A Method for Inferring Polymers Based on Linear Regression and Integer Programming.

A novel framework has recently been proposed for designing the molecular structure of chemical compounds with a desired chemical property using both artificial neural networks and mixed integer linear programming. In this paper, we design a new method for inferring a polymer based on the framework. For this, we introduce a new way of representing a polymer as a form of monomer and define new descriptors that feature the structure of polymers. We also use linear regression as a building block of constructing a prediction function in the framework. The results of our computational experiments reveal a set of chemical properties on polymers to which a prediction function constructed with linear regression performs well. We also observe that the proposed method can infer polymers with up to 50 nonhydrogen atoms in a monomer form.

Molecular Design Based on Integer Programming and Splitting Data Sets by Hyperplanes.

A novel framework for designing the molecular structure of chemical compounds with a desired chemical property has recently been proposed. The framework infers a desired chemical graph by solving a mixed integer linear program (MILP) that simulates the computation process of two functions: a feature function defined by a two-layered model on chemical graphs and a prediction function constructed by a machine learning method. To improve the learning performance of prediction functions in the framework, we design a method that splits a given data set C into two subsets C(i),i=1,2 by a hyperplane in a chemical space so that most compounds in the first (resp., second) subset have observed values lower (resp., higher) than a threshold θ. We construct a prediction function ψ to the data set C by combining prediction functions ψi,i=1,2 each of which is constructed on C(i) independently. The results of our computational experiments suggest that the proposed method improved the learning performance for several chemical properties to which a good prediction function has been difficult to construct.

An Inverse QSAR Method Based on Linear Regression and Integer Programming.

BACKGROUND: Drug design is one of the important applications of biological science. Extensive studies have been done on computer-aided drug design based on inverse quantitative structure activity relationship (inverse QSAR), which is to infer chemical compounds from given chemical activities and constraints. However, exact or optimal solutions are not guaranteed in most of the existing methods. METHOD: Recently a novel framework based on artificial neural networks (ANNs) and mixed integer linear programming (MILP) has been proposed for designing chemical structures. This framework consists of two phases: an ANN is used to construct a prediction function, and then an MILP formulated on the trained ANN and a graph search algorithm are used to infer desired chemical structures. In this paper, we use linear regression instead of ANNs to construct a prediction function. For this, we derive a novel MILP formulation that simulates the computation process of a prediction function by linear regression. RESULTS: For the first phase, we performed computational experiments using 18 chemical properties, and the proposed method achieved good prediction accuracy for a relatively large number of properties, in comparison with ANNs in our previous work. For the second phase, we performed computational experiments on five chemical properties, and the method could infer chemical structures with around up to 50 non-hydrogen atoms. CONCLUSIONS: Combination of linear regression and integer programming is a potentially useful approach to computational molecular design.

A Novel Image Encryption Scheme Based on Elliptic Curves over Finite Rings.

Image encryption based on elliptic curves (ECs) is emerging as a new trend in cryptography because it provides high security with a relatively smaller key size when compared with well-known cryptosystems. Recently, it has been shown that the cryptosystems based on ECs over finite rings may provide better security because they require the computational cost for solving the factorization problem and the discrete logarithm problem. Motivated by this fact, we proposed a novel image encryption scheme based on ECs over finite rings. There are three main steps in our scheme, where, in the first step, we mask the plain image using points of an EC over a finite ring. In step two, we create diffusion in the masked image with a mapping from the EC over the finite ring to the EC over the finite field. To create high confusion in the plain text, we generated a substitution box (S-box) based on the ordered EC, which is then used to permute the pixels of the diffused image to obtain a cipher image. With computational experiments, we showed that the proposed cryptosystem has higher security against linear, differential, and statistical attacks than the existing cryptosystems. Furthermore, the average encryption time for color images is lower than other existing schemes.

A Novel Method for Inferring Chemical Compounds With Prescribed Topological Substructures Based on Integer Programming.

Drug discovery is one of the major goals of computational biology and bioinformatics. A novel framework has recently been proposed for the design of chemical graphs using both artificial neural networks (ANNs) and mixed integer linear programming (MILP). This method consists of a prediction phase and an inverse prediction phase. In the first phase, an ANN is trained using data on existing chemical compounds. In the second phase, given a target chemical property, a feature vector is inferred by solving an MILP formulated from the trained ANN and then a set of chemical structures is enumerated by a graph enumeration algorithm. Although exact solutions are guaranteed by this framework, the types of chemical graphs have been restricted to such classes as trees, monocyclic graphs, and graphs with a specified polymer topology with cycle index up to 2. To overcome the limitation on the topological structure, we propose a new flexible modeling method to the framework so that we can specify a topological substructure of graphs and a partial assignment of chemical elements and bond-multiplicity to a target graph. The results of computational experiments suggest that the proposed system can infer chemical graphs with around up to 50 non-hydrogen atoms.

A novel method for inference of acyclic chemical compounds with bounded branch-height based on artificial neural networks and integer programming.

Analysis of chemical graphs is becoming a major research topic in computational molecular biology due to its potential applications to drug design. One of the major approaches in such a study is inverse quantitative structure activity/property relationship (inverse QSAR/QSPR) analysis, which is to infer chemical structures from given chemical activities/properties. Recently, a novel two-phase framework has been proposed for inverse QSAR/QSPR, where in the first phase an artificial neural network (ANN) is used to construct a prediction function. In the second phase, a mixed integer linear program (MILP) formulated on the trained ANN and a graph search algorithm are used to infer desired chemical structures. The framework has been applied to the case of chemical compounds with cycle index up to 2 so far. The computational results conducted on instances with n non-hydrogen atoms show that a feature vector can be inferred by solving an MILP for up to [Formula: see text], whereas graphs can be enumerated for up to [Formula: see text]. When applied to the case of chemical acyclic graphs, the maximum computable diameter of a chemical structure was up to 8. In this paper, we introduce a new characterization of graph structure, called "branch-height" based on which a new MILP formulation and a new graph search algorithm are designed for chemical acyclic graphs. The results of computational experiments using such chemical properties as octanol/water partition coefficient, boiling point and heat of combustion suggest that the proposed method can infer chemical acyclic graphs with around [Formula: see text] and diameter 30.

An Inverse QSAR Method Based on a Two-Layered Model and Integer Programming.

A novel framework for inverse quantitative structure-activity relationships (inverse QSAR) has recently been proposed and developed using both artificial neural networks and mixed integer linear programming. However, classes of chemical graphs treated by the framework are limited. In order to deal with an arbitrary graph in the framework, we introduce a new model, called a two-layered model, and develop a corresponding method. In this model, each chemical graph is regarded as two parts: the exterior and the interior. The exterior consists of maximal acyclic induced subgraphs with bounded height, the interior is the connected subgraph obtained by ignoring the exterior, and the feature vector consists of the frequency of adjacent atom pairs in the interior and the frequency of chemical acyclic graphs in the exterior. Our method is more flexible than the existing method in the sense that any type of graphs can be inferred. We compared the proposed method with an existing method using several data sets obtained from PubChem database. The new method could infer more general chemical graphs with up to 50 non-hydrogen atoms. The proposed inverse QSAR method can be applied to the inference of more general chemical graphs than before.

An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops.

Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers n≥1 and Δ≥0, we propose a method to count all non-isomorphic trees with n vertices, Δ self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, Δ self-loops and no multi-edges, in O(n2(n+Δ(n+Δ·min{n,Δ}))) time and O(n2(Δ2+1)) space, since every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any "cycle rank".

Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank.

Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.