A rapid and efficient algorithm for packing polypeptide chains by energy minimization

An improved algorithm for packing polypeptide chains with fixed geometry, which converges to a local energy minimum rapidly and efficiently, is described. The speed of convergence of the new algorithm is comparable to that of existing algorithms for minimizing the energies of single polypeptide chains, and it is several times greater than the speed of convergence of previous algorithms for minimizing the energy of structures consisting of several polypeptide chains. The algorithm has been used to minimize the energy of three-stranded (L -Ala)₈ β-sheets, three-stranded (L -Val)₆ β-sheets, and five-stranded (L -Ile)₆ β-sheets, starting from regular structures found previously; of the three-stranded regular and truncated (Gly-L -Pro-L -Pro)₄ structures used in earlier work to model collagen; and of the stacked β-sheet (L -Ala-GLy)₆ structures used to model silk. The antiparallel L -Ala β-sheet, and Gly-Pro-Pro triple helices, and the silk II structure remained nearly regular after energy minimization, but by contrast with results from earlier computations the other structures became significantly irregular. © 1994 by John Wiley & Sons, Inc.     

An adaptive immune optimization algorithm for energy minimization problems

Based on the immune theory of biology, a novel evolutionary algorithm, adaptive immune optimization algorithm (AIOA), is proposed. In AIOA, density regulation and immune selection is adopted to control the individual diversity and the convergence adaptively. By an application of the algorithm to the optimization of test functions, it is shown that the algorithm is a highly efficient optimization method compared with other stochastic optimization methods. The algorithm was also applied to the optimization of Lennard-Jones clusters, and the results show that the method can find the optimal structure of N相似文献   

The ellipsoid algorithm as a method for the determination of polypeptide conformations from experimental distance constraints and energy minimization

A new method for constrained nonlinear optimization known as the ellipsoid algorithm is evaluated as a means of determining and refining the conformations of peptides. Advantages of the ellipsoid algorithm over conventional optimization methods include that it avoids many local minima that other methods would be trapped by, and that it is sometimes able to find optimum solutions in which the constraints are satisfied exactly. The dihedral angles about single bonds were used as variables to keep the dimensionality low (the rate of convergence decreases rapidly with increasing dimensionality of the problem). The method is evaluated on problems involving distance constraints, and for minimization of conformational energy functions. In an initial application, conformations consistent with an experimental set of NMR distance constraints were obtained in a problem involving 48 variable dihedral angles.     

Global energy minimization by rotational energy embedding

Given a sufficiently good empirical potential function for the internal energy of molecules, prediction of the preferred conformations is nearly impossible for large molecules because of the enormous number of local energy minima. Energy embedding has been a promising method for locating extremely good local minima, if not always the global minimum. The algorithm starts by locating a very good local minimum when the molecule is in a high-dimensional Euclidean space, and then it gradually projects down to three dimensions while allowing the molecule to relax its energy throughout the process. Now we present a variation on the method, called rotational energy embedding, where the descent into three dimensions is carried out by a sequence of internal rotations that are the multidimensional generalization of varying torsion angles in three dimensions. The new method avoids certain kinds of difficulties experienced by ordinary energy embedding and enables us to locate conformations very near the native for avian pancreatic polypeptide and apamin, given only their amino acid sequences and a suitable potential function.     

An efficient newton-like method for molecular mechanics energy minimization of large molecules

Techniques from numerical analysis and crystallographic refinement have been combined to produce a variant of the Truncated Newton nonlinear optimization procedure. The new algorithm shows particular promise for potential energy minimization of large molecular systems. Usual implementations of Newton's method require storage space proportional to the number of atoms squared (i.e., O(N²)) and computer time of O(N³). Our suggested implementation of the Truncated Newton technique requires storage of less than O(N^1.5) and CPU time of less than O(N²) for structures containing several hundred to a few thousand atoms. The algorithm exhibits quadratic convergence near the minimum and is also very tolerant of poor initial structures. A comparison with existing optimization procedures is detailed for cyclohexane, arachidonic acid, and the small protein crambin. In particular, a structure for crambin (662 atoms) has been refined to an RMS gradient of 3.6 × 10^?6 kcal/mol/Å per atom on the MM2 potential energy surface. Several suggestions are made which may lead to further improvement of the new method.     

