The behavior of the collapse transition of a single chain is also related to the
chain length. Figure4.9ademonstrates the collapse transition of single chains in
dynamic Monte Carlo simulations (Hu 1998 ). The figure shows the curves of mean-
square radius of gyration<s^2 >/nas a function of the thermodynamic condition
B/kT(Bis the mixing energy parameter) with various chain lengths. One can see
that the theta point locates right at the onset of the collapse transition for all the
chain lengths. With the increase of the chain length, the collapse transitions appear
sharper, approaching a first-order phase transition. Chi Wu and coworkers reported
the experimental observation on the molten-globule state of single chain during the
collapse transition (Wu and Zhou 1996 ). If we observe the internal structure of
single coil during this process, the radial distribution of the local-average
concentrationsof chain units will reveal a core-shell structure, as shown
in Fig.4.9b(Hu 1998 ). In the range ofB/kTaround 0.05, there exists an intermedi-
ate state with two-phase coexistence. The core contains an enhanced density and the
shell behaves like an expanded coil. This implies that there exists a surface pre-
dissolution on the single collapsed coil. Partial release of the conformation entropy
at the surface will benefit the stability of the interface between the condensed chain
units and the solvent. Apparently, the shorter is the chain, the smaller is the core.
Therefore, the surface pre-dissolution of a shorter chain will be more significant,
and hence its collapse transition becomes less dramatic. The interesting point is
that, along with their folding into the native state, many protein molecules exhibit
such a molten-globule state driven by their hydrophobic interactions. The core-shell
structures are similar and appear as a key state on the fast path of protein folding, as
further introduced in Sect.11.5.
At the theta state, the contributions of the intra-molecular attraction and repul-
sion to the coil size compensate with each other. Flory first gave a thermodynamic
treatment to the theta state (Flory 1953 ). He assumed that the solvent molar mass
wasN 1 , and the excess chemical potentialDm 1 e∂DFme/∂N 1 contained two parts
of contributions, i.e., enthalpy and entropy
Fig. 4.9 Dynamic Monte Carlo simulation results of single-chain collapse transition. (a) The
curves of mean square radius of gyration<s^2 >/(N1) vs.B/kTfor varying chain lengthsNas
labeled. (0.032, 0.26) is the theta point. (b) Radial distributions of local-average concentrations
of chain units in 512-mer at various temperatures (Hu 1998 ) (Reprinted with permission)
56 4 Scaling Analysis of Real-Chain Conformations