Top on a smooth plane.

We investigate the dynamics of a sliding top that is a rigid body with an ideal sharp tip moving in a perfectly smooth horizontal plane, so no friction forces act on the body. We prove that this system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem. The cases with the constant gravity field with acceleration g≠0 and without external field g=0 are considered. The non-integrability proof for g≠0 is based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. We show that the integrability of the classical top is a necessary condition for the integrability of the sliding top. Among four integrable classical top cases, the corresponding two cases for the sliding top are also integrable, and for the two remaining cases, we prove their non-integrability by analyzing the differential Galois group of variational equations along a certain particular solution. In the absence of a constant gravitational field g=0, the integrability is much more difficult. First, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied the Ziglin theorem concerning the splitting of separatrices phenomenon. Then, we prove the non-integrability of the symmetric sliding top using the differential Galois group of variational equations except two the same as for g≠0 cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.

Destructive relativity.

The description of dynamics for high-energy particles requires an application of the special relativity theory framework, and analysis of properties of the corresponding equations of motion is very important. Here, we analyze Hamilton equations of motion in the limit of weak external field when potential satisfies the condition 2V(q)≪mc2. We formulate very strong necessary integrability conditions for the case when the potential is a homogeneous function of coordinates of integer non-zero degrees. If Hamilton equations are integrable in the Liouville sense, then eigenvalues of the scaled Hessian matrix γ-1V″(d) at any non-zero solution d of the algebraic system V'(d)=γd must be integer numbers of appropriate form depending on k. As it turns out, these conditions are much stronger than those for the corresponding non-relativistic Hamilton equations. According to our best knowledge, the obtained results are the first general integrability necessary conditions for relativistic systems. Moreover, a relation between the integrability of these systems and corresponding non-relativistic systems is discussed. The obtained integrability conditions are very easy to use because the calculations reduce to linear algebra. We show their strength in the example of Hamiltonian systems with two degrees of freedom with polynomial homogeneous potentials. It seems that the only integrable relativistic systems with such potentials are those depending only on one coordinate or having a radial form.

Modulation of hippocampal neuronal resilience during aging by the Hsp70/Hsp90 co-chaperone STI1.

Chaperone networks are dysregulated with aging, but whether compromised Hsp70/Hsp90 chaperone function disturbs neuronal resilience is unknown. Stress-inducible phosphoprotein 1 (STI1; STIP1; HOP) is a co-chaperone that simultaneously interacts with Hsp70 and Hsp90, but whose function in vivo remains poorly understood. We combined in-depth analysis of chaperone genes in human datasets, analysis of a neuronal cell line lacking STI1 and of a mouse line with a hypomorphic Stip1 allele to investigate the requirement for STI1 in aging. Our experiments revealed that dysfunctional STI1 activity compromised Hsp70/Hsp90 chaperone network and neuronal resilience. The levels of a set of Hsp90 co-chaperones and client proteins were selectively affected by reduced levels of STI1, suggesting that their stability depends on functional Hsp70/Hsp90 machinery. Analysis of human databases revealed a subset of co-chaperones, including STI1, whose loss of function is incompatible with life in mammals, albeit they are not essential in yeast. Importantly, mice expressing a hypomorphic STI1 allele presented spontaneous age-dependent hippocampal neurodegeneration and reduced hippocampal volume, with consequent spatial memory deficit. We suggest that impaired STI1 function compromises Hsp70/Hsp90 chaperone activity in mammals and can by itself cause age-dependent hippocampal neurodegeneration in mice. Cover Image for this issue: doi: 10.1111/jnc.14749.

Dynamics of constrained many body problems in constant curvature two-dimensional manifolds.

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Molecular basis for the interaction between stress-inducible phosphoprotein 1 (STIP1) and S100A1.

