Notes: Using the scanning simulation method we study the tricritical behavior at the Flory θ-point of self-avoiding walks (SAWs) with nearest neighbors attractions ε (ε〈0) on a simple cubic lattice (in the following paper we investigate tricritical trails on the same lattice). The tricritical temperature Tt is −ε/kBTt=0.274±0.006 (one standard deviation). The results for the radius of gyration G and the end-to-end distance R are consistent with the theoretical prediction νt=0.5 and with a logarithmic correction to scaling; the ratio G2/R2 =0.1659±0.0001 (calculated without taking into account correction to scaling) is only slightly smaller than the theoretical asymptotic value 1/6=0.1666.... The results for the partition function Z at Tt lead to γt=1.005±0.017 in accord with theory and to μt=5.058±0.014, where μt is the growth parameter; the correction to scaling in Z is found to be relatively small. For the chain length studied the divergence of the specific heat at Tt (αt(approximately-equal-to)0.3) is significantly larger than that predicted by theory, (ln N)3/11 (i.e., αt=0). Also, at Tt our data are affected by strong correction to scaling and are thus not consistent with the theoretical value of the crossover exponent φt=0.5.

Notes: A self-attracting trail is a walk on a lattice which may intersect itself but two bonds are not allowed to overlap; an interaction energy ε (ε〈0) is associated with each self-intersection. Using the scanning simulation method, we study the tricritical behavior at the collapse transition of self-attracting trails of N≤250 steps on a simple cubic lattice. In the preceding paper (paper I) tricritical self-avoiding walks (SAWs) on the same lattice have been investigated. The tricritical temperature of trails is −ε/kBTt=0.550±0.004 (one standard deviation). The results for the radius of gyration, G, and the end-to-end distance, R, lead to νt=0.515±0.003, which is larger than νt=1/2, the theoretical prediction for SAWs. The ratio G2/R2=0.1676±0.0001 is slightly larger than 1/6=0.1666 ... predicted by theory for SAWs; The results for the partition function at Kt lead to γt=1.040±0.005 (as compared to the theoretical prediction for SAWs γt=1) and to the growth parameter value μt=5.0023±0.0020. The crossover exponent, φt, is approximately 0.5 as expected for SAWs at tricriticality; this value is significantly smaller than that found for SAWs in paper I. The results of G, R, and Z at Kt are found to be inconsistent with logarithmic corrections to scaling. However, we do not think that the above differences between trails and SAWs are sufficient to suggest unequivocally that the two models belong to different universality classes.

Notes: The double scanning method (DSM) is a computer simulation technique suggested recently by Meirovitch [J. Chem. Phys. 89, 2514 (1988)]. This method is a variant of the usual or "single'' scanning method (SSM) of the same author, which was extended by us to polypeptides [Biopolymers 27, 1189 (1988); this paper is designated here as paper II]. The two methods are step-by-step construction procedures from which the entropy and the free energy can be estimated. The transition probabilities are obtained by scanning the so-called "future'' chains, which are continuations of the chain in future steps up to a maximum of b steps. With the SSM, the process is carried out by exact enumeration of the future chains; this is time consuming, and therefore b is limited to small values. With the DSM, on the other hand, only a relatively small sample of the future chains is generated by applying an additional scanning procedure. This enables one to increase b at the expense of approximating the transition probabilities. Increasing of b, however, is important in order to treat medium- and long-range interactions more properly. In this paper (as in our paper II), we apply the DSM to a model of decaglycine without solvent, described by the potential energy function ECEPP at 100 and 300 K. Using the SSM with the maximal value, b=4, we found in paper II that, at 100 K, the α helix rather than the statistical coil is the most stable state. The present DSM simulation at T=100 K (based on b=5) is more efficient than the SSM, and a structure with significantly lower energy than that of the α helix is found. It is argued that b can be increased further to 7 at this temperature. At 300 K the DSM, like the SSM, shows that the statistical coil is the most stable state of decaglycine. However, the DSM is found to be less efficient than the SSM. It is argued, however, that the DSM is expected to be advantageous (even at 300 K) to simulate more complex polypeptides that are stable in small regions of phase space (such as the α-helical state). Finally, it should be pointed out that the present method can be employed to treat a wide range of macromolecular models, such as those for synthetic polymers and nucleic acids.

