out that the primary chemical structure of proteins decided their native tertiary
structure with minimum free energy (Anfinsen 1973 ). Protein molecules in the
random coil state contain very large conformational entropy, while at ambient
conditions their native states get quite little folding energy, normally the free energy
for a few of hydrogen bonds. How protein folding can find the sole native state from
all the possible random conformation states by overcoming large entropy barriers
has been a challenge question. The answer to this question is of essential impor-
tance for us to control the physic-chemical process of the living macromolecules.
To this end, Levinthal suggested a paradox: proteins should not have enough time to
search over all the possible kinetic paths during their folding process, and there
must be some fast paths (Levinthal 1968 ). Kauzmann proposed that the hydropho-
bic interactions of amino acid segments are the dominant thermodynamic driving
forces for fast folding of proteins (Kauzmann 1959 ). The hydrophobic residues tend
to assemble themselves in the inner core of native proteins. Even though the
sequences of proteins with similar functions could be different in various species,
their hydrophobic cores are similar. The hydrophobic interactions benefiting the
stacking of alkali-group pairs are also important to maintain the stability of DNA
double helix (the hydrogen bonding in each base pair is comparable with the
hydrogen bonding between the base groups and water molecules; therefore, its
free energy contributions to the stability is negligible.). In addition, the hydrophobic
interactions are crucial for the recognition between the antigen and its counterpart.
We can roughly separate the various interactions in the units of protein
molecules into two parts, according to their corresponding roles in phase
transitions: the first part plays a role analogous to the mixing interactionB, which
drives the hydrophobic collapse transition of the single chain, corresponding to the
liquid-liquid phase separation; and the second part plays a role analogous to the
parallel-packing interactionEp, which drives chain packing in the beta-sheet,
corresponding to polymer crystallization. Thus, one can recognize that the
corresponding contributions of these two interaction parameters control the inter-
play of phase transitions in the prototypical single-chain system.
By employing the so-called biased sampling algorithm and the parallel temper-
ing method in dynamic Monte Carlo molecular simulations of single lattice chain,
we can compute the free energy change of a homopolymer chain during the process
of crystalline ordering (Hu et al.2003b). Figure11.9ademonstrates free energy
curves at the equilibrium melting points with varying strengths of mixing
interactions. One can see that the height of free energy barriers for crystallization
changes with the solvent quality, reflecting the relative difficulty in crystal nucle-
ation. Figure11.9bsummarizes the height of free energy barriers at the equilibrium
melting points, together with the simulated phase diagrams for collapse transition
and crystallization of the single-chain systems. One can clearly see that near the
triple point, the prior collapse transition can effectively decrease the free energy
barrier for intramolecular crystal nucleation, resulting in a significant acceleration
to the self-assembly of the single chain with chain folding (Hu and Frenkel 2006 ).
11.5 Accelerated Crystal Nucleation in the Single-Chain Systems 233