Stress-inducible phosphoprotein 1 (STIP1) is a cellular co-chaperone, which regulates heat-shock protein 70 (Hsp70) and Hsp90 activity during client protein folding. Members of the S100 family of dimeric calcium-binding proteins have been found to inhibit Hsp association with STIP1 through binding of STIP1 tetratricopeptide repeat (TPR) domains, possibly regulating the chaperone cycle. Here, we investigated the molecular basis of S100A1 binding to STIP1. We show that three S100A1 dimers associate with one molecule of STIP1 in a calcium-dependent manner. Isothermal titration calorimetry revealed that individual STIP1 TPR domains, TPR1, TPR2A and TPR2B, bind a single S100A1 dimer with significantly different affinities and that the TPR2B domain possesses the highest affinity for S100A1. S100A1 bound each TPR domain through a common binding interface composed of α-helices III and IV of each S100A1 subunit, which is only accessible following a large conformational change in S100A1 upon calcium binding. The TPR2B-binding site for S100A1 was predominately mapped to the C-terminal α-helix of TPR2B, where it is inserted into the hydrophobic cleft of an S100A1 dimer, suggesting a novel binding mechanism. Our data present the structural basis behind STIP1 and S100A1 complex formation, and provide novel insights into TPR module-containing proteins and S100 family member complexes.

Spectral and Photophysical Behavior of Cytisine in n-Hexane. Experimental Evidence for the S1(n,π*) → S0 Fluorescence.

Spectral and photophysical properties of (-)-cytisine (the compound used as a smoking cessation aid and a potential drug in Alzheimer's and Parkinson's diseases) were investigated. The two conformers of cytisine, whose presence in the S0 state has been earlier proved by the NMR and IR methods as well as in theoretical calculation, in nonpolar n-hexane show a rarely observed prompt fluorescence from the S1(n,π*) excited state. This observation is unambiguously evidenced by very small radiative rate constants of these two emitting conformers, kF = 7.4 × 105 and 3.0 × 105 s-1. Their lifetimes in the S1(n,π*) state are relatively long, τS1 = 1.9 and 6.7 ns; therefore, their fluorescence quantum yield is relatively high ϕF ∼ 10-3. The long-wavelength band in the cytisine absorption originates from the excitation to the S2(π,π*) state, while the S1(n,π*) state is not observed in this spectrum. Thus, the excited state S2(π,π*) is manifested only in the absorption spectrum, while the excited state S1(n,π*) is seen only in the fluorescence spectrum, so cytisine in n-hexane is characterized by close lying (n,π*) and (π,π*) excited singlet states.

Domains of STIP1 responsible for regulating PrPC-dependent amyloid-ß oligomer toxicity.

Soluble oligomers of amyloid-beta peptide (AßO) transmit neurotoxic signals through the cellular prion protein (PrP(C)) in Alzheimer's disease (AD). Secreted stress-inducible phosphoprotein 1 (STIP1), an Hsp70 and Hsp90 cochaperone, inhibits AßO binding to PrP(C) and protects neurons from AßO-induced cell death. Here, we investigated the molecular interactions between AßO and STIP1 binding to PrP(C) and their effect on neuronal cell death. We showed that residues located in a short region of PrP (90-110) mediate AßO binding and we narrowed the major interaction in this site to amino acids 91-100. In contrast, multiple binding sites on STIP1 (DP1, TPR1 and TPR2A) contribute to PrP binding. DP1 bound the N-terminal of PrP (residues 23-95), whereas TPR1 and TPR2A showed binding to the C-terminal of PrP (residues 90-231). Importantly, only TPR1 and TPR2A directly inhibit both AßO binding to PrP and cell death. Furthermore, our structural studies reveal that TPR1 and TPR2A bind to PrP through distinct regions. The TPR2A interface was shown to be much more extensive and to partially overlap with the Hsp90 binding site. Our data show the possibility of a PrP, STIP1 and Hsp90 ternary complex, which may influence AßO-mediated cell death.

Penning trap with an inclined magnetic field.

A modified Penning trap with a spatially uniform magnetic field B inclined with respect to the axis of rotational symmetry of the electrodes is considered. The inclination angle can be arbitrary. Canonical transformation of phase variables transforming the Hamiltonian of the considered system into a sum of three uncoupled harmonic oscillators is found. We determine the region of stability in space of two parameters controlling the dynamics: the trapping parameter κ and the squared sine of the inclination angle ϑ0. If the angle ϑ0 is smaller than 54°, a charge occupies a finite spatial volume within the processing chamber. A rigid hierarchy of trapping frequencies is broken if B is inclined at the critical angle: the magnetron frequency reaches the modified cyclotron frequency while the axial frequency exceeds them. Apart from this resonance, we reveal the family of resonant curves in the region of stability. In the relativistic regime, the system is not linear. We show that it is not integrable in the Liouville sense. The averaging over the fast variable allows to reduce the system to two degrees of freedom. An analysis of the Poincaré cross-sections of the averaged systems shows the regions of effective stability of the trap.

