Generation, Ranking and Unranking of Ordered Trees with Degree Bounds

We study the problem of generating, ranking and unranking of unlabeled ordered trees whose nodes have maximum degree of $\Delta$. This class of trees represents a generalization of chemical trees. A chemical tree is an unlabeled tree in which no node has degree greater than 4. By allowing up to $\Delta$ children for each node of chemical tree instead of 4, we will have a generalization of chemical trees. Here, we introduce a new encoding over an alphabet of size 4 for representing unlabeled ordered trees with maximum degree of $\Delta$. We use this encoding for generating these trees in A-order with constant average time and O(n) worst case time. Due to the given encoding, with a precomputation of size and time O(n^2) (assuming $\Delta$ is constant), both ranking and unranking algorithms are also designed taking O(n) and O(nlogn) time complexities.Comment: In Proceedings DCM 2015, arXiv:1603.0053

Computational exploration of the chemical structure space of possible reverse tricarboxylic acid cycle constituents

The reverse tricarboxylic acid (rTCA) cycle has been explored from various standpoints as an idealized primordial metabolic cycle. Its simplicity and apparent ubiquity in diverse organisms across the tree of life have been used to argue for its antiquity and its optimality. In 2000 it was proposed that chemoinformatics approaches support some of these views. Specifically, defined queries of the Beilstein database showed that the molecules of the rTCA are heavily represented in such compound databases. We explore here the chemical structure space, e.g. the set of organic compounds which possesses some minimal set of defining characteristics, of the rTCA cycle's intermediates using an exhaustive structure generation method. The rTCA's chemical space as defined by the original criteria and explored by our method is some six to seven times larger than originally considered. Acknowledging that each assumption in what is a defining criterion making the rTCA cycle special limits possible generative outcomes, there are many unrealized compounds which fulfill these criteria. That these compounds are unrealized could be due to evolutionary frozen accidents or optimization, though this optimization may also be for systems-level reasons, e.g., the way the pathway and its elements interface with other aspects of metabolism

Herndon\u27 s Structure-Resonance Theory. On the Valence Structure Count for Conjugated Radical Cations

Enumeration of valence structures for conjugated radical cations is of interest in estimating the first ionization potentials of conjugated hydrocarbons. Alternative routes for enumeration are reviewed and a novel method, based on the properties of the acyclic polynomials of conjugated structures, is outlined. The method is illustrated for several molecules and results reported for a number of conjugated systems involving either 14 or less carbon atoms

Formula Periodic Table for the Isomer Classes of Acyclic Hydrocarbons - Enumerative and Asymptotic Characteristics

The overall set of acyclic hydrocarbons CnH2m with classical valence structures is considered, the structural isomers are enumerated, and the results displayed in the form of a »periodic table« with the C atom count n and H atom half-count m respectively identifying rows and columns. Asymptotic n → ∞ behaviors of these enumerations are developed, first for fixed degree u ≡ n + 1 - m of unsaturation and second for fixed number 2m of H-atoms. The first-set isomer classes increase in size exponentially fast with n, whereas with the second set, the isomer-class sizes increase sub-exponentially, as a power of n

Exploring the GDB-13 chemical space using deep generative models

Recent applications of recurrent neural networks (RNN) enable training models that sample the chemical space. In this study we train RNN with molecular string representations (SMILES) with a subset of the enumerated database GDB-13 (975 million molecules). We show that a model trained with 1 million structures (0.1% of the database) reproduces 68.9% of the entire database after training, when sampling 2 billion molecules. We also developed a method to assess the quality of the training process using negative log-likelihood plots. Furthermore, we use a mathematical model based on the “coupon collector problem” that compares the trained model to an upper bound and thus we are able to quantify how much it has learned. We also suggest that this method can be used as a tool to benchmark the learning capabilities of any molecular generative model architecture. Additionally, an analysis of the generated chemical space was performed, which shows that, mostly due to the syntax of SMILES, complex molecules with many rings and heteroatoms are more difficult to sample

Quantum Isomer Search

Isomer search or molecule enumeration refers to the problem of finding all the isomers for a given molecule. Many classical search methods have been developed in order to tackle this problem. However, the availability of quantum computing architectures has given us the opportunity to address this problem with new (quantum) techniques. This paper describes a quantum isomer search procedure for determining all the structural isomers of alkanes. We first formulate the structural isomer search problem as a quadratic unconstrained binary optimization (QUBO) problem. The QUBO formulation is for general use on either annealing or gate-based quantum computers. We use the D-Wave quantum annealer to enumerate all structural isomers of all alkanes with fewer carbon atoms (n < 10) than Decane (C10H22). The number of isomer solutions increases with the number of carbon atoms. We find that the sampling time needed to identify all solutions scales linearly with the number of carbon atoms in the alkane. We probe the problem further by employing reverse annealing as well as a perturbed QUBO Hamiltonian and find that the combination of these two methods significantly reduces the number of samples required to find all isomers.Comment: 20 pages, 9 figure

Fullerene‐like structures of Cretaceous crinoids reveal topologically limited skeletal possibilities

