Unambiguous determination of the ionization state of a glycoside hydrolase active site lysine by 1H-15N heteronuclear correlation spectroscopy

We have investigated the lysine side chain amines in the 34 kDa catalytic domain from Cellulomonas fimi beta-(1,4)-glycosidase Cex (or CfXyn10A) using 1H-detected 15N heteronuclear correlation NMR spectroscopy. Signals from the 1Hzeta ( approximately 8 ppm) and 15Nzeta ( approximately 35 ppm) of Lys302 in the unmodified enzyme and Lys47 in a trapped cellobiosyl-enzyme intermediate were detected in a 1H-15N HMQC spectrum (pH 6.5 and 30 degrees C). The amine of Lys302 forms a buried ion pair, and that of Lys47 is hydrogen bonded to the cellobioside. Both lysines are positively charged, as unambiguously demonstrated by the splitting of their 15Nzeta signals into quartets (|1JNH| approximately 75 Hz) in a 1H-15N HSQC spectrum recorded without 1H decoupling during 15N evolution. Qualitative insights into the dynamic properties of these lysines are also provided by the deviations of their quartet intensity ratios from that of approximately 3:1:1:3 expected for a highly mobile amine. On the basis of the observed ratios of approximately 1:1:1:1 for the quartet of Lys302 and approximately 0.5:1:1:0.5 for Lys47, the amine of the latter active site residue is most rigidly positioned. Signals from at least 8 and 10 additional positively charged, mobile amines in Cex were observed at 10 degrees C and pH 6.5 and 5.6, respectively. By using conditions of reduced temperature, slightly acidic pH, and low general base concentrations, as well as water flipback pulses to minimize the effects of hydrogen exchange, 1H-15N correlation experiments provide a sensitive route to directly investigate the charge states and dynamic properties of the N-terminal and side chain amines in proteins and protein complexes.     

Automerization reaction of cyclobutadiene and its barrier height: an ab initio benchmark multireference average-quadratic coupled cluster study

The problem of the double bond flipping interconversion of the two equivalent ground state structures of cyclobutadiene (CBD) is addressed at the multireference average-quadratic coupled cluster level of theory, which is capable of optimizing the structural parameters of the ground, transition, and excited states on an equal footing. The barrier height involving both the electronic and zero-point vibrational energy contributions is 6.3 kcal mol(-1), which is higher than the best earlier theoretical estimate of 4.0 kcal mol(-1). This result is confirmed by including into the reference space the orbitals of the CC sigma bonds beyond the standard pi orbital space. It places the present value into the middle of the range of the measured data (1.6-10 kcal mol(-1)). An adiabatic singlet-triplet energy gap of 7.4 kcal mol(-1) between the transition state (1)B(tg) and the first triplet (3)A(2g) state is obtained. A low barrier height for the CBD automerization and a small DeltaE((3)A(2g),(1)B(1g)) gap bear some relevance on the highly pronounced reactivity of CBD, which is briefly discussed.     

Thermodynamics and statistical mechanics of dense granular media

By detailed molecular dynamics and Monte Carlo simulations of a model system we show that granular materials at rest can be described as thermodynamics systems. First, we show that granular packs can be characterized by few parameters, as much as fluids or solids. Then, in a second independent step, we demonstrate that these states can be described in terms of equilibrium distributions which coincide with the statistical mechanics of powders first proposed by Edwards. We also derive the system equation of state as a function of the "configurational temperature," its new intensive thermodynamic parameter.     

Magic of centroids

In this paper, we show some properties of centroids of geometric figures, such as triangles, quadrilaterals and tetrahedra. In particular, we will prove the properties by means of geometric transformations and by introducing extensions of triangles and quadrilaterals, i.e. by adding one, two or three new vertices to the figure. The study of these properties can be used, with profit, in a classroom activity supported by a dynamic geometry system.     

A Neyman–Pearson problem with ambiguity and nonlinear pricing

We consider a problem of the Neyman–Pearson type arising in the theory of portfolio choice in the presence of probability weighting, such as in markets with Choquet pricing (as in Araujo et al. in Econ Theory 49(1):1–35, 2011; Cerreia-Vioglio et al. in J Econ Theory 157(1):730–762, 2015; Chateauneuf and Cornet in Submodular financial markets with frictions. Working Paper, 2015; Chateauneuf et al. in Math Finance 6(3):323–330, 1996) and ambiguous beliefs about the payoffs of contingent claims (see Gilboa and Marinacci, in: Acemoglu, Arellano, Dekel (eds) Advances in economics and econometrics: theory and applications, tenth world congress of the econometric society, Cambridge University Press, Cambridge, 2013). Specifically, we consider a problem of optimal choice of a contingent claim so as to minimize a non-linear pricing functional (or a distortion risk measure), subject to a minimum expected performance measure (or a minimum expected return or utility), where expectations with respect to distorted probabilities are taken in the sense of Choquet. Such contingent claims are called cost-efficient. We give an analytical characterization of cost-efficient contingent claims under very mild assumptions on the probability weighting functions, thereby extending some of the results of Ghossoub (Math Financ Econ 10(1):87–111, 2016), and we provide examples of some special cases of interest. In particular, we show how a cost-efficient contingent claim exhibits a desirable monotonicity property: It is anti-comonotonic with the random mark-to-market value (or return, etc.) of the underlying financial position, and it is hence a hedge against such variability.     

On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

In this article we study the validity of the Whitney \(C^1\) extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the \(C^1\) extension property. We conclude by showing that the \(C^1\) extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.     

Optimally ending an epidemic

A stochastic and controlled version of the classic three-dimensional Kermack–McKendrick model for the spread of epidemics is considered. The aim is to end the epidemic as soon as possible, taking the quadratic control costs into account. An exact and explicit solution is found in a particular case by making use of the method of similarity solutions to solve the partial differential equation satisfied by the value function, subject to the appropriate conditions.