Free energy landscape parameters

Fig. 6. Free energy landscape in which during a transition from A to B both the variable q and q change systematically left hand side). For this case, the variable q is a good order parameter and suffices to distinguish the equilibrium fluctuations in the stable states A and B. By following g as a function of time t we can detect transitions between the stables states right hand. side). However, this variable does not capture all essential aspects of the transition and is therefore not a good reaction coordinate. In the definition of such of a good reaction coordinate capable of describing the complete course of the reaction the variable q can not be neglected...

Fig. 5. Free energy landscape of a lattice model protein (see Sect. 2.2), as a function of two order parameters, the number of contacts C and the number of native contacts Qo (see Sect. 2.3). Unlike the energy landscape funnel picture, the free energy shows two stable states separated by a barrier (the transition state). Extended unfolded conformers quickly collapse to the molten globule, and have to overcome a barrier to folding to the native state. The funnel picture is thus reconciled with the two-state concept of a free energy barrier. Reprinted from Dinner et ah. Trends Biochem. Sci. 25, 331, (2000) with permission from Elsevier...

Fig. 7. Generic free energy landscape illustrating the hysteresis problem. If q is the only order parameter used to describe the reaction, but other parameters such as q are also important slow variables, hysteresis might occur. Solid curve starting from the initial state A, the ensemble is slowly biased along q, until the transition to B suddenly occurs. Dashed curve When the simulation starts in B and q is slowly reduced a different part of phase space is sampled, leading to hysteresis...

Fig. 11. Representations of the folding event in 3 different order parameter planes. The free energy landscape from replica exchange is given by thin solid contour lines separated by 0.2 ksT. A few smoothed paths in the F-H ensemble are denoted by a scatter plot [small dots). Each dot represents a time slice along a path. Also given are the different committor ensembles pB < 0.2 light gray, 0.4 < pB < 0.6 in dark gray and pB > 0.9 in black. The apparent transition state saddle points in the FE landscape are indicated by arrows...

This function represents for given system parameters T, V, and AT the free-energy landscape in dependence of the components of the vector of relevant degrees of freedom Q. Minima in this landscape correspond to locally stable (metastable) equilibrium system states. Peaks in this landscape represent free-energy barriers. A structural transition requires the system to circumvent the barrier or to overcome it by a fluctuation with thermal energy that exceeds the barrier height. 

Another, less costly, unique alternative is the introduction of an angular overlap order parameter. Before we define it, let us first discuss the characteristic features of free-energy landscape parametrized by angular degrees of freedom. 

Multicanonical histograms Wmuca (f/ 0) of energy f and angular overlap parameter Q and free-energy landscapes F Q) at different temperatures for the three seguences (see Table 8.1) (a) 20.1, (b) 20.4, and (c) 20.3. The reference folds reside at (3 = landf = fmin. Pseudophases are symbolized by D (denatured states), N (native folds),... 

The two-state behavior is confirmed by analyzing the temperature dependence of the minima in the free-energy landscape. The free energy as a function of the order parameter Q at fixed temperature can be suitably defined as ... 

The classification of the heteropolymer with sequence 20.6 as a two-state folder arises from the analysis of the free-energy landscape. We assume that gr is a suitable parameter that describes the macrostate of the system adequately. Considering this parameter as a constraint, we can formally average out the conformational degrees of freedom, and the probability for a conformation in a macrostate with contact parameter q reads... 

The map of all possible free-energy minima in the range of external parameters T [0,10] andi 6 [—2,10] is shown in Fig, 14,2 for the peptide in the vicinity of a substrate that is equally attractive for both hydrophobic and polar monomers. Solid lines visualize paths through the free-energy landscape when changing temperature under constant solvent (i = const) conditions. Let us follow the exemplified trajectory for 5 = 2.5. 

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