Estimating the asymptotics of solid partitions

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $\lim_{n\rightarrow\infty} n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001$. This shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $\log p_3(n)$. In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to $n^{1/4}$, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.Comment: 21 pages, 8 figure

Energy required to pinch a DNA plectoneme

DNA supercoiling plays an important role on a biological point of view. One of its consequences at the supra-molecular level is the formation of DNA superhelices named plectonemes. Normally separated by a distance on the order of 10 nm, the two opposite double-strands of a DNA plectoneme must be brought closer if a protein or protein complex implicated in genetic regulation is to be bound simultaneously to both strands, as if the plectoneme was locally pinched. We propose an analytic calculation of the energetic barrier, of elastic nature, required to bring closer the two loci situated on the opposed double-strands. We examine how this energy barrier scales with the DNA supercoiling. For physically relevant values of elastic parameters and of supercoiling density, we show that the energy barrier is in the $k_{\rm B} T$ range under physiological conditions, thus demonstrating that the limiting step to loci encounter is more likely the preceding plectoneme slithering bringing the two loci side by side.Comment: Published version (new title to conform to editorial policy

DNA denaturation bubbles: free-energy landscape and nucleation/closure rates

The issue of the nucleation and slow closure mechanisms of non superhelical stress-induced denaturation bubbles in DNA is tackled using coarse-grained MetaDynamics and Brownian simulations. A minimal mesoscopic model is used where the double helix is made of two interacting bead-spring rotating strands with a prescribed torsional modulus in the duplex state. We demonstrate that timescales for the nucleation (resp. closure) of an approximately 10 base-pair bubble, in agreement with experiments, are associated with the crossing of a free-energy barrier of $22~k_{\rm B}T$ (resp. $13~k_{\rm B}T$) at room temperature $T$. MetaDynamics allows us to reconstruct accurately the free-energy landscape, to show that the free-energy barriers come from the difference in torsional energy between the bubble and duplex states, and thus to highlight the limiting step, a collective twisting, that controls the nucleation/closure mechanism, and to access opening time scales on the millisecond range. Contrary to small breathing bubbles, these more than 4~base-pair bubbles are of biological relevance, for example when a preexisting state of denaturation is required by specific DNA-binding proteins.Comment: 11 pages (5 pages and Appendix), 13 figures, published in Journal of Chemical Physic

Mixed lipid bilayers with locally varying spontaneous curvature and bending

A model of lipid bilayers made of a mixture of two lipids with different average compositions on both leaflets, is developed. A Landau hamiltonian describing the lipid-lipid interactions on each leaflet, with two lipidic fields $\psi_1$ and $\psi_2$, is coupled to a Helfrich one, accounting for the membrane elasticity, via both a local spontaneous curvature, which varies as $C_0+C_1(\psi_1-\psi_2)/2$, and a bending modulus equal to $\kappa_0+\kappa_1(\psi_1+\psi_2)/2$. This model allows us to define curved patches as membrane domains where the asymmetry in composition, $\psi_1-\psi_2$, is large, and thick and stiff patches where $\psi_1+\psi_2$ is large. These thick patches are good candidates for being lipidic rafts, as observed in cell membranes, which are composed primarily of saturated lipids forming a liquid-ordered domain and are known to be thick and flat nano-domains. The lipid-lipid structure factors and correlation functions are computed for globally spherical membranes and planar ones. Phase diagrams are established, within a Gaussian approximation, showing the occurrence of two types of Structure Disordered phases, with correlations between either curved or thick patches, and an Ordered phase, corresponding to the divergence of the structure factor at a finite wave vector. The varying bending modulus plays a central role for curved membranes, where the driving force $\kappa_1C_0^2$ is balanced by the line tension, to form raft domains of size ranging from 10 to 100~nm. For planar membranes, raft domains emerge via the cross-correlation with curved domains. A global picture emerges from curvature-induced mechanisms, described in the literature for planar membranes, to coupled curvature- and bending-induced mechanisms in curved membranes forming a closed vesicle