Dynamic behavior of stochastic gene expression models in the presence of bursting

This paper considers the behavior of discrete and continuous mathematical models for gene expression in the presence of transcriptional/translational bursting. We treat this problem in generality with respect to the distribution of the burst size as well as the frequency of bursting, and our results are applicable to both inducible and repressible expression patterns in prokaryotes and eukaryotes. We have given numerous examples of the applicability of our results, especially in the experimentally observed situation that burst size is geometrically or exponentially distributed.Comment: 22 page

Ornstein-Zernike equation and Percus-Yevick theory for molecular crystals

We derive the Ornstein-Zernike equation for molecular crystals of axially symmetric particles and apply the Percus-Yevick approximation to this system. The one-particle orientational distribution function has a nontrivial dependence on the orientation and is needed as an input. Despite some differences, the Ornstein-Zernike equation for molecular crystals has a similar structure as for liquids. We solve both equations for hard ellipsoids on a sc lattice. Compared to molecular liquids, the tensorial orientational correlators exhibit less structure. However, depending on the lengths a and b of the rotation axis and the perpendicular axes of the ellipsoids, different behavior is found. For oblate and prolate ellipsoids with b >= 0.35 (units of the lattice constant), damped oscillations in distinct directions of direct space occur for some correlators. They manifest themselves in some correlators in reciprocal space as a maximum at the Brillouin zone edge, accompanied by maxima at the zone center for other correlators. The oscillations indicate alternating orientational fluctuations, while the maxima at the zone center originate from nematic-like orientational fluctuations. For a <= 2.5 and b <= 0.35, the oscillations are weaker. For a >= 3.0 and b <= 0.35, no oscillations occur any longer. For many of the correlators in reciprocal space, an increase of a at fixed b leads to a divergence at the zone center q = 0, consistent with nematic-like long range fluctuations, and for some oblate and prolate systems with b ~< 1.0 a simultaneous tendency to divergence of few other correlators at the zone edge is observed. Comparison with correlators from MC simulations shows satisfactory agreement. We also obtain a phase boundary for order-disorder transitions.Comment: 20 pages, 13 figures, submitted to Phys. Rev.

Microscopic theory of glassy dynamics and glass transition for molecular crystals

We derive a microscopic equation of motion for the dynamical orientational correlators of molecular crystals. Our approach is based upon mode coupling theory. Compared to liquids we find four main differences: (i) the memory kernel contains Umklapp processes, (ii) besides the static two-molecule orientational correlators one also needs the static one-molecule orientational density as an input, where the latter is nontrivial, (iii) the static orientational current density correlator does contribute an anisotropic, inertia-independent part to the memory kernel, (iv) if the molecules are assumed to be fixed on a rigid lattice, the tensorial orientational correlators and the memory kernel have vanishing l,l'=0 components. The resulting mode coupling equations are solved for hard ellipsoids of revolution on a rigid sc-lattice. Using the static orientational correlators from Percus-Yevick theory we find an ideal glass transition generated due to precursors of orientational order which depend on X and p, the aspect ratio and packing fraction of the ellipsoids. The glass formation of oblate ellipsoids is enhanced compared to that for prolate ones. For oblate ellipsoids with X <~ 0.7 and prolate ellipsoids with X >~ 4, the critical diagonal nonergodicity parameters in reciprocal space exhibit more or less sharp maxima at the zone center with very small values elsewhere, while for prolate ellipsoids with 2 <~ X <~ 2.5 we have maxima at the zone edge. The off-diagonal nonergodicity parameters are not restricted to positive values and show similar behavior. For 0.7 <~ X <~ 2, no glass transition is found. In the glass phase, the nonergodicity parameters show a pronounced q-dependence.Comment: 17 pages, 12 figures, accepted at Phys. Rev. E. v4 is almost identical to the final paper version. It includes, compared to former versions v2/v3, no new physical content, but only some corrected formulas in the appendices and corrected typos in text. In comparison to version v1, in v2-v4 some new results have been included and text has been change

Evaluating Signs of Determinants Using Single-Precision Arithmetic

We propose a method of evaluating signs of 2×2 and 3×3 determinants with b-bit integer entries using only b and (b + 1)-bit arithmetic, respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and compared with other exact computation methods