Dynamics of a relativistic charge in the Penning trap.

We are interested in the motion of a classical charge within a processing chamber of a Penning trap. We examine the relativistic Lagrangian and Hamiltonian dynamics without any approximations. We show that the radial and axial motions are non-linearly coupled to each other whenever the special relativity is taken into account. As the restoring quadruple potential has the axial symmetry, the dynamics of the system can be reduced to two degrees of freedom. If all the energy of a charge belongs to the axial oscillating mode, its time evolution is described by the nonlinear equation of motion for a simple pendulum. If the whole energy is accumulated in radial oscillating mode, the dynamical system resembles a double pendulum. We demonstrate that the Hamiltonian system is not integrable in the Liouville sense in the class of functions meromorphic in coordinates and momenta. Using Poincaré sections, we show that, in spite of the non-integrability, a large part of the phase space is filled by quasi-periodic solutions that encircle some periodic solutions. We determine numerically characteristic frequencies of these periodic solutions.

The prion protein ligand, stress-inducible phosphoprotein 1, regulates amyloid-ß oligomer toxicity.

In Alzheimer's disease (AD), soluble amyloid-ß oligomers (AßOs) trigger neurotoxic signaling, at least partially, via the cellular prion protein (PrP(C)). However, it is unknown whether other ligands of PrP(C) can regulate this potentially toxic interaction. Stress-inducible phosphoprotein 1 (STI1), an Hsp90 cochaperone secreted by astrocytes, binds to PrP(C) in the vicinity of the AßO binding site to protect neurons against toxic stimuli. Here, we investigated a potential role of STI1 in AßO toxicity. We confirmed the specific binding of AßOs and STI1 to the PrP and showed that STI1 efficiently inhibited AßO binding to PrP in vitro (IC50 of ∼70 nm) and also decreased AßO binding to cultured mouse primary hippocampal neurons. Treatment with STI1 prevented AßO-induced synaptic loss and neuronal death in mouse cultured neurons and long-term potentiation inhibition in mouse hippocampal slices. Interestingly, STI1-haploinsufficient neurons were more sensitive to AßO-induced cell death and could be rescued by treatment with recombinant STI1. Noteworthy, both AßO binding to PrP(C) and PrP(C)-dependent AßO toxicity were inhibited by TPR2A, the PrP(C)-interacting domain of STI1. Additionally, PrP(C)-STI1 engagement activated α7 nicotinic acetylcholine receptors, which participated in neuroprotection against AßO-induced toxicity. We found an age-dependent upregulation of cortical STI1 in the APPswe/PS1dE9 mouse model of AD and in the brains of AD-affected individuals, suggesting a compensatory response. Our findings reveal a previously unrecognized role of the PrP(C) ligand STI1 in protecting neurons in AD and suggest a novel pathway that may help to offset AßO-induced toxicity.

Deactivation of 6-aminocoumarin intramolecular charge transfer excited state through hydrogen bonding.

This paper presents results of the spectral (absorption and emission) and photophysical study of 6-aminocoumarin (6AC) in various aprotic hydrogen-bond forming solvents. It was established that solvent polarity as well as hydrogen-bonding ability influence solute properties. The hydrogen-bonding interactions between S1-electronic excited solute and solvent molecules were found to facilitate the nonradiative deactivation processes. The energy-gap dependence on radiationless deactivation in aprotic solvents was found to be similar to that in protic solvents.

An intermolecular hydrogen-bonding effect on spectral and photophysical properties of 6-aminocoumarin in protic solvents.

The paper presents results of the spectral (absorption and emission) and photophysical study of 6-aminocoumarin (6AC) in strong protic solvents with different hydrogen-bond acidity. The measurements of the fluorescence lifetime and quantum yield clearly show that in protic solvents intermolecular hydrogen-bonding interactions between solute and solvent molecules facilitate the nonradiative processes. It is suggested that in these solvents very efficient internal conversion due to intermolecular hydrogen-bonding is induced. A large isotope effect is found for 6AC in deuterated water.

Solute - solvent repulsion effects on the absorption spectra of anthracene in n-hexane investigated under high pressure.

