Folding free energy barrier

If the two representations are equivalent then Eqs. (3.79) and (3.80) describe how A s and B s must be transformed in terms of a s and /Ts. (These identities are performed explicitly by Sanchez and Di Marzio, [49]. Frank and Tosi [105] further show that if a s and /Ts are chosen to satisfy detailed balance conditions, that is equilibrium behaviour, then the occupation numbers of the two representations are only equivalent if the nv s are in an equilibrium distribution within each stage. This is likely to be true if there is a high fold free energy barrier at the end of each stem deposition, and thus will probably be a good representation for most polymers. In particular, the rate constant for the deposition of the first stem, A0 must contain the high fold free energy term, i.e. ... 

Figure 18 The various free energy terms involved in <()-value analysis.The free energy as a function of a reaction coordinate q is plotted for the wild-type (wt) (solid line) and a mutated protein (dashed line). Mutations can affect both the stability of the native state AGo = Gn — Gu and the height of the folding free energy barrier AGt = G+ — Gu. The relative change in these quantities AAG /AAGq upon mutation is the <()-value.

Figure 20 The method of Vendruscolo et recognizes the fact that transition state structures are difficult to observe because they are located at the top of the folding free energy barrier and are thus rarely visited in simulations. To increase the statistical weight of these structures, a biasing potential is introduced that has a minimum in the TS region and drives the system away from both native and unfolded states. As a result, the free energy profile undergoes a transformation from a bimodal shape with populated native and denatured states to a unimodal shape where mosdy TS structures are present.

The particular models used to demonstrate the theory obviously have many drawbacks as true representations of polymer crystals. These could include the lack of a fold energy, no distinction between new molecules and those already attached, neglect of chain ends, a somewhat arbitrary choice of pinning rules etc. However, they all serve their purpose in that they show that an energetic free energy barrier is not necessary to obtain the experimental curves. A truly representative growth picture can probably only be achieved via molecular dynamics. 

In a later study (DeKimpe et al., 1964) they demonstrated that it was necessary that the Al be in six-fold coordination before it would combine with Si to form kaolinite. Starting with the Al in six-fold coordination presumably decreased the free-energy barrier necessary for the formation of kaolinite. Gibbsite, Al(OH)3, was used as a starting material but was too stable to be attacked by depolymerized silica it was necessary for the gibbsite to be in the process of reorganizing ( dynamic stage) into boehmite, AIO(OH). 

In this theory it is assumed that a chain stem, one fold period long, is laid down on the lateral growth face of the crystal. This is the slowest step because the stem has only one surface face on which to sit. Once this stem is in place, however, an adjacent stem is more easily laid down (i.e., has a lower free energy barrier to cross), because it can now contact two surfaces, the crystal substrate and the side face of the first stem. The row therefore quickly fills up once the first stem is deposited. The growth rate of the crystal (primary crystallization) is thus largely determined by secondary nucleation (Figure 10-31). 

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...

By analyzing experimental data Akmal and Munoz [18] conclude that the top of the free energy barrier is reached when the protein in its search for the native state reaches a critical native density , i.e. is close enough to the native state to expel the interstitial water molecules and form a folding nucleus. At that point the stabilization energy starts to overcome the decrease of conformational entropy. In addition, the expelled interstitial solvent gains translational entropy as well [19]. 

A. Akmal, V. Munoz (2004) The nature of the free energy barriers to two-state folding. Proteins Struc. Fund. Bio. 47, pp. 142-152... 

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