Gradient SHAKE: An improved method for constrained energy minimization in macromolecular simulations

Current macromolecular energy minimization algorithms become inefficient and prone to failure when bond length constraints are imposed. They are required to relieve steric stresses in biomolecules prior to a molecular dynamics simulation. Unfortunately, the latter often require constraints, leading to difficulties in initiating trajectories from unconstrained energy minima. This difficulty was overcome by requiring that the components of the energy gradient vanish along the constrained bonds. The modified energy minimization algorithm converges to a lower energy in a fewer number of iterations and is more robust than current implementations. The method has been successfully applied to the Dickerson DNA dodecamer, CGCGAATTCGCG. © 1995 John Wiley & Sons, Inc.     

Predicting RNA secondary structure by free energy minimization

RNA structure is hierarchical. Secondary structure contacts, i.e. the canonical base pair contacts, are generally stronger and form faster than the tertiary structure. Therefore, RNA secondary structures can be predicted independently of tertiary structure prediction. Furthermore, the stability of a given RNA secondary structure can be quantified using nearest neighbor free energy parameters. These parameters are the basis of a number of free energy minimization algorithms that predict RNA secondary structure for either a single sequence or multiple sequences. This article reviews the progress of RNA secondary structure prediction by free energy minimization and describes many of the algorithms that have been developed.     

Revised algorithms for the build-up procedure for predicting protein conformations by energy minimization

The build-up procedure for predicting low-energy conformations of polypeptides has been extended to cover the case of peptides in aqueous solutions. The revised procedure consists of five steps to be applied to each stage of the build-up. I. All low-energy minima of each of the two fragments to be joined are combined as starting points for energy minimization of the enlarged fragment, and those minima of the enlarged fragment within a certain upper bound of the lowest energy are retained. II. Whenever one of the combinations in Step I leads to an atomic overlap, the minimization is started again using a pseudoenergy function which remains finite everywhere and becomes equal to the standard energy function when no atoms overlap. III. The minima generated in Steps I and II are culled by ignoring side-chain conformations and retaining only those minima whose backbone conformations differ significantly. IV. The rotameric states of the side chains are optimized, by testing their energy of interaction with the rest of the molecule, and subjecting the whole molecule to a further round of energy minimization if the test indicates that this would reduce the energy. V. The energies of all minima are recomputed with inclusion of a term for solvation and with a smaller upper bound as the criterion for retention. The original build-up procedure consisted of Steps I and III only. Examples are presented showing the effectiveness of the new Steps II and IV in locating low-energy minima, and the problems that remain to be solved, chiefly concerning Step V, are discussed.     

Empirical energy functions for energy minimization and dynamics of nucleic acids

An improved empirical energy function for energy minimization and dynamics calculations of nucleic acids is developed and evaluated by an examination of its representation of both static and dynamic properties of model systems. Among the properties studied and used for parameter optimization are base pairing interactions, sugar and phosphate energy surfaces, small crystal heats of sublimation, base, phosphate and sugar analogue vibration spectra, and the overall behavior of a DNA hexamer duplex in vacuum molecular dynamics simulations. The results obtained are compared with those from two other energy functions that have been used recently for nucleic acids. Parameters for two energy functions are given; one includes heavy atoms and only polar hydrogens and the other includes all atoms.     

A method for constrained energy minimization of macromolecules

Two algorithms for the local energy minimization of the structure of macromolecules in the presence of constraints are proposed. They are a combination of the method of steepest descents and the method of conjugate gradients with the procedure SHAKE, by which distance constraints can be satisfied. The two algorithms are tested by applying them to a small protein, the bovine pancreatic trypsin inhibitor (BPTI), and compared with the penalty function method for conserving constraints. The efficiency of the proposed methods depends on the level of interdependence of the constraints. For bond-length constraints, the use of SHAKE is superior to the penalty function method. However, when bond-angle constraints are included, SHAKE is more efficient only if the curvature of the penalty function is considerably greater than that of the potential function being minimized. The results indicate that with bond-length constraints the minimization behavior is similar to that without constraints. However, the simultaneous application of bond-length and bond-angle constraints appears to confine the molecule to a very limited part of configuration space, very different from the part covered by an unconstrained minimization. This conclusion calls into question energy minimizations of protein systems in which only the dihedral angles are allowed to vary.     