A molecular dynamics simulation for a 2d odic phase

The importance of the translation-rotation coupling effect on the dynamics of some ODIC crystal is shown through the study of a two dimensional model which can be viewed as a schematisation of real alkali cyanide. The molecular dynamics simulation of this model enabled us to study namely the structural aspects of the disorder, the individual rotational dynamics (librations and reorientations) and the influence of translation-rotation coupling effect on the collective excitations (phonon lattice mode and reorientational processes) of the crystal

Orientational disorder in plastic molecular crystals II. — Neutron scattering and orientational dynamics

In this paper, it is shown how the formalism of the canonical symmetry adapted functions can be used to analyse neutron scattering data related to orientationally disordered molecular crystals when the rotational dynamics of the molecule is assumed to be independent of the lattice dynamics. In this approximation, the rotational contribution to the intermediate scattering function is expanded in terms of correlations between the symmetry adapted functions of the molecular orientations. The site and molecular symmetry are taken into account to derive the complete set of independent correlation functions and to find which of them can be measured through neutron coherent and incoherent scattering experiments performed on single crystals or powder sample. Besides, it is shown that the orientational probability density function can be fully determined from the elastic part of incoherent neutron scattering.La diffusion des neutrons par les cristaux moléculaires en phase désordonnée est analysée à l'aide d'une base canonique de fonctions adaptée à la symétrie du site et des molécules. Les mouvements rotationnels des molécules sont supposés indépendants de la dynamique du réseau cristallin. Dans cette approximation, la contribution rotationnelle aux spectres de neutrons est explicitée en termes de corrélations des fonctions canoniques des orientations moléculaires. En tenant compte de la symétrie du site et de la molécule, on précise quelles sont les fonctions de corrélations indépendantes, et lesquelles peuvent être mesurées par des expériences de diffusion cohérente ou incohérente de neutrons effectuées sur monocristaux ou sur poudres. En outre, on montre que la densité de probabilité des orientations moléculaires peut être déterminée complètement à partir de la partie élastique des spectres de diffusion incohérente

Orientational disorder in plastic molecular crystals I. — Group theory and ODIC description

A group theory method for deriving a complete and non redundant canonical set of basis functions adapted to the description of Orientational Disorder In (plastic molecular) Crystal (ODIC) is developed. The method takes full advantage of the properties of the site and molecular symmetry groups. In particular, it is shown that, when both groups contain improper rotations, the canonical basis includes functions which have not been previously considered.Cet article présente une méthode de théorie des groupes pour construire une base canonique (complète et non redondante) de fonctions adaptée à la description du désordre orientationnel dans les cristaux moléculaires plastiques (ODIC). Cette méthode prend complètement en compte les propriétés des groupes de symétrie du site et de la molécule. En particulier, on montre que lorsque les deux groupes contiennent des rotations impropres, la base canonique comporte certaines fonctions qui jusqu'alors n'ont pas été prises en considération

Orientational disorder in plastic molecular crystals - III. — Infrared and Raman spectroscopy of internal modes

The formalism of symmetry adapted functions for molecular orientations introduced previously [5, 6] is here applied to the analysis of the Raman and infrared spectroscopy of internal modes in orientationally disordered molecular crystals. In some favorable cases, the contribution to the Raman and infrared lineshapes arising from the rotational dynamics of the molecules can be disentangled from other contributions arising from the vibrational lifetime or various coupling effects. Here, the rotational lineshapes are analysed in terms of independent, symmetry adapted rotational self-correlation functions. Furthermore, it is shown that the integrated intensity of internal modes provides a measurement of the first symmetry independent coefficients in the development of the orientational probability density function.La spectroscopie Raman et infrarouge des modes internes dans les cristaux moléculaires à désordre d'orientation est analysée dans le formalisme, précédemment développé [5, 6], des fonctions de base adaptées aux symétries du site et de la molécule. Dans certains cas la contribution rotationnelle aux profils Raman et infrarouge des modes internes peut être déconvoluée de la largeur purement vibrationnelle ou d'autres effets dus à des couplages variés. La partie purement rotationnelle du profil spectral est ici analysée à l'aide d'un petit nombre de fonctions d'autocorrélations rotationnelles indépendantes du point de vue de la symétrie. On montre de plus que l'intensité intégrée des raies internes permet de déterminer expérimentalement les premiers coefficients indépendants de la densité de probabilité d'orientation des molécules