Structural and biochemical characterization of Malaria parasites

The Apicomplexa phylum includes more than 6000 species, and some of them are pathogens of humans and animals of socio-economic importance. The most representative parasites are Plasmodium spp. and Toxoplasma gondii. Plasmodium spp. cause malaria, a parasitic infection with a wide distribution in tropical and subtropical areas. These parasites display a unique mode of cell motility called gliding motility. A macromolecular motor complex, the glideosome, is indispensable for parasite locomotion and host cell infection. The core of the glideosome is formed by an actomyosin motor comprised of actin as well as myosin (Myo) A and its two light chains, essential light chain (ELC) and MyoA tail interacting protein (MTIP). Plasmodium spp. have six myosins classified into three classes (VI, XII, and XIV). MyoA from class XIV is the most studied of these. The force for parasite gliding motility is produced by the hydrolysis of ATP, which promotes the movement of MyoA along actin filaments. Plasmodium spp. have two actin isoforms, of which the major isoform, ActI, is the most studied. In this work, a biochemical and structural characterization was performed on Plasmodium falciparum actins, focusing on the minor isoform, ActII. In vitro experiments were performed to understand the polymerization properties of these unconventional actins. The critical concentration, the kinetics of the elongation phase, and spontaneous depolymerization were studied. This thesis work showed that the filament stability of Plasmodium actins is different between the isoforms and from canonical actins. Especially for ActI, which forms shorter filaments than ActII. Atomic structures were determined by cryogenic electron microscopy (cryo-EM) of PfActII in the ADP-Mg2+ form in the absence and presence of the stabilizing agent jasplakinolide (JAS) at resolutions of 3.5 and 3.2 Å, respectively. The structures reveal monomer interactions along the filament, the effect of JAS on the filaments, and conformational changes in the actin protomers upon polymerization, including the D-loop conformation. In addition, this work contributed to obtaining the first high-resolution structure of the Plasmodium actomyosin motor complex at an average resolution of 3.1 Å, and the structure of filamentous PfActI with a resolution of 2.6 Å showing details of the nucleotide and JAS binding sites. Besides, a preliminary characterization of class VI myosins was performed. A molecular tool was generated, which can be used to study protein-protein interactions in order to find the interacting light chains for other Plasmodium myosins. Understanding the structure and biochemical properties of the glideosome components and other actomyosin complexes provides a basis for developing new treatments against these devasting pathogens.Doktorgradsavhandlin

Discussion On Some Important Topics in Graph Theory

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Some Topics of Special Interest in Graph Theory

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Games on graphs, visibility representations, and graph colorings

In this thesis we study combinatorial games on graphs and some graph parameters whose consideration was inspired by an interest in the symmetry of hypercubes. A capacity function f on a graph G assigns a nonnegative integer to each vertex of V(G). An f-matching in G is a set M ⊆ E(G) such that the number of edges of M incident to v is at most f(v) for all v ⊆ V(G). In the f-matching game on a graph G, denoted (G,f), players Max and Min alternately choose edges of G to build an f-matching; the game ends when the chosen edges form a maximal f-matching. Max wants the final f-matching to be large; Min wants it to be small. The f-matching number is the size of the final f-matching under optimal play. We extend to the f-matching game a lower bound due to Cranston et al. on the game matching number. We also consider a directed version of the f-matching game on a graph G. Peg Solitaire is a game on connected graphs introduced by Beeler and Hoilman. In the game, pegs are placed on all but one vertex. If x, y, and z form a 3-vertex path and x and y each have a peg but z does not, then we can remove the pegs at x and y and place a peg at z; this is called a jump. The goal of the Peg Solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1. Beeler and Rodriguez proposed a variant where we want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph G when no jumps remain is the Fool's Solitaire number F(G). We determine the Fool's Solitaire number for the join of any graphs G and H. For the cartesian product, we determine F(G ◻ K_k) when k ≥ 3 and G is connected. Finally, we give conditions on graphs G and H that imply F(G ◻ H) ≥ F(G) F(H). A t-bar visibility representation of a graph G assigns each vertex a set that is the union of at most t horizontal segments ("bars") in the plane so that vertices are adjacent if and only if there is an unobstructed vertical line of sight (having positive width) joining the sets assigned to them. The visibility number of a graph G, written b(G), is the least t such that G has a t-bar visibility representation. Let Q_n denote the n-dimensional hypercube. A simple application of Euler's Formula yields b(Q_n) ≥ ⌈(n+1)/4⌉. To prove that equality holds, we decompose Q_{4k-1} explicitly into k spanning subgraphs whose components have the form C_4 ◻ P_{2^l}. The visibility number b(D) of a digraph D is the least t such that D can be represented by assigning each vertex at most t horizontal bars in the plane so that uv ∈ E(D) if and only if there is an unobstructed vertical line of sight (with positive width) joining some bar for u to some higher bar for v. It is known that b(D) ≤ 2 for every outerplanar digraph. We give a characterization of outerplanar digraphs with b(D)=1. A proper vertex coloring of a graph G is r-dynamic if for each v ∈ V (G), at least min{r, d(v)} colors appear in N_G(v). We investigate r-dynamic versions of coloring and list coloring. We give upper bounds on the minimum number of colors needed for any r in terms of the genus of the graph. Two vertices of Q_n are antipodal if they differ in every coordinate. Two edges uv and xy are antipodal if u is antipodal to x and v is antipodal to y. An antipodal edge-coloring of Q_n is a 2-coloring of the edges in which antipodal edges have different colors. DeVos and Norine conjectured that for n ≥ 2, in every antipodal edge-coloring of Q_n there is a pair of antipodal vertices connected by a monochromatic path. Previously this was shown for n ≤ 5. Here we extend this result to n = 6. Hovey introduced A-cordial labelings as a simultaneous generalization of cordial and harmonious labelings. If S is an abelian group, then a labeling f: V(G) → A of the vertices of a graph G induces an edge-labeling on G; the edge uv receives the label f(u) + f(v). A graph G isA-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most 1, and (2) the induced edge label classes differ in size by at most 1. The smallest non-cyclic group is V_4 (also known as Z_2×Z_2). We investigate V_4-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. Finally, we introduce a generalization of A-cordiality involving digraphs and quasigroups, and we show that there are infinitely many Q-cordial digraphs for every quasigroup Q

