Neurodegenerative diseases including Parkinson’s, Alzheimer’s, Huntington’s, and Prion’s were found they all featured amyloid fibrils [1-6]. Amyloid is characterized by a cross-β sheet quaternary structure and recent X-ray diffraction studies of microcrystals revealed atomistic details of core region of amyloid [7,8]. All the quaternary structures of amyloid cross-β spines can be reduced to one of the 8 classes of steric zippers of [8], with strong van der Waals (vdW) interactions between β- sheets and hydrogen bonds (HBs) to maintain the β- strands. A new era in the structural analysis of amyloids started from the “steric zipper”-β-sheets [7]. As the two β-sheets zip up, Hydrophobic Packings (HPs & vdWs) have been formed. The extension of the “steric zipper” above and below (i.e. the β-strands) is maintained by Hydrogen Bonds (HBs) (but usually there is no HB between the two β-sheets). This is the common structure associated with some 20 neurodegenerative amyloid diseases.

We first do some mathematical analysis for the common structure. The vdW contacts of the two atoms can be described by the Lennard-Jones (LJ) potential energy, where the parameters on the depth of the potential well and the atom diameter can be fitted to reproduce experimental data or deduced from results of accurate quantum chemistry calculations. There are two parts describing LJ potential energy: the repulsion and the attraction (Figure 1). The minimization of LJ potential energy is an optimization problem, which is a well-known and challenging test problem for global optimization [10-14]. It is very hard for global optimization to directly solve this problem even with a small number of atoms. Similarly,

the potential energy for the HBs between β-strands has a similar mathematical formula, and usually most of the HBs are still kept during the phase of molecular modelling of amyloid fibrils. Thus, the amyloid fibril molecular modelling problem can be reduced to solve the optimization problem of minimizing LJ potential energy though it is not easy to accurately solve for a large molecule.

Alternatively, we have found another way to solve the problem of minimizing LJ potential energy [15]. Seeing Figure 1, we may know that this optimization problem reaches its optimal value at the bottom of the LJ potential well, where the distance between two atoms equals to the sum of vdW radii of the two atoms. Hence, the amyloid fibril molecular modelling problem can be looked as a molecular distance geometry problem (MDGP) [16-18]. As an example to explain MDGP, the problem of locating sensors in telecommunication networks is a DGP. In such a case, the positions of some sensors are known (which are called anchors) and some of the distances between sensors (which can or cannot be anchors) are known. The DGP is to locate the positions of all the sensors. The MDGP looks sensors as atoms and their telecommunication network as a molecule. In this paper, we aim to solve the alternative MDGP problem (i.e. Eq.11 of [15]) for modelling amyloid fibril molecular 3D structures.

Neurodegeneration is the progressive loss of structure or function of neurons, including death of neurons. A prion is a misshapen protein that acts like an infectious agent (but not requiring either DNA, RNA, or both) to cause a number of fatal diseases. Prion diseases are rich in β-sheets (compared with the normal prion protein PrPC in rich of α-helices) and are so-called “protein structural conformational” diseases. The normal hydrophobic region 113-120 AGAAAAGA peptide of prion proteins is an inhibitor/blocker of prion diseases. PrP lacking this palindrome could not convert to prion diseases. Brown et al. pointed out that the AGAAAAGA peptide was found to be necessary (though not sufficient) for blocking the toxicity and amyloidogenicity of PrP 106-126, and the peptide AGAA does not form fibrils [19]. The minimum sequence necessary for fibril formation should be AGAAA, AGAAAA, AGAAAAG, AGAAAAGA and GAAAAGA, but the molecular structures of these fibrils have not known yet. This paper addresses an important problem on modelling the 3D molecular structures of prion AGAAAAGA amyloid fibrils of neurodegenerative diseases. The rest of this paper is arranged as follows. In the next section, i.e. Section 2, an improved LBFGS Quasi-Newtonian method is presented for solving Eq.11 of [5]. Section 3 implements this Quasi-Newtonian method by constructing a 3D molecular structure of prion AGAAAAGA amyloid fibrils of neurodegenerative prion diseases. Numerical results of computations show that the method designed in Section 2 is very effective and successful. This concluding remark will be made in the last section, i.e. Section 4.

In a (macro) molecular system, if it is very far from equilibrium, then the forces may be excessively large, a robust energy minimization (EM) is required; another reason to perform an EM is the removal of all kinetic energy from the system: EM reduces the thermal noise in the structures and potential energies [20]. EM, with the images at the endpoints fixed in space, of the total system energy provides a minimum energy path. EM can be done using steepest descent (SD), conjugate gradient (CG), and Limited-memory Broyden Fletcher Goldfarb Shanno (LBFGS) methods.

