It is important to know the rate of intra-molecular contact formation in proteins in order to understand how proteins fold clearly. Here we investigate the rate of intra-molecular contact formation in short two-dimensional compact polymer chains by calculating the probability distribution p（r） of end-to-end distance r using the enumeration calculation method and HP model on two-dimensional square lattice. The probability distribution of end-to-end distance p（r） of short two-dimensional compact polymers chains may consist of two parts, i.e. p（r） = p1（r） ＋ p2（r）, where p1（r） and p2（r） are different for small r. The rate of contact formation decreases monotonically with the number of bonds N, and the rate approximately conforms to the scaling relation of k（N）∝ N^-α. Here the value of α increases with the contact radius a and it also depends on the percentage of H （hydrophobic） residues in the sequences of compact chains and the energy parameters of ^εНН、 ^εНН and ^εpp. Some comparisons of theoretical predictions with experimental results are also made. This investigation may help us to understand the protein folding.
The conformational properties and elastic behaviors of protein-like single chains in the process of tensileelongation were investigated by means of Monte Carlo method.The sequences of protein-like single chains contain two typesof residues:hydrophobic(H)and hydrophilic(P).The average conformations and thermodynamics statistical properties ofprotein-like single chains with various elongation ratio λ were calculated.It was found that the mean-square end-to-enddistance(?).increases with elongation ratio λ.The tensor eigenvalues ratio of:decreases with elongationratio λ for short(HP)_x protein-like polymers,however,the ratio of:increases with elongation ratio λ,especially for long (H)_x sequence.Average energy per bond increases with elongation ratio λ,especially for (H)_xprotein-like single chains.Helmholtz free energy per bond also increases with elongation ratio λ.Elastic force(f),energycontribution to force(f_U)and entropy contribution to force(fs)for different protein-like single chains were also calculated.These investigations may provide some insights into elastic behaviors of proteins.
In protein molecules each residue has a different ability to form contacts.In this paper,we calculated the number of contacts per residue and investigated the distribution of residue-residue contacts from 495 globular protein molecules using Contacts of Structural Units(CSU)software.It was found that the probability P(n)of amino acid residues having n pairs of contacts in all contacts fits Gaussian distribution very well.The distribution function of residue-residue contacts can be expressed as:P(n)=P_0+aexp[-b(n-n_c)~2].In our calculation,P_0=-0.06,α=11.4,b=-0.04 and n_c=9.0.According to distribution function,we found that those hydrophobic(H)residues including Leu,Val,Ile,Met,Phe,Tyr,Cys,and Trp residues have large values of the most probable number of contact n_c,and hydrophilic(P)residues including Ala,Gly,Thr, His,Glu,Gln,Asp,Asn,Lys,Ser,Arg,and Pro residues have the small ones.We also compare with Fauchere-Pliska hydrophobicity scale(FPH)and the most probable number of contact n_c for 20 amino acid residues,and find that there exists a linear relationship between Fauchere-Pliska hydrophobicity scale(FPH)and the most probable number of contact n_c, and it is expressed as:n_c=a+b×FPH,here α=8.87,and b=1.15.It is important to further explain protein folding and its stability from residue-residue contacts.
Short two-dimensional compact chains adsorbed on the attractive surface at different temperatures were investigated by using the enumeration calculation method. First we investigate the chain size and shape of adsorbed chains, such as characteristic ratios of mean-square radii of gyration 〈S^2〉x/N and 〈S^2〉y/N, shape factor 〈δ〉, and the orientation of chain bonds 〈cos^2 θ〉 to illuminate how the size and shape of adsorbed compact chains change with increasing temperatures. There are some special behaviors for the chain size and shape at low temperature, especially for strong attraction interaction. In the meantime, adsorbed compact chains have different behaviors from general adsorbed polymer chains. Some thermodynamics properties are also discussed here. Heat capacity changes non-monotonously, first increases and then reduces. The transition temperature Tc is nearly 1.0, 1.4, 2.0 and 4.2 （in the unit of To） for the case of ε = 0, -1, -2 and -4 （in the unit of kTo）, respectively. Average energy per bond increases while average Helmholtz free energy per bond decreases with increasing temperatures. From these two thermodynamics parameters we can also get another transition temperature Tc＇, and it is close to 0.7, 1.1, 1.5 and 3.4 for ε= 0, -1, -2, and -4, respectively. Therefore, Tc is greater than Tc＇ under the same condition. These investigations may provide some insights into the thermodynamics behaviors of adsorbed protein-like chains.
