Lattice Monte Carlo Simulation Of Protein Folding

Theoretical and simulation research for protein folding is an interdisciplinary field between molecular biology, polymer science and statistical physics. In the light of polymer physics, protein folding can be viewed as the process that a special type of polymer chain proceeds from random coil state to compact and ordered globular state. The folding process is complicate, accompanied with large change of entropy, thus far away from a unified theoretical treatment; on the other hand, it happens usually in a nano-meter spatial scale and nano-to-micro-second temporal scale, thus resulting in much difficulty of direct experimental detection. As an alternate, computer simulation has been an important and useful method in this field.Compared with all-atom model in continuous space, the coarse-grained lattice model is, at the cost of spatial resolution, beneficial for CPU time. Lattice model is specifically suitable for Monte Carlo simulation.α-helix is a type of fundamental secondary structure, holding the largest content in proteins, α-helix constitutes the major subject of the present Ph. D. thesis. The thesis developed corresponding lattice model, and reproduced the helix-coil transition via dynamic Monte Carlo (DMC) simulation. We have also expanded the statistic order parameter for helix structure, and revisited the classical Zimm-Bragg theory for helix formation.The main achievements and original contributions of this thesis are summarized as follows:1. A coarse-grained model has been constructed for generating α-helix in the simple cubic lattice space. Though lattice model have been widely employed in polymer research and protein's globule-coil transition, it is difficult for embedding α-helix into simple cubic lattice space because of the requirement of spatial packing and warping. In this thesis, a single-unit lattice model was suggested, in which one amino acid residue is dealt as the basic unit for movement and interaction. This model combines the eight-site bond fluctuation model originally used in polymer simulationand the interactions for polypeptide and α-helix. Thus, the helix-coil transition has been reproduced in our DMC, and the regular α-helix with period of integer 4. In comparison with traditional one-site cubic lattice space, the eight-site cubic lattice space allows much more bond lengths and bond orientations, and also the branching point, thus can be considered as a quasi-continuous space, but still holds the advantage of effective computing for lattice model.. What's more, the simplified chirality and hydrogen bonding interaction ensure the corresponding single-unit model simple but effective.2. The single-unit model has been improved in the simple cubic lattice space, resulting in α-helix structure with non-integer period; and the four-unit lattice model has also been constructed. Based on the single-unit lattice model, the inclusion of virtual-imino group and virtual-carboxyl group makes an improved single-unit model, thus an α-helix with period of about 3.6 can be embedded into the simple cubic lattice space. As a further step, the explicit representation of α-carbon, imino group, carboxyl group and side-chain group as basic unit for movement and interaction makes a four-unit lattice model with intermediate resolution, which holds the highest resolution in simple cubic lattice space for polypeptide at present. The four-unit model includes conformational entropy into residue and packing effect for the unit, and makes a good compromise between computation consumption and spatial resolution. The formed α-helix is more "realistic", with imino group and carboxyl group inside and side-chain group outside. Around the transition point, the thermal fluctuation of the distance between imino group and carboxyl group is the biggest, indicating the leading role of hydrogen bond for helix formation; after the formation, the thermal fluctuation of the side-chain group's distance is the biggest, indicating the probability for performing biological function; the thermal fluctuation of the α-carbon's distance is the smallest in the whole process, indicating it's role as polypeptide's backbone.3. A spatial orientational correlation function for helix structure and the related conception of persistent length have been suggested. This function can quantificationally describe the period and correlation length of helix. The correlation length is independent on helix length, much like the persistent length in polymer science, thus essentially describing the regularity of helix structure. Most importantly, this function works not only for irregular helix structure, but also for the polypeptidechain containing more than one helix segment.4. The classical Zimm-Bragg (ZB) theory for helix-coil transition has been revisited via DMC simulation and theoretical deduction, thus resulting in a few medications and one new formula. DMC simulations based on the single-unit model in three-dimensional lattice space indicated that the ZB theory captured the main physics of helix-coil transition though it is essentially a one-dimensional Ising model. On the other hand, the traditional large-N approximation for the quantificational treatment of the ZB theory was found not suitable for the polypeptide with short or medium length; as the helical segment in natural protein is largely of medium length, another large-λ approximation and associated simple formula has been suggested by us. A multi-residue-nucleus or block-nucleus assumption has been put forward, which could describe helix nucleation process more precisely than the single-residue-nucleus assumption in the original ZB theory. A detailed comparison has been made between different method for nucleation constant and propagation constant, which indicated that, if both helical ratio and mean length of helix were set as input, the output would be better than the case with only helical ratio as input.5. The role of non-native hydrogen bonding interaction in the process of α-helix formation has been explored. Though the non-native hydrogen bonds (i.e., sequence interval is larger than four) can not be found in the natural α-helix structure, they hold a big ratio in all hydrogen bonds during α-helix formation. The inclusion of non-native hydrogen bond results in the formation of intermediate-like conformation in the helix-coil transition, thus another arrangement of conformation for the formation of α-helix. The non-native hydrogen bond also complicates the relationship between propagation constant for helix and temperature, changes the exponential property especially around the transition temperature. Thus, all those "abnormal" phenomena may provide a possible explanation for the latest experimental result about the multi- or stretched- exponential kinetics.