Three kinds of possible EM methods are: 1) Derivative-free methods—that require only function evaluations, e.g. the simplex method and its variants; 2) derivative information methods—the partial derivatives of the potential energy with respect to all coordinates are known and the forces are minimized, e.g. SD, CG methods; and 3) second derivative information methods, e.g. LBFGS method. “SD is based on the observation that if the real-valued function is defined and differentiable in a neighbourhood of a point x₀ then decreases fastest if one goes from x₀ in the direction of the negative gradient of at x₀ and SD local search method converges fast [21]. SD is robust and easy to implement but it is not most efficient especially when closer to minimum; at this moment, we may use the efficient CG. CG is slower than SD in the early stages but more efficient when closer to minimum. CG algorithm adds an orthogonal vector to the current direction of the search, and then moves them in another direction nearly perpendicular to this vector. The hybrid of SD-CG will make SD or CG more efficient than SD or CG alone. However, CG cannot be used to find the EM path, for example, when “forces are truncated according to the tangent direction, making it impossible to define a Lagrangian” [22,23]. In this case, the powerful and faster quasiNewtonian method (e.g. the LBFGS quasi-Newtonian minimiser) can be used [22,24-28]. We briefly introduce the LBFGS quasi-Newtonian method as follows.

Newton’s method in optimization explicitly calculates the Hessian matrix of the second-order derivatives of the objective function and the reverse of the Hessian matrix [29]. The convergence of this method is quadratic, so it is faster than SD or CG. In high dimensions, finding the inverse of the Hessian is very expensive. In some cases, the Hessian is a non-invertible matrix, and furthermore in some cases, the Hessian is symmetric indefinite. QuasiNewton methods thus appear to overcome all these shortcomings.

Quasi-Newton methods (a special case of variable metric methods) are to approximate the Hessian. Currently, the most common quasi-Newton algorithms are the SR1 formula, the BHHH method, the widespread BFGS method and its limited/low-memory extension LBFGS, DFP, MS, and Broyden’s methods [30-33]. In Amber [23] and Gromacs [20], LBFGS is used, and the hybrid of LBFGS with CG—a Truncated Newton linear CG method with optional LBFGS Preconditioning [25]— is used in Amber [23].

For BFGS method, whether it converges at all on nonconvex problems is still an open problem. In fact, Powell (1984) gave a counter-example that shows that BFGS with an inexact line-search search may fail to converge [34-36]. Li and Fukushima (2001) proposed a modified BFGS method for non-convex objective function [37]. Basing on [24,25,37-39], in this paper we present an improved LBFGS method described as follows [40]— which presents the non-monotone line search technique [41,42] for the Wolfe-type search. The improved LBFGS method presented in this paper is much better than the standard BFGS method in view of the CPU time (see Figure 2) tested through more than 30 nonlinear programming problems (where each selected problem is regular, that is, its first and second derivatives exist and are continuous everywhere, and each problem is with different dimensions, i.e., 100, 500, 1000 and 10,000 dimensions) and its mathematical theory to support this algorithm can be seen from [40]. This paper implements the Wolfe-type search by the approximation technique of piecewise linear or quadratic function [43].

Algorithm 1: An Improved LBFGS Optimization Method for minimizing a non-convex function [40].

Step 0: Choose an initial point, an initial positive definite matrix H₀, and choose constants σ₁, σ₂ such that 0 < σ₁ < σ₂ < 1, and choose an positive integer m₁. Let k = 0.

Step 1: If, then output x_k and stop; otherwise,

go to Step 2.

Step 2: Solve the following linear equation to get d_k:

Step 3: Find a step-size λ_k > 0 satisfying the Wolfetype line search conditions:

,

.

Step 4: Let be the next iteration. Calculate g_k+1 and ||g_k+1||.

Step 5: Let, , , then.

Step 6: Let ḿ = min{k + 1, m₁}. Update H_k following the formula

where (when k = 0,

, ,

Step 7: Let k = k + 1 and go to Step 1.

In Algorithm 1, g_k denotes the gradient of at x_k, and the convergence and the R-linear convergent rate of Algorithm 1 are guaranteed by the following assumptions: (A1) f is twice continuously differentiable, (A2) f has Lipschitz continuous gradients and Hessians, (A3) the Hessian at the stationary point is always positively definite, and (A4) the level set of f is bounded. The detailed proof of the convergence and R-linear convergent rate of Algorithm 1 can be found in [40].

