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1. Deep Analysis of Residue Constraints (DARC): identifying determinants of protein functional specificity

   https://doi.org/10.1038/s41598-019-55118-6  

   Tondnevis, Farzaneh; Dudenhausen, Elizabeth E.; Miller, Andrew M.; McKenna, Robert; Altschul, Stephen F.; Bloom, Linda B.; Neuwald, Andrew F. (February 2020, Scientific Reports)

   Abstract Protein functional constraints are manifest as superfamily and functional-subgroup conserved residues, and as pairwise correlations. Deep Analysis of Residue Constraints (DARC) aids the visualization of these constraints, characterizes how they correlate with each other and with structure, and estimates statistical significance. This can identify determinants of protein functional specificity, as we illustrate for bacterial DNA clamp loader ATPases. These load ring-shaped sliding clamps onto DNA to keep polymerase attached during replication and contain one δ, three γ, and one δ’ AAA+ subunits semi-circularly arranged in the order δ-γ₁-γ₂-γ₃-δ’. Only γ is active, though both γ and δ’ functionally influence an adjacent γ subunit. DARC identifies, as functionally-congruent features linking allosterically the ATP, DNA, and clamp binding sites: residues distinctive of γ and of γ/δ’ that mutually interact in trans, centered on the catalytic base; several γ/δ’-residues and six γ/δ’-covariant residue pairs within the DNA binding N-termini of helices α2 and α3; and γ/δ’-residues associated with the α2 C-terminus and the clamp-binding loop. Most notable is a trans-acting γ/δ’ hydroxyl group that 99% of other AAA+ proteins lack. Mutation of this hydroxyl to a methyl group impedes clamp binding and opening, DNA binding, and ATP hydrolysis—implying a remarkably clamp-loader-specific function. 

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2. The Complexity of Approximating the Matching Polynomial in the Complex Plane

   https://doi.org/10.1145/3448645  

   Bezáková, Ivona; Galanis, Andreas; Goldberg, Leslie Ann; Štefankovič, Daniel (June 2021, ACM Transactions on Computation Theory)

   We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function. 

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3. Coarse models of homogeneous spaces and translations-like actions

   D. B. McReynolds, Mark Pengitore (January 2019, Preprint)

   For finitely generated groups G and H equipped with word metrics, a translation-like action of H on G is a free action where each element of H moves elements of G a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group G that is not isogenous to SL(2,ℝ) admit translation-like actions by ℤ2. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical N of the Borel subgroup AN of G acts translation-like on any cocompact lattice in G. We also prove that for noncompact simple Lie groups G,H with H 

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4. Testing effects of Lorentz invariance violation in the propagation of astroparticles with the Pierre Auger Observatory

   https://doi.org/10.1088/1475-7516/2022/01/023  

   Abreu, P.; Aglietta, M.; Albury, J.M.; Allekotte, I.; Almeida Cheminant, K.; Almela, A.; Alvarez-Muñiz, J.; Alves Batista, R.; Anastasi, G.A.; Anchordoqui, L.; et al (January 2022, Journal of Cosmology and Astroparticle Physics)

   Abstract Lorentz invariance violation (LIV) is often described by dispersion relations of the form E i 2  =  m i 2 + p i 2 +δ i,n E 2+n with delta different based on particle type i , with energy E , momentum p and rest mass m . Kinematics and energy thresholds of interactions are modified once the LIV terms become comparable to the squared masses of the particles involved. Thus, the strongest constraints on the LIV coefficients δ i,n tend to come from the highest energies. At sufficiently high energies, photons produced by cosmic ray interactions as they propagate through the Universe could be subluminal and unattenuated over cosmological distances. Cosmic ray interactions can also be modified and lead to detectable fingerprints in the energy spectrum and mass composition observed on Earth. The data collected at the Pierre Auger Observatory are therefore possibly sensitive to both the electromagnetic and hadronic sectors of LIV. In this article, we explore these two sectors by comparing the energy spectrum and the composition of cosmic rays and the upper limits on the photon flux from the Pierre Auger Observatory with simulations including LIV. Constraints on LIV parameters depend strongly on the mass composition of cosmic rays at the highest energies. For the electromagnetic sector, while no constraints can be obtained in the absence of protons beyond 10 19 eV, we obtain δ γ,0  > -10 -21 , δ γ,1  > -10 -40 eV -1 and δ γ,2  > -10 -58 eV -2 in the case of a subdominant proton component up to 10 20 eV. For the hadronic sector, we study the best description of the data as a function of LIV coefficients and we derive constraints in the hadronic sector such as δ had,0  < 10 -19 , δ had,1  < 10 -38 eV -1 and δ had,2  < 10 -57 eV -2 at 5σ CL. 

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5. Dimension bounds for escape on average in homogeneous spaces

   https://doi.org/10.3934/dcds.2024141  

   Kleinbock, Dmitry; Mirzadeh, Shahriar (January 2025, Discrete and Continuous Dynamical Systems)

   Let X = G/Γ, where G is a Lie group and Γ is a uniform lattice in G, and let O be an open subset of X. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape O on average with frequency δ, where 0 < δ ≤ 1. 

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