Grand-canonical simulation of DNA condensation with two salts, affect of divalent counterion size

The problem of DNA$-$DNA interaction mediated by divalent counterions is studied using a generalized Grand-canonical Monte-Carlo simulation for a system of two salts. The effect of the divalent counterion size on the condensation behavior of the DNA bundle is investigated. Experimentally, it is known that multivalent counterions have strong effect on the DNA condensation phenomenon. While tri- and tetra-valent counterions are shown to easily condense free DNA molecules in solution into toroidal bundles, the situation with divalent counterions are not as clear cut. Some divalent counterions like Mg$^{+2}$ are not able to condense free DNA molecules in solution, while some like Mn$^{+2}$ can condense them into disorder bundles. In restricted environment such as in two dimensional system or inside viral capsid, Mg$^{+2}$ can have strong effect and able to condense them, but the condensation varies qualitatively with different system, different coions. It has been suggested that divalent counterions can induce attraction between DNA molecules but the strength of the attraction is not strong enough to condense free DNA in solution. However, if the configuration entropy of DNA is restricted, these attractions are enough to cause appreciable effects. The variations among different divalent salts might be due to the hydration effect of the divalent counterions. In this paper, we try to understand this variation using a very simple parameter, the size of the divalent counterions. We investigate how divalent counterions with different sizes can leads to varying qualitative behavior of DNA condensation in restricted environments. Additionally a Grand canonical Monte-Carlo method for simulation of systems with two different salts is presented in detail.Comment: Final revision, published online at J. Chem. Phys. arXiv admin note: text overlap with arXiv:0912.359

The inviscid limit of Navier-Stokes equations for vortex-wave data on R2\mathbb{R}^2

We establish the inviscid limit of the incompressible Navier-Stokes equations on the whole plane $\mathbb{R}^2$ for initial data having vorticity as a superposition of point vortices and a regular component. In particular, this rigorously justifies the vortex-wave system from the physical Navier-Stokes flows in the vanishing viscosity limit, a model that was introduced by Marchioro and Pulvirenti in the early 90s to describe the dynamics of point vortices in a regular ambient vorticity background. The proof rests on the previous analysis of Gallay in his derivation of the vortex-point system.Comment: 27 page

The inviscid limit of Navier-Stokes for analytic data on the half-space

In their classical work Caflisch and Sammartino proved the inviscid limit of the incompressible Navier-Stokes equations for well-prepared data with analytic regularity in the half-space. Their proof is based on the detailed construction of Prandtl's boundary layer asymptotic expansions. In this paper, we give a direct proof of the inviscid limit for general analytic data without having to construct Prandtl's boundary layer correctors. Our analysis makes use of the boundary vorticity formulation and the abstract Cauchy-Kovalevskaya theorem on analytic boundary layer function spaces that capture unbounded vorticity.Comment: to appear on Archive for Rational Mechanics and Analysi

Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers

This is a continuation and completion of the program (initiated in \cite{GrN1,GrN2}) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a generic stationary boundary layer profile. This is done via a spectral analysis approach and a careful study of the Orr-Sommerfeld equations, or equivalently the Navier-Stokes resolvent operator $(\lambda - L)^{-1}$. The earlier work (\cite{GrN1,GrN2}) treats the Orr-Sommerfeld equations away from critical layers: this is the case when the phase velocity is away from the range of the background profile or when $\lambda$ is away from the Euler continuous spectrum. In this paper, we study the critical case: the Orr-Sommerfeld equations near critical layers, providing pointwise estimates on the Green function as well as carefully studying the Dunford's contour integral near the critical layers. As an application, we obtain pointwise estimates on the Green function and sharp bounds on the semigroup of the linearized Navier-Stokes problem near monotonic boundary layers that are spectrally stable to the Euler equations, complementing \cite{GrN1,GrN2} where unstable profiles are considered.Comment: 84 page

Sharp bounds for the resolvent of linearized Navier Stokes equations in the half space around a shear profile

In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary. The novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticity that are uniform in the inviscid limit.Comment: this greatly revised and shortened the previous versio

Green function of Orr Sommerfeld equations away from critical layers

The classical Orr-Sommerfeld equations are the resolvent equations of the linearized Navier Stokes equations around a stationary shear layer profile in the half plane. In this paper, we derive pointwise bounds on the Green function of the Orr Sommerfeld problem away from its critical layers.Comment: title changed, Green function construction greatly simplified. To appear SIM

On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows

In this paper, we study Prandtl's boundary layer asymptotic expansion for incompressible fluids on the half-space in the inviscid limit. In \cite{Gr1}, E. Grenier proved that Prandtl's Ansatz is false for data with Sobolev regularity near Rayleigh's unstable shear flows. In this paper, we show that this Ansatz is also false for Rayleigh's stable shear flows. Namely we construct unstable solutions near arbitrary stable monotonic boundary layer profiles. Such shear flows are stable for Euler equations, but not for Navier-Stokes equations: adding a small viscosity destabilizes the flow

Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate

This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier-Stokes flows. The stationary flows, with small viscosity, are considered on $[0,L]\times \mathbb{R}_{+}$, assuming a no-slip boundary condition over a moving plate at $y=0$. We establish the validity of the Prandtl boundary layer expansion and its error estimates.Comment: 46 pages, no figur

A model of HIV budding and self-assembly, role of cell membrane

Budding from the plasma membrane of the host cell is an indispensable step in the life cycle of the Human Immunodeficiency Virus (HIV), which belongs to a large family of enveloped RNA viruses, retroviruses. Unlike regular enveloped viruses, retrovirus budding happens {\em concurrently} with the self-assembly of retrovirus protein subunits (Gags) into spherical virus capsids on the cell membrane. Led by this unique budding and assembly mechanism, we study the free energy profile of retrovirus budding, taking into account of the Gag-Gag attraction energy and the membrane elastic energy. We find that if the Gag-Gag attraction is strong, budding always proceeds to completion. During early stage of budding, the zenith angle of partial budded capsids, $\alpha$, increases with time as $\alpha \propto t^{1/3}$. However, when Gag-Gag attraction is weak, a metastable state of partial budding appears. The zenith angle of these partially spherical capsids is given by $\alpha_0\simeq(\tau^2/\kappa\sigma)^{1/4}$ in a linear approximation, where $\kappa$ and $\sigma$ are the bending modulus and the surface tension of the membrane, and $\tau$ is a line tension of the capsid proportional to the strength of Gag-Gag attraction. Numerically, we find $\alpha_0<0.3\pi$ without any approximations. Using experimental parameters, we show that HIV budding and assembly always proceed to completion in normal biological conditions. On the other hand, by changing Gag-Gag interaction strength or membrane rigidity, it is relatively easy to tune it back and forth between complete budding and partial budding. Our model agrees reasonably well with experiments observing partial budding of retroviruses including HIV

L∞L^\infty instability of Prandtl layers

In $1904$, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$ in the inviscid limit. In this paper we prove that, for a class of smooth solutions of Navier Stokes equations, namely for shear layer profiles which are unstable for Rayleigh equations, this Ansatz is false if we consider solutions with Sobolev regularity, in strong contrast with the analytic case, pioneered by R.E. Caflisch and M. Sammartino \cite{SammartinoCaflisch1,SammartinoCaflisch2}. Meanwhile we address the classical problem of the nonlinear stability of shear layers near a boundary and prove that if a shear flow is spectrally unstable for Euler equations, then it is non linearly unstable for the Navier Stokes equations provided the viscosity is small enough.Comment: revised version, removing order one forcing and including time-dependent boundary layers. Annals of PDE, to appea