In [40], the non-monotone line search technique for step-size λ_k is used. This is a difference between the algorithm of [40] and this paper. All in all, it is well known that quasi-Newton method is an efficient solution method for unconstrained and continuously differentiable minimization problem [45-47]. However, it needs computing and storage of the updated matrix which is an approximation to the Hessian matrix in each iteration of the method. Hence, its efficiency may decrease when it is applied to large scale optimization problem. To overcome the drawback, limited memory quasi-Newton method is proposed [48]. The main ideal of this method is nearly identical to that of the standard BFGS method, and the only difference is that the inverse Hessian approximation is not formed explicitly, but defined by a small number, say ḿ, of BFGS updates. This technique received much attention in recent years and numerical experiments show that it is very competitive [24,49], and its global convergence and R-linear convergence rate with Wolfe line search are established for the uniformly convex case [24,26]. Since the limited memory BFGS method may suffer from ill-conditions for small value of ḿ, Al-Baali (1999) [50] made some modifications to the method and establish its global convergence based on the same assumptions, and Byrd et al. (1994) [51] derives new representation of limited memory quasi-Newton matrices for the benefit of computing the updated matrix. Recently, a non-monotone line search is introduced, see e.g., [41,42]. Then it is showed to be more competitive and practical for solving nonlinear optimization problems, and [52] established the global convergence of this line search applied to limited memory BFGS method based on the uniformly convex assumption. Motivated by the above observation, it turns out that in two respects the limited memory BFGS method is much less effective. First, we note that the convergence analysis of the method is focused on the uniformly convex assumption and little is known for non-convex case. Second, numerical experiments have suggested the main weakness of limited memory method is that it may converge very slowly in terms of number of iterations for ill-conditioned problems. The purpose of the above Algorithm 1 is to reduce these defects and Figure 2 shows the effectiveness of the proposed algorithm. We will apply it into the molecular modelling of prion AGAAAAGA amyloid fibrils in the next section.

From their research of prion, scientists found that the cross-β structure of peptides is with the nature of selfaggregation, the self-aggregating to form fibers. This provides us a new research idea for nanomaterials. HBs can be formed between peptide β-strands, and one peptide monomer connects together with another in accordance with the specific structure to form fibers. Many laboratories in the world are synthesizing peptides that can self-aggregate to form fibers, and want to be able to control the growth of the fiber to find out new functional materials [53,54]. The studies of this paper not only benefit nanometerials research, but also benefit the research on neurodegenerative amyloid fibril diseases. Prion AGAAAAGA peptide has been reported to own an amyloid fibril forming property (initially described in 1992 by Gasset et al. of Prusiner’s Group) [19,55-78], but there has not been experimental structural bioinformatics for this segment yet due to the unstable, noncrystalline and insoluble nature of this region. Furthermore, Zhang (2011) did accurate calculations to confirm the amyloid fibril property at this region (Figure 3) [79] in the use of the fibril prediction program of [80].

In a (macro) molecular system, a robust energy mini-

mization is very necessarily required. Mathematical optimization minimization methods find a place to apply in these systems. Because in physics the (macro) molecular system usually is not a simple two-body problem of system, local search optimization methods are very useful in the applications to the (macro) molecular system. On anther sense, when a protein is unstable, noncrystalline or insoluble and very difficult to detect its 3D structure by the expensive and costly NMR and X-ray, theoretical mathematical or physical computational method can be used to produce the 3D structure of the protein. Moreover, even though the X-ray crystallography finds the X-ray final structure of a protein, we still need refinements using theoretical protocols in order to produce a better structure. The theoretical computational method—an improved LBFGS Quasi-Newtonian mathematical optimization method-presented in this paper and other mathematical optimization methods mentioned in this paper should be very useful in the protein molecular modeling research field.

This paper also shows the effectiveness of the improved LBFGS mathematical optimization method presented. Prion AGAAAAGA amyloid fibrils have not much structural information. This paper presents some bioinformatics on the molecular structures of prion AGAAAAGA amyloid fibrils in the sense of theoretical emphasis. The structures may be helpful in the advance in the biochemical knowledge of prion protein misfolding or instability and in the future applications for therapeutic agent design.

This research was supported by the National Natural Science Foundation of China (No. 11171180), the National Basic Research Program of China (2010CB732501), and a Victorian Life Sciences Computation Initiative (VLSCI) grant numbered VR0063 & 488 on its Peak Computing Facility at the University of Melbourne, an initiative of the Victorian Government (Australia). The first author appreciates the financial support by Professors Changyu Wang and Yiju Wang for his visits to Qufu Normal University (China), and by Professor Xiangsun Zhang for his visits to Chinese Academy of Sciences (Beijing); the first author also appreciates Professor Adil M. Bagirov (University of Ballarat) for his instructions to implement the Wolfe-type search. This paper is dedicated to Professor Xiangsun Zhang (Academia Sinica, Beijing) on the occasion of his 70th birthday. The authors appreciate the anonymous referees for their comments on this paper.