There are few cases where numbers or types of possible phenotypes are known, although vast state spaces have been postulated. Rarely applied in this context, graph theory and topology enable enumeration of possible phenotypes and evolutionary transitions. Here, we generate polyhedral calyx graphs for the Late Cretaceous, stemless crinoids Marsupites testudinarius and Uintacrinus socialis (Uintacrinoidea Zittel) revealing structural similarities to carbon fullerene and fulleroid molecules (respectively). The U. socialis calyx incorporates numerous plates (e.g. graph vertices |V| ≥ 197), which are small, light, low‐density and have four to eight sides. Therefore, the corresponding number of possible plate arrangements (number of polyhedral graphs) is large (≫1 × 1014). Graph vertices representing plates with sides >6 introduce negative Gaussian curvature (surface saddle points) and topological instability, increasing buckling risk. However, observed numbers of vertices for Uintacrinus do not allow more stable pentaradial configurations. In contrast, the Marsupites calyx dual graph has 17 faces that are pentagonal or hexagonal. Therefore, it is structurally identical to a carbon fullerene, specifically C30‐D5h. Corresponding graph restrictions result in constraint to only three structural options (fullerene structures C30‐C2v 1, C30‐C2v 2 and C30‐D5h). Further restriction to pentaradial symmetry allows only one possibility: the Marsupites phenotype. This robust, stable topology is consistent with adaptation to predation pressures of the Mesozoic marine revolution. Consequently, the most plausible evolutionary pathway between unitacrinoid phenotypes was a mixed heterochronic trade‐off to fewer, larger calyx plates. Therefore, topological limitations radically constrained uintacrinoid skeletal possibilities but thereby aided evolution of a novel adaptive phenotype

Beyond mitigation : potential options for counter-balancing the climatic and environmental consequences of the rising concentrations of greenhouse gases

Global climate change is occurring at an accelerating pace, and the global greenhouse gas (GHG) emissions that are forcing climate change continue to increase. Given the present pace of international actions, it seems unlikely that atmospheric composition can be stabilized at a level that will avoid"dangerous anthropogenic interference"with the climate system, as called for in the UN Framework Convention on Climate Change. Complicating the situation, as GHG emissions are reduced, reductions in the offsetting cooling influence of sulfate aerosols will create an additional warming influence, making an early transition to climate stabilization difficult. With significant reductions in emissions (mitigation) likely to take decades, and with the impacts of projected climate change-even with proactive adaptation-likely to be quite severe over the coming decades, additional actions to offset global warming and other impacts have been proposed as important complementary measures. Although a number of possible geoengineering approaches have been proposed, each has costs and side effects that must be balanced against the expected benefits of reduced climate impacts. However, substantial new research is needed before comparison of the relative benefits and risks of intervening is possible. A first step in determining whether geoengineering is likely to be a useful option is the initiation of research on four interventions to limit the increasing serious impacts: limiting ocean acidification by increasing the removal of carbon dioxide from the atmosphere and upper ocean; limiting the increasing intensity of tropical cyclones; limiting the warming of the Arctic and associated sea level rise; and sustaining or enhancing the existing sulfate cooling influence. In addition, in depth consideration is needed regarding the governance structure for an international geoengineering decision-making framework in the event that geoengineering becomes essential.Montreal Protocol,Climate Change,Energy Production and Transportation,Energy and Environment,Environment and Energy Efficiency

Application of the Dualist Model. Generation of Kekule Structures and Resonant Sextets of Benzenoid Hydrocarbons

The dualist model of Balaban is used for the enumeiration and dis.play of Kelmle struotmres K aind .resooant sextet numbers rCG; k) O·f la.rge cata-condenrsed bemmnoi:d hydrncarbons. The key steps are: (1) Transfo.rm the berw;enoid graph into the corresponding dualist and associate with it a linear-angular, L-A, sequence, (2) Fragment the dualist into subgraphs after each L-A pair. The resulting subgraphs are called fragment graiphs. (3) Colour each fragment graph, containing v veritices, v + 1 times such ·that each co1ourin.g contaiins a.t mo&t on black vertex (the rest being white). ( 4) Re-assemble the coloured fragments into theiir initial geometry, preserved in the dual\u27ist, to produce a set o.f c10l0>ured dua!ists such that no coloured dualist has more than one black vertex in each linear segment. The number of such coloured duali:sts is K, the Kek•ule count. By convention, each blac.k dualist vertex corresponds to a propex resona,nt sextet. This, plus the fac.t that a lineair segment can have at most one resonant sexteit, completely defines all o.f the individual VB Kekule structures and their resonant sextets. The method is an illustration o.f data reduction schemes and is quite suited for large benzenoid hydrocarbons. A numbe•r O<f fo:rlIIlllllae for com,putiing the number of Kekule striuctures of vairious families of cata-condensed benzenoid hydrocarbons are derived. In addition, the abnve apprnach is applicable to large benzenoid systems consisting 0if c.ata- condensed f.ra,gments and thin peri-condensed fragments