Notes: As in the preceding paper (Paper I) we study here a model of chains with excluded volume enclosed in a "box'' on a square lattice. The system is simulated by the Metropolis Monte Carlo method and the entropy is extracted from the samples by using the "hypothetical scanning method.'' With this method each system configuration is treated as if it has been generated step by step with the scanning method (studied in Paper I). The transition probabilities are reconstructed and three approximations of the entropy are obtained. Thus the pressure and the chemical potential are calculated directly from the results of the entropy as in Paper I using standard thermodynamic relations. These results are found to be in a very good agreement with those obtained in Paper I, which are considered to be exact within the statistical error.

Notes: Using the scanning simulation method we study a system of many chains with excluded volume contained in a "box'' on a square lattice. With this method an initially empty box is filled with the chains monomers step by step with the help of transition probabilities. The probability of construction, P of the whole system is the product of the transition probabilities used and hence the entropy S is known, (S∼ln P). Thus the pressure and the chemical potential can be calculated with high accuracy directly from the entropy using standard thermodynamic relations. In principle, all these quantities can be obtained from a single sample without the need to carry out any thermodynamic integration. Various alternatives for performing the scanning construction are discussed and their efficiency is examined. This is important due to the fact that for lattice polymer models the scanning method is ergodic (unlike some dynamical Monte Carlo techniques). The computer simulation results are compared to the approximate theories of Flory, Huggins, Miller, and Guggenheim and to the recent improved theories of Freed and co-workers.

Notes: The Helmholtz free energy F (rather than the energy) is the correct criterion for stability; therefore, calculation of F is important for peptides and proteins that can populate a large number of metastable states. The local states (LS) method proposed by H. Meirovitch [(1977) Chemical Physics Letters, Vol. 45, p. 389] enables one to obtain upper and lower bounds of the conformational free energy, FB (b, l) and FA (b, l), respectively, from molecular dynamics (MD) or Monte Carlo samples. The correlation parameter b is the number of consecutive dihedral or valence angles along the chain that are taken into account explicitly. The continuum angles are approximated by a discretization parameter l; the larger are b and l, the better the approximations; while FA can be estimated efficiently, it is more difficult to estimate FB. The method is further developed here by applying it to MD trajectories of a relatively large molecule (188 atoms), the potent “Asp4-Dpr10” antagonist [cyclo(4/10)-(Ac-Δ3Pro1-D-pFPhe2-D-Trp3-Asp4-Tyr-5-D-Nal6-Leu7-Arg8-Pro9-Dpr10-NH2)] of gonadotropin releasing hormone (GnRH). The molecule was simulated in vacuo at T = 300 K in two conformational states, previously investigated [J. Rizo et al. Journal of the American Chemical Society, (1992) Vol. 114, p. 2860], which differ by the orientation of the N-terminal tail, above (tail up, TU) and below (tail down. TD) the cyclic heptapeptide ring. As in previous applications of the LS method, we have found the following: (1) While FA is a crude approximation for the correct F, results for the difference, ΔFA = FA(TD) - FA(TU) converge rapidly to 5.6(1) kcal/mole as the approximation is improved (i.e., as b and l are increased), which suggests that this is the correct value for ΔF; therefore TD is more stable than TU. (The corresponding difference in entrophy. TΔSA = 1.3(2) kcal/mole, is equal to the value obtained by the harmonic approximation.) (2) The lowest approximation, which has the minimal number of local states, i.e., based on b = 0 (no correlations) and l = 1 (the angle values are distributed homogeneously), also leads to the correct value of ΔF, within the error bars. This is important since the lowest approximation can be applied even to large proteins. (3) The method enables one to define the entropy of a part of the molecule and thus to measure the flexibility of this part. We have verified that the results for T[SA(TU) - SA(TD)] of the tail alone converged to 2.4(1) kcal/mole, which demonstrates the relatively high flexibility of the tail in the TU state. In order to study the random coil state, the Asp4-Dpr10 analogue and its linear version were simulated by MU at 1000 K. We have been able to calculate a lower bound, ∼ 25 kcal/mole for T[S(linear) - S(cyclic)], which is the reduction in the conformational entropy caused by the ring closure. © 1994 John Wiley & Sons, Inc.