Combinations of histone post-translational modifications

Histones are essential proteins that package the eukaryotic genome into its physiological state of nucleosomes, chromatin, and chromosomes. Post-translational modifications (PTMs) of histones are crucial to both the dynamic and persistent regulation of the genome. Histone PTMs store and convey complex signals about the state of the genome. This is often achieved by multiple variable PTM sites, occupied or unoccupied, on the same histone molecule or nucleosome functioning in concert. These mechanisms are supported by the structures of ‘readers’ that transduce the signal from the presence or absence of PTMs in specific cellular contexts. We provide background on PTMs and their complexes, review the known combinatorial function of PTMs, and assess the value and limitations of common approaches to measure combinatorial PTMs. This review serves as both a reference and a path forward to investigate combinatorial PTM functions, discover new synergies, and gather additional evidence supporting that combinations of histone PTMs are the central currency of chromatin-mediated regulation of the genome.

NEW COUNTING FORMULA FOR DOMINATING SETS IN PATH AND CYCLE GRAPHS

Let G=(V(G), E(G)) be a path or cycle graph. A subset D of V(G) is a dominating set of G if for every u element of V(G)\D, there exists v element of D such that uv element of E(G), that is, N[D]=V(G). The domination number of G, denoted by gamma(G), is the smallest cardinality of a dominating set of G. A set D_1 subset of V(G) is a set containing dominating vertices of degree 2, that is, each vertex is internally stable. A set D_2 subset of V(G) is a set containing dominating vertices where one of the element say a element of D_2, and the rest are of degree 2. A set D_3 subset of V(G) is a set containing dominating vertices in which two of the elements say b, c element of D_3, deg(b)=deg(c)=1. This paper developed a new combinatorial formula that determines the number of ways of putting a dominating set in a path and cycle graphs of order n>=1 and n>=3, respectively. Further, a combinatorial function P^1_G(n), P^2_G(n) and P^3_G(n) that determines the probability of getting the set D_1, D_2, and D_3, respectively in graph G of order n were constructed.

Top Coefficients of the Denumerant

International audience For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately. Considérons une liste $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ de $N+1$ entiers positifs. Le dénumérant $E(\alpha)(t)$ est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation $\sum^{N+1}_{i=1}x_i\alpha_i=t$, où $t$ varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de $t$, de degré $N$. Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé $k$ (mais $N$ n’est pas fixé, les $k+1$ plus hauts coefficients du quasi-polynôme $E(\alpha)(t)$ en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de $E(\alpha)(t)$. Les $k+1$ plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à $k$.

Markov's principle, isols and Dedekind finite sets

Markov's principle is more than a convenience in constructive arithmetic and analysis; it is absolutely essential to significant areas of constructive cardinal arithmetic. In turn, logical relations among intuitively appealing principles of constructive cardinal arithmetic parallel relations between MPS and other “problematic axioms” for constructive mathematics, such as the limited principle of omniscience. Finally, simple closure properties on the Dedekind finite sets provide ready examples of statements which are strictly weaker than Markov's principle and yet are independent of extensions of IZF.MPS, Markov's principle with variables over sets, is equivalent to each of these elementary properties of ⊿, the class of Dedekind finite sets:1. ∀A(A Є ⊿ ↔ CP(A)).2. [*]A Є ⊿ ↔ ∀B (A + B is infinite ↔ B is infinite).3. [**]A Є ⊿ ↔ ∀B((A + 1) x B is infinite ↔ B is infinite).CP is Tarski's cancellation property. Consequently, MPS is tantamount, in constructive mathematics, to the standard classical characterizations of ⊿ in terms of cardinality.[*] and [**] imply that ⊿ is closed under addition and multiplication. Closure under addition is, in turn, constructively equivalent to the closure of ⊿ under all combinatorial functions and to the fact that ⊿ is closed under each strict combinatorial function individually. Generally speaking, a function on P(ω) is combinatorial whenever it preserves finiteness, respects cardinality and has a “moduluslike” associate function. It is strict when it is strictly increasing relative to the subset ordering, [cf. §6 for precise definitions.] From this, we see that MPS is foundational for cardinal arithmetic on the constructive Dedekind finite sets.