Optimization of quantum Monte Carlo wave functions by energy minimization

We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.     

A powerful truncated Newton method for potential energy minimization

With advances in computer architecture and software, Newton methods are becoming not only feasible for large-scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second-derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.     

B-spline method for energy minimization in grid-based molecular mechanics calculations

A method is described for molecular mechanics calculations based on a cubic B-spline approximation of the potential energy. This method is useful when parts of the system are allowed to remain fixed in position so that a potential energy grid can be precalculated and used to approximate the interaction energy between parts of a molecule or between molecules. We adapted and modified the conventional B-spline method to provide an approximation of the Empirical Conformational Energy Program for Peptides (ECEPP) potential energy function. The advantage of the B-spline method over simpler approximations is that the resulting B-spline function is C2 continuous, which allows minimization of the potential energy by any local minimization algorithm. The standard B-spline method provides a good approximation of the electrostatic energy; but in order to reproduce the Lennard–Jones and hydrogen-bonding functional forms accurately, it was necessary to modify the standard B-spline method. This modification of the B-spline method can also be used to improve the accuracy of trilinear interpolation for simulations that do not require continuous derivatives. As an example, we apply the B-spline method to rigid-body docking energy calculations using the ECEPP potential energy function. Energies are calculated for the complex of Phe-Pro-Arg with thrombin. For this system, we compare the performance of the B-spline method to that of the standard pairwise summation in terms of speed, accuracy, and overhead costs for a variety of grid spacings. In our rigid-body docking calculations, the B-spline method provided an accurate approximation of the total energy of the system, and it resulted in an 180-fold reduction in the time required for a single energy and gradient calculation for this system. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 71–85, 1998     

Analysis of an energy minimization method for locating transition states on potential energy hypersurfaces

Conditions are given for the successful search for a transition state by an energy minimization method. Proofs for these guidelines are presented. Advantages of this method are discussed, including its use in establishing lower bounds to transition state energies. Comparisons are made with other searching methods.     

Allostery of the two-state model of hemoglobin studied by ECEPP energy minimization

Recently, a powerful parallel-vector processor became available for molecular science. A new FORTRUN program was coded to treat the whole hemoglobin molecule with twofold symmetry. Using the X-ray coordinates of deoxyhemoglobin and oxyhemoglobin, minimum energy conformations were obtained for both the T-state and the R-state on the two-state model of allostery. From them, further energy minimization was performed with simple perturbation to bring the proximal region close to the heme group instead of oxygen binding, and the structural changes and energy changes were investigated. The difference of calculated energy changes between T and R was semiquantitatively in agreement with the experimental value 2.7 kcal/mol for one oxygen binding. When the perturbation was exerted on the alpha-subunits, the energy change within the perturbed alpha-subunits in the T-state gave a main contribution, and in the R-state, the structural changes of the alpha-subunits were specifically large. When the perturbation was exerted on the beta-subunits, the intersubunit interaction energy between alpha1 and beta2 (alpha2 and beta1) was dominant in the difference of the energy changes between T and R.     

气流床粉煤气化的Gibbs自由能最小化模拟

用Gibbs自由能最小化方法对粉煤气化过程进行了热力学平衡分析。对一混合煤种,在3.0 MPa和气化温度限制在1 200 ℃～1 450 ℃时,研究了氧-煤比、蒸气-煤比对气化炉出口气体组成、温度和有效气产率的影响,并由此确定了可行的操作域是氧-煤比545m3/t～605 m3/t、蒸气-煤比为152.64 kg/t～313.92 kg/t及其对应的工艺指标。从操作域中选择有代表性的工艺条件为氧-煤比578 m3/t、蒸气-煤比为187 kg/t,对应的气化炉出口温度1 358 ℃,CO+H2干基体积分数为91.5%,有效气产率为2.123(CO+H2)m3/kg。同时,研究了碳转化率和热损失对气化工艺指标的影响,其影响是显著的。