The band positions in the UV-VIS absorption spectra of compressed solution of anthracene in n-hexane significantly depend not only on the dispersive but also on the repulsive solute-solvent interactions, what has so far been omitted. Their strength is determined not only by the solvent polarity but also by Onsager cavity radius changing with pressure. The results obtained for anthracene show that repulsive interactions should be included in the interpretation of barochromic and solvatochromic results of aromatic compounds. We show that the barochromic studies in the liquid solvent can be an alternative to solvatochromic studies, e.g. to determine the polarizability of organic molecules in the electronic excited state. The pressure-induced polarity change in n-hexane exceeds that induced by the exchange of n-alkane solvents between n-pentane and n-hexadecane.

Radiationless deactivation of 6-aminocoumarin from the S1-ICT state in nonspecifically interacting solvents.

This paper presents the results of a spectral (absorption and emission) and photophysical study of 6-aminocoumarin (6AC) in the solvents with which this molecule interacts only nonspecifically (n-alkanes, tetrachloromethane and 1-chloro-n-alkanes) and in nitriles. The strong effects of the solvents on the emission spectra, fluorescence quantum yield and lifetime of 6AC were observed. The results of the steady-state and time-resolved photophysical study suggest the presence of very fast nonradiative deactivation processes. It is concluded that besides fluorescence, the efficient S(1)-ICT → S(0) internal conversion in nonpolar aprotic solvents arises from vibronic interactions between close-lying S(1)-ICT(π,π*) and S(2)(n,π*) states. Moreover, unexpectedly efficient triplet state formation occurs. In nitriles the intermolecular hydrogen-bonding interactions with solvent molecules also facilitate the nonradiative decay process involving the S(1)-exciplex.

Hydrogen bond effects in the ground and excited singlet states of 4H-1-benzopyrane-4-thione in water--theory and experiment.

The hydrogen bond energies for 4H-1-benzopyrane-4-thione (BPT) in its ground and two lowest excited singlet electronic states have been determined using ab initio methods. It was shown that the BPT molecule can form, as an acceptor, four relatively strong hydrogen bonds with water molecules, leading to a stable complex in the ground electronic state S(0). The hydrogen bonds involving the sulfur atom from the thiocarbonyl group were found to be stronger than those involving the oxygen atom from the benzopyran moiety. The former hydrogen bonds were predicted to become significantly weaker upon excitation to the S(1) state and, in contrast, stronger upon excitation to the S(2) state. Calculated changes in the hydrogen bond energy due to the S(0)→ S(1) and S(0)→ S(2) excitation are in very good agreement with the experimental values obtained from the absorption solvatochromic study, according to a procedure proposed by us in [E. Krystkowiak, et al., J. Photochem. Photobiol. A: Chem., 2006, 184, 250]. The maxima of absorption spectra of the BPT-water hydrogen-bonded complex, calculated taking into consideration nonspecific solute-solvent interactions, are also in good agreement with the experimental results.

Influence of hydrogen bonds and nonspecific interactions on the spectral and photophysical properties of the excited singlet states of 4-aminophthalimide in amine solution.

The hydrogen-bond and nonspecific interaction energies for 4-aminophthalimide (4-AP), often used as a probe, in the ground electronic and excited singlet states are determined using ab initio computational methods. It is shown that the 4-AP molecule can form three relatively strong hydrogen bonds with trimethylamine (TMA) and triethylamine (TEA), which leads to the formation of S(0)-complexes between the solute and solvent molecules. Only two of the hydrogen bonds with the amine group of 4-AP change significantly their energies upon excitation and deactivation. The theoretical results are necessary to explain the spectral and unusual photophysical properties of 4-AP in amine solutions.

Mechanism and dynamics of intramolecular triplet state decay of 1-propyl-4-thiouracil and its α-methyl-substituted derivatives studied in perfluoro-1,3-dimethylcyclohexane.