Graph parameters and the speed of hereditary properties

In this thesis we study the speed of hereditary properties of graphs and how this defines some of the structure of the properties. We start by characterizing several graph parameters by means of minimal hereditary classes. We then give a global characterization of properties of low speed, before looking at properties with higher speeds starting at the Bell number. We then introduce a new parameter, clique-width, and show that there are an infinite amount of minimal hereditary properties with unbounded clique-width. We then look at the factorial layer in more detail and focus on P7-free bipartite graphs. Finally we discuss word-representable graphs

Some Investigations in the Theory of Graphs

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Colourings of P5P_5-free graphs

For a set of graphs H, we call a graph G H-free if G-S is non-isomorphic to H for each S⊆V(G) and each H∈H. Let f_H^* ∶N_(>0)↦N_(>0 )be the optimal χ-binding function of the class of H-free graphs, that is, f_H^* (ω)=max⁡{χ(G): ω(G)=ω,G is H-free} where χ(G),ω(G) denote the chromatic number and clique number of G, respectively. In this thesis, we mostly determine optimal χ-binding functions for subclasses of P_5-free graphs, where P_5 denotes the path on 5 vertices. For multiple subclasses we are able to determine them exactly and for others we prove the right order of magnitude. To achieve those results we prove structural results for the graph classes and determine colourings. We sometimes obtain those results by researching the prime graphs and combining the two decomposition methods by homogeneous sets and clique-separators. Additionally, we use the Strong Perfect Graph Theorem and analyse the neighbourhood of holes. For some of these subclasses we characterise all graphs G with χ(G)>χ(G-\{u\}), for each u∈V(G) and use those to determine the function

Forced symmetry breaking of Euclidean equivariant partial differential equations, pattern formation and Turing instabilities

Many natural phenomena may be modelled using systems of differential equations that possess symmetry. Often the modelling process introduces additional symmetries that are only approximately present in the real physical system. This thesis investigates how the inclusion of small symmetry breaking effects changes the behaviour of the original solutions, such a process is called forced symmetry breaking. Part I introduces the general equivariant bifurcation theory required for the rest of this work. In particular, we generalise previous techniques used to study forced symmetry breaking to a certain class of Euclidean invariant problems. This allows the study of the effects of forced symmetry breaking on spatially periodic solutions to differential equations. Part II considers spatially periodic solutions in two dimensions that are supported by the square or hexagonal lattices. The methods of Part I are applied to investigate how the translation free solutions, supported by these lattices, are altered when the perturbation term possesses certain symmetries. This leads to a partial classification theorem, describing the behaviour of these solutions. This classification is extended in Part III to three-dimensional solutions. In particular, the cubic lattices: simple, face centred, and body centred cubic, are considered. The analysis follows the same lines as Part II, but is necessarily more complex. This complexity is also present in the results, there are much richer dynamical possibilities. Parts II and III lead to a partial classification of the behaviour of spatially periodic solutions to differential equations in two and three dimensions. Finally in Part IV the results of Part III, concerning the body centred cubic lattice, are applied to the black-eye Turing instability. In particular, the model of Gomes [39] is cast in a new light where forced symmetry breaking is present, leading to several qualitative predictions. Nonlinear optical systems and the Polyacrylamide-Methylene Blue-Oxygen reaction are also discussed

Symbolic Automation and Numerical Synthesis for Robot Kinematics

This research analyzes three topics in robot arm kinematics. First, the direct kinematics which determines the Cartesian position and orientation of the end effector for the specified values of joint parameters is analyzed. Second, the differential motions concerning the differential relationships between the command variables in position and orientation of the end effector and the joint-controlled variables are studied. Finally, the inverse kinematics which determines the joint variables for a specified Cartesian position and orientation of the end effector is considered. This dissertation presents a methodology for incorporating the artificial intelligence types of knowledge into automating solutions for the direct kinematics problem and the manipulator Jacobian matrix. Furthermore, the dissertation utilizes the backward recursive techniques, the trigonometric identity rules, and a set of heuristic rules for implementing this methodology. To expedite computation efforts, a new algorithm is developed to obtain a differential relationship of a robotic manipulator via the vector kinematics method. Moreover, the speed control model for general robotic manipulators, together with the inverse Jacobian regarding cases of under-determined and over-determined of joint-controlled variables, are also discussed. Three mathematical approaches are proposed for solving the inverse kinematics problem: the inverse homogeneous transformation matrices approach, the geometric approach, and the arm-wrist partitioned synthesis approach. The first two approaches yield the symbolic closed-form solutions; the last approach, based on the iterative technique, provides a maximum of 16 distinct solutions of joint motion variables for any given position and orientation of the end effector in the workspace