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The WNT10A gene, which codes for the WNT Family Member 10A (WNT-10a) protein and regulates WNT signaling pathways, has been linked to dental anomaly. WNT-10a protein binds with Frizzled (FZD (FZD1, FZD5 and FZD10) receptors and is involved in tooth development. Mutations in the WNT10A gene cause Oligodontia, tooth agenesis, microdontia and root maldevelopment by impairing WNT signaling. In addition, mutations in the WNT10A gene affect its structure and functional behavior of interaction with FZD receptors. However, the structural changes and interaction behavior of WNT-10 protein upon mutation at the molecular level are still unclear. Hence, in this study, the structural consequences of WNT-10a mutations at the atomic level were elucidated using a molecular simulation approach. Furthermore, docking simulations and MM-GBSA approaches were applied to investigate the interaction pattern of FZD proteins upon mutation in the WNT-10a protein. This study demonstrates that when the WNT-10a protein loses stability, the G213S mutant becomes more flexible, whereas R293C and T357I mutants become more rigid than the wild-type protein. This structural loss affects the interaction between WNT-10a and FZD1, FZD5 and FZD10 receptors which dysregulate the WNT signaling pathway in tooth development. Understanding this mechanism at a molecular level will be beneficial for treating dental anomalies.

Using an argument from statistical mechanics, V. Jones has given a new method for constructing pairs of links with identical skein polynomials. We give a more general construction and use it to provide a simple proof of a theorem of Traczyk's involving rotors of links. Several examples are given of pairs of knots with coincident skein polynomials and small crossing number.

A modification of the Bak–Sneppen model to include simple elements of Darwinian evolution is used to check the survival of prey and predators in long food chains. Mutations, selection, and starvation resulting from depleted prey are incorporated in this model.

A standard Genetic Algorithm is applied to a set of test problems, three of them taken from physics and the rest analytical expressions explicitly constructed to test search procedures. The relation between mutation rate and population size in the search for optimum performance is obtained showing similar behavior in these problems.

The core in most genetic algorithms is the bitwise manipulations of bit strings. We show that one can directly manipulate the bits in floating point numbers. This means the main bitwise operations in genetic algorithm mutations and crossings are directly done inside the floating point number. Thus the interval under consideration does not need to be known in advance. For applications, we consider the roots of polynomials and finding solutions of linear equations.

We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators. We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.

In the framework of the bit-string model of biological ageing we show that the survival chance of a small population in an environment of limited carrying capacity grows exponentially with the size of the habitat. Extinction is usually preceded by a gradual decline of the genetic condition of the population. With death due to senescence coming earlier, the typical population size shrinks, and at some stage the population dies out due to the fluctuations.

Evolution, based on the principles of mutation and selection, is a powerful basis for microscopic changes which can account for the evolution of a species and macroscopic speciation where there is splitting of a species into two distinct new species. We show that a single species evolves into distinct species after several generations in an unrestricted genome space.

This paper presents a multi-objective co-evolutionary population migration algorithm based on Good Point Set (GPSMCPMA) for multi-objective optimization problems (MOP) in view of the characteristics of MOPs. The algorithm introduces the theory of good point set (GPS) and dynamic mutation operator (DMO) and adopts the entire population co-evolutionary migration, based on the concept of Pareto nondomination and global best experience and guidance. The performance of the algorithm is tested through standard multi-objective functions. The experimental results show that the proposed algorithm performs much better in the convergence, diversity and solution distribution than SPEA2, NSGA-II, MOPSO and MOMASEA. It is a fast and robust multi-objective evolutionary algorithm (MOEA) and is applicable to other MOPs.

Abnormal cells in the human body that keep on mutating are termed to be cancer in medical terms. There are multiple types of cancer identified in human beings. It is very much essential to identify and classify the type of cancer in its earlier stage. This objective can be satisfied by artificial intelligence which has a subfield of machine learning to create a generalized model that could identify and classify cancer with increased performance. To perform the identification and classification of various cancer types, in this paper, two techniques are adopted. The optimized feature set computation was done using the Kernel-Induced Matriarch path tracking Elephant Herding Optimization (KIM-EHO) and the classification for the given samples was done using the Support Vector Machines (SVM). The proposed techniques are implemented with the benchmark datasets and the results proved that the proposed methodologies outperformed the existing methods in terms of accuracy, specificity, sensitivity and time complexity.

We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing mutants. Then we study the distinction of mutants, with particular regard to the symmetric mutants. We discuss the determination of braid index as another application of our computational methods.

Polynomial mutation is widely used in evolutionary optimization algorithms as a variation operator. In previous work on the use of evolutionary algorithms for solving multi-objective problems, two versions of polynomial mutations were introduced. The first is non-highly disruptive that is not prone to local optima and the second is highly disruptive polynomial mutation. This paper looks at the two variants and proposes a dynamic version of polynomial mutation. The experimental results show that the proposed adaptive algorithm is doing well for three evolutionary multiobjective algorithms on well known multiobjective optimization problems in terms of convergence speed, generational distance and hypervolume performance metrics.

We introduce a new approach to Vassiliev invariants. This approach deals with Vassiliev invariants directly on knots and does not make use of diagrams. We give a series of applications of this approach, (re)proving some new and known facts on Vassiliev invariants.

Using the new approach of braiding sequences we give a proof of the Lin-Wing conjecture, stating that a Vassiliev invariant ν of degree k has a value O_ν (c(K)^k) on a knot K, where c(K) is the crossing number of K and O_ν depends on ν only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniquely a Vassiliev invariant of degree k. This also makes orientation and mutation sensitivity of Vassiliev invariants decidable by testing them on alternating/positive knots/mutants only.

We give an exponential upper bound for the number of Vassiliev invariants on a special class of closed braids.

S-equivalence of classical knots is investigated, as well as its relationship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon's double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot.

We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We also show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-manifolds which have the same quantum SU(2)-invariants but are not homology cobordant. These results are obtained by investigating the effect of mutation on the Milnor -invariants, or equivalently the Massey products.

We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsváth and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ⊔ K with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author.

This paper contains two remarks about the application of the $d$-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math.192(3) (2013) 717–750] concerning Conway mutation of alternating links.

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.