The absorption, phosphorescence and phosphorescence excitation spectra, phosphorescence quantum yields, and T(1) excited state lifetimes of four 4-thiouracil derivatives were measured for the first time in chemically inert and very weakly interacting perfluoro-1,3-dimethylcyclohexane at room temperature. The set of the 4-thiouracil derivatives comprises 1-propyl-4-thiouracil (PTU) and the related compounds having a methyl substituent at the position α to the thiocarbonyl group, namely 1-propyl-4-thiothymine (PTT), 1,3-dimethyl-4-thiouracil (DMTU), and 1-methyl-3-trideuteriomethyl-4-thiouracil ([D(3)]DMTU). Quantitative information on the intramolecular decay of the T(1) excited state of the four 4-thiouracil derivatives is presented, and the mechanism and dynamics of this process are discussed. In the absence of self quenching and solvent induced deactivation, the T(1) decay of the four 4-thiouracil derivatives was dominated by intramolecular nonradiative processes (NR). The values of the rate constant k(NR) in DMTU and [D(3)]DMTU are about 4 times larger than that in PTT and about 3 times larger than that in PTU. The reasons for the enhanced nonradiative rate constant in DMTU are discussed. It is concluded that the faster rate of the nonradiative processes in DMTU is related to a larger contribution from mixing of the T(2) (nπ*) state into the lowest energy T(1) (ππ*) state, as compared to the analogous coupling in PTU and PTT. This conclusion is supported by ab initio calculations performed at the EOM-CC2/aug-cc-pVDZ level of theory. The energy spacing between the T(2) (nπ*) and T(1) (ππ*) states is estimated to be about 500, 1100, and 2000 cm(-1) for DMTU, PTU, and PTT, respectively. Among the three compounds in question, the predicted energy spacing is thus the smallest for DMTU.

Changes in energy of three types of hydrogen bonds upon excitation of aminocoumarins determined from absorption solvatochromic experiments.

Absorption spectra of 6-aminocoumarin (6AC) and 7-aminocoumarins (C120 and C151) were studied in polyfluorinated alcohols: (1,1,1,3,3,3-hexafluoroisopropanol (HFIP), 2,2,2-trifluoroethanol (TFE)), in water and in methanol, and compared to those taken in 1-chloro-n-alkanes. According to our results, the observed unusual blue-shift of a long-wavelength band in absorption spectra in strong protic solvents is direct evidence of significant weakening of a NH-O hydrogen bond. The results obtained for the aminocoumarins in HFIP, which in contrast to aliphatic alcohols does not form hydrogen bonds of the acceptor type, prove that the decrease in the energy of the NH-O hydrogen bond upon excitation to the lowest S(1)-LE state is significantly greater than the increase in the energy of hydrogen bonds made by the oxygen atom of carbonyl group OH-O. It is in contrast to theoretical calculations for C151 [Y. Liu, J. Ding, R. Liu, D. Shi and J. Sun, J. Photochem. Photobiol. A, 2009, 201, 203-207]. A comparison of the absorption spectra measured in DMSO and in 1-chloro-n-alkanes shows that the energy of two N-HO hydrogen bonds considerably increases as a result of excitation. These results are consistent with those of the theoretical calculations [Y. Liu, J. Ding, R. Liu, D. Shi and J. Sun, J. Photochem. Photobiol. A, 2009, 201, 203-207; P. Zhou, P. Song, J. Liu, K. Han and G. He, Phys. Chem. Chem. Phys., 2009, 11, 9440-9449]. In this study we applied the procedure proposed by us in J. Photochem. Photobiol. A, 2006, 184, 250-264 for the determination of changes in hydrogen bond energy as a result of electronic excitation based on analysis of the absorption spectra of the probe studied in the solvents interacting with it exclusively nonspecifically and in those forming hydrogen bonds with it.

Dynamics of dipole in a stationary non-homogeneous electromagnetic field.

The non-relativistic equations of motion for a dipole in a stationary non-homogeneous electromagnetic field are derived and analysed. It is shown that they are Hamiltonian with respect to a certain degenerated Poisson structure. Described by them dynamics is complex because the motion of the centre of mass of the dipole is coupled with its rotational motion. The problem of the existence of linear in momenta first integrals which can be useful for the separation of rotational motion is discussed. The presence of such first integral appears to be related with a linear symmetry of electric and magnetic fields. Also results of search of quadratic in momenta first integrals for uniform and stationary electromagnetic fields are reported. Deriving equations of motion of a dipole in arbitrary stationary electromagnetic fields and analysis of described by them dynamics is important for the construction of electromagnetic traps for polar particles.

Dynamics of a dipole in a stationary electromagnetic field.

The non-relativistic dynamics of an electric dipole in a uniform and stationary electromagnetic field is considered. The equations of motion are derived ab initio. It is shown that they are Hamiltonian with respect to a certain degenerated Poisson structure. The system has a 'hidden' symmetry which allows its dimension to be reduced. The reduced system is also Hamiltonian with respect to a degenerated Poisson structure. We show how to perform this reduction in the framework of the Lagrange formalism. Integrability of the reduced system is investigated. It was proved that the system is non-integrable except for two cases when, for specific values of parameters, the system admits an additional first integral.