Discrete Models of Biochemical Networks: The Toric Variety of Nested Canalyzing Functions

Author(s):  
Abdul S. Jarrah ◽  
Reinhard Laubenbacher
2021 ◽  
Vol 12 (05) ◽  
pp. 449-469
Author(s):  
Samaneh Gholami ◽  
Silvana Ilie

Author(s):  
Marek Capinski ◽  
Ekkehard Kopp

Author(s):  
Shiwei Wang ◽  
Anton Chavez ◽  
Simil Thomas ◽  
Hong Li ◽  
Nathan C. Flanders ◽  
...  

This work reports on the assembly of imine-linked macrocycles that serve as models of two-dimensional covalent organic frameworks (2D COFs). Interlayer interactions play an important role in the formation of 2D COFs, yet the effect of monomer structure on COF formation, crystallinity, and susceptibility to exfoliation are not well understood. For example, monomers with both electron-rich and electron-poor π-electron systems have been proposed to strengthen interlayer inter-actions and improve crystallinity. Here we probe these effects by studying the stacking behavior of imine-linked macrocycles that represent discrete models of 2D COFs. <div><br></div><div>Specifically, macrocycles based on terephthaldehyde (PDA) or 2,5-dimethoxyterephthaldehyde (DMPDA) stack upon cooling molecularly dissolved solutions. Both macrocycles assemble cooperatively with similar ΔHe values of -97 kJ/mol and -101 kJ/mol, respectively, although the DMPDA macrocycle assembly process showed a more straightforward temperature dependence. Circular dichroism spectroscopy performed on macrocycles bearing chiral side chains revealed a helix reversion process for the PDA macrocycles that was not observed for the DMPDA macrocycles. <br></div><div><br></div><div>Given the structural similarity of these monomers, these findings demonstrate that the stacking processes associated with nanotubes derived from these macrocycles, as well as for the corresponding COFs, are complex and susceptible to kinetic traps, casting doubt on the relevance of thermodynamic arguments for improving materials quality. <br></div>


Author(s):  
Ugo Bruzzo ◽  
William D. Montoya

AbstractFor a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ i ∗ : H p ( P Σ ) → H p ( X ) induced by the inclusion is injective for $$p=\dim X$$ p = dim X and an isomorphism for $$p<\dim X-1$$ p < dim X - 1 . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ NL β as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ β such that $$i^*$$ i ∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ dim P Σ = 2 k + 1 and $$k\beta -\beta _0=n\eta $$ k β - β 0 = n η , $$n\in \mathbb {N}$$ n ∈ N , where $$\eta $$ η is the class of a 0-regular ample divisor, and $$\beta _0$$ β 0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ β satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ n + 1 ⩽ codim Z ⩽ h k - 1 , k + 1 ( X ) .


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