Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line

Recently, Mori [Phys. Rev. E 84, 031128 (2011)] has conjectured that the free energy of Ising spin glass models with the Kac potential in the non-additive limit, such as the power-law potential in the non-additive regime, is exactly equal to that of the Sherrington-Kirkpatrick model in the thermodynamic limit. In this study, we prove that his conjecture is true on the Nishimori line at any temperature in any dimension. One of the key ingredients of the proof is the use of the Gibbs-Bogoliubov inequality on the Nishimori line. We also consider the case in which the probability distribution of the interaction is symmetric, where his conjecture is true at any temperature in one dimension but is an open problem in the low-temperature regime in two or more dimensions.Comment: 12pages, 0 figur

Exact solution of free entropy for matrix-valued geometric Brownian motion with non-commutative matrices via the replica method

Geometric Brownian motion (GBM) is a standard model in stochastic differential equations. In this study, we consider a matrix-valued GBM with non-commutative matrices. Introduction of non-commutative matrices into the matrix-valued GBM makes it difficult to obtain an exact solution because the existence of noise terms prevents diagonalization. However, we show that the replica method enables us to overcome this difficulty. We map the time evolution operator of the matrix-valued GBM with non-commutative matrices into the partition function of the isotropic Lipkin-Meshkov-Glick model used in quantum spin systems. Then, solving the eigenvalue problem of the isotropic Lipkin-Meshkov-Glick model, we obtain an analytical expression of the free entropy. Numerical simulation is consistent with our analytical result. Thus, our expression is the exact solution of the free entropy for the matrix-valued GBM with non-commutative matrices.Comment: 9 pages, 4 figure

Gibbs-Bogoliubov inequality on Nishimori line

The Gibbs-Bogoliubov inequality states that the free energy of a system is always lower than that calculated by a trial function. In this study, we show that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori line for Ising spin-glass models with Gaussian randomness. Our inequality states that the quenched free energy of a system is always lower than that calculated using a quenched trial function. The key component of the proof is the convexity of the pressure function $\mathbb{E}\left[\log Z_{} \right]$ with respect to the parameters along the Nishimori line, which differs from the conventional convexity with respect to the inverse temperature. When our inequality was applied to mean-field models, such as the Sherrington-Kirkpatrick model and $p$-spin model, the bound coincided with the replica-symmetric solution indicating that the equality holds.Comment: 9 pages, 0 figur

Replacement of the Catalytic Nucleophile Aspartyl Residue of Dextran Glucosidase by Cysteine Sulfinate Enhances Transglycosylation Activity

Dextran glucosidase from Streptococcus mutans (SmDG) catalyzes the hydrolysis of an α-1,6-glucosidic linkage at the nonreducing end of isomaltooligosaccharides and dextran. This enzyme has an Asp-194 catalytic nucleophile and two catalytically unrelated Cys residues, Cys-129 and Cys-532. Cys-free SmDG was constructed by replacement with Ser (C129S/C532S (2CS), the activity of which was the same as that of the wild type, SmDG). The nucleophile mutant of 2CS was generated by substitution of Asp-194 with Cys (D194C-2CS). The hydrolytic activity of D194C-2CS was 8.1 × 10⁻⁴ % of 2CS. KI-associated oxidation of D194C-2CS increased the activity up to 0.27% of 2CS, which was 330 times higher than D194C-2CS. Peptide-mapping mass analysis of the oxidized D194C-2CS (Ox-D194C-2CS) revealed that Cys-194 was converted into cysteine sulfinate. Ox-D194C-2CS and 2CS shared the same properties (optimum pH, pI, and substrate specificity), whereas Ox-D194C-2CS had much higher transglucosylation activity than 2CS. This is the first study indicating that a more acidic nucleophile (-SOO−) enhances transglycosylation. The introduction of cysteine sulfinate as a catalytic nucleophile could be a novel approach to enhance transglycosylation