manifold. Displacement along the <i>M₁</i> coordinate of the manifold coincides with the transformation of the 5) nearly native and 6) partially unfolded mesostates that lack the tertiary hydrophobic core and the native β-turn, but retain a frayed α-helix and the tertiary salt bridge. Displacement along the <i>M₂</i> coordinate coincides in part with the transformation of the α-helix from mesostate 7) that possesses a near native β-turn and hydrophobic cores and a non-α-helical but compact N-terminus, and mesostate 8) that lacks the native hydrophobic cores and has a non-native β-turn centered at K⁸ that is part of the N-terminal α-helix in the NMR structure. Displacement along the <i>M₃</i> coordinate coincides with the transformation of the β-turn, including mesostates 9) that have a near native β-turn and tertiary salt bridge but have an unfolded α-helix and hydrophobic cores, and 10) possess a near native topology and α-helix but lack a native β-turn.</p

Folding cooperativity in the unfolded state ensemble of TC5b

<p>Conditional probabilities of forming pairs of native interactions, as listed, with <i>P_pair</i> = <i>P</i> (<i>i</i>+<i>j</i>|<i>i</i> ; <i>j</i>), the probability of forming both interactions <i>i</i> and <i>j</i> under the condition that either <i>i</i> or <i>j</i> is formed. The overall probabilities of forming individual interactions <i>i</i> and <i>j</i> are defined by <i>P_i</i> and <i>P_j</i>, respectively, and the product <i>P_i</i><i>P_j</i> expresses the probability of forming both interactions <i>i</i> and <i>j</i> in the absence of any cooperativity between them. This cooperativity is expressed by the ratio <i>P_pair</i>/<i>P_i</i><i>P_j</i>.</p