The nature of phospholipid membranes repulsion at nm-distances

Why hydrophobic membranes repel at small distances? We apply recently developed phenomenological theory of polar liquids to calculate the repulsive pressure between two hydrophilic membranes at nm-distances. We find that the repulsion does show up in the model and the solution to the problem fits the published experimental data well both qualitatively and quantitatively. Moreover, we find that the repulsion is practically independent on temperature, and thus put some extra weight in favor of the so called hydration over entropic hypothesis for the membranes interactions explanation. The calculation is a good “proof of concept” example a continuous water model application to non-trivial interactions on nm-size bodies in water arising from long-range correlations between the water molecules.
More details can be found here:

arXiv:0908.0632 [pdf, other]

The nature of phospholipid membranes repulsion at nm-distances

Authors: P. O. Fedichev, L.I. Menshikov

Molecular polarization on a polar liquid interface: the structure of a water surface

The orientations of water molecules within the liquid depend on interplay of long-range dipole-dipole interaction and short range hydrogen-bonding interactions. At room temperature water is in paraelectric (disordered) phase and thus average dipole moment of any large enough fraction of the liquid vanishes.

This observation is not necessarily true at the liquid boundary. Since no molecules "like" to point at a hydrophobic interface direction (this would imply many uncompensated hydrogen bonds), most of the molecules orient along the liquid surface, the boundary between the liquid and a hydrophobic may become polarized (become essentially ferroelectric). Practically this amounts to a formation of stable network of hydrogen bonds on the interface.

The solution corresponding to such polarized boundary can be obtained in the very latest "incarnation" of the QUANTUM water model (See the attached figures). The first one demonstrates the density of the liquid starting from the gas phase on the left (model liquid density 0.3) and to the liquid phase (model density 1). The transition between the two phases is similar to that in classical model of van der Waals liquid.

The second graph corresponds to the polarization density (mean dipole moment of a liquid volume). Comparing the two graphs we see that the interface is indeed polarized and the polarization decays quickly into the bulk both of the gas and the liquid phase and, as the detailed calculation shows, contributes to the surface tension coefficient considerably.

In short we observe that depending on the ordering state of the water layer next to a molecular surface the effective surface tension may be different by a large number. Another observation suggests that the water density depletion next to a fully hydrophobic (and thus ordering) surface can be large (up to 30-50%)

Solvation energy of a large atom cluster: continuous solvation energy test - II

As it has been already stated here, binding energy calculation of a small molecule to a large protein poses a difficult problem: a method for molecular electrostatic energy calculation should work well both for the protein ligand complex, the protein and the ligand at infinite separation. The protein and the complex are large molecules, whereas the ligand is, by definition, small.

Not every computational approach for the solvation energy calculation is fit for the job though. To elucidate the nature of the problems at hand we performed the following model calculation:
- we prepared a spherical "protein" of a large (but realistic) radius
- we placed a single-atom ligand with a charge at a given distance from the "protein" center (see the Figure)
- we calculated the solvation energy of the system as a function of the ligand distance both when the protein is neutral and charged (in the latter case the protein charge is opposite to that of the "ligand")

We used four different methods for the electrostatic contribution to the solvation energy calculation: Poisson equation solver (in its surface electrostatic incarnation, blue) QUANTUM MGB (cyan) and the two "classic" GB methods, based on the Coulomb approximation: HCT (magenta) and AGBNP (yellow).

The first Figure, corresponding to an overall electrically neutral cluster, shows absolute deficiency of HCT approach deep enough inside the "protein". The problem is caused by unrealistic assumptions with regard to the overlap integrals calculations is occurs pretty frequently in realistic proteins. AGBNP method represents one of the latest GB approaches and is practically free of these difficulties. However, AGBNP is based on Coulomb approximation and thus fails to recover correct behavior of the solvation energy close to the "protein" boundary: AGBNP energy is off by a large number from both QMGB and the exact solution. QMGB and Poisson solutions agree very well everywhere!

The last Figure shows the same calculation for a charged model "protein-ligand" complex. Once again, HCT fails entirely, AGBNP does not work properly at the "protein" boundary and both Poisson solver and QMGB agree very well, though QMGB is about one order of magnitude faster than the Poisson solver!

Practically all this means that QMGB represents a fast and accurate approximation to the Poisson equation solution. QMGB approach does not rely on Coulomb approximation and is shown to work both for small molecules and large molecular clusters involving molecules of very different sizes. Therefore, with QMGB one can find a single transferable set of GB parameters capable of describing large and small molecules on the same footing.

Solvation energy of a diatomic molecule: continuous solvation energy test - I

Diatomic molecule is the simplest example of a realistic solvation energy calculation. Indeed, any reasonable solvation energy model gives exact value for a single atom.

Depending on the radii of the atoms involved the solvation energy of a pair may be a very good test of a solvation energy model and transferability of its parameters. In what follows we show the results of our Modified GB (MGB) approach for the test system. The graph below represents the solvation energies for similar and opposite charges pairs.

The upper (blue) solid curve represents the atom pair with opposite charges, whereas the lower (red) curve corresponds to the atoms with similar charges. First of all, the behavior of the two energies is easy to understand. At infinite separation both curves saturate at -0.125 (which is the Born solvation energy of the two charges of 0.5 and bare radii 2.). If the total charge is 0 (the opposite charges case, blue curve), at r=0 we have Es=0 as well. If the total charge is 2x0.5=1 (the red curve), then at r=0 we have Es=-0.25, as it should be for a combined charge within the sphere of radius 2.

Although the asymptotic values are OK, this does not mean the whole curve is fine. To compare our approach with true electrostatic we performed the calculation of the model system solving the Poisson equation as well as by two "classic" GB models (that of HCT and AGNP). The results for a diatomic molecule with zero total charge are represented on the last graph.

The electrostatic part of the solvation energy corresponds to the blue curve of the previous graph and is calculated either by a (surface-electrostatic) Poisson equation solver (blue), QUANTUM's MGB (cyan), AGBNP (yellow) and HCT GB model (yellow). As it is clear from here, all the approaches give very similar results for the "small" molecule and are practically indistinguishable. Indeed, it is well known that practically any sort of GB approximation gives good results for solvation energies of small molecules.

The difference between QMGB method and "classic" GB approaches and its relation to the exact solution becomes more obvious if we consider a charged diatomic molecule (a molecular ion with total charge, say, 1 placed on one of the atoms). The exact (blue) and QMGB (cyan), once again, are both in agreement with each other, whereas both "classic" GB approaches, HCT and AGBNP fail to recover correct asymptotic value at zero inter-atomic separation. The latter difference between GB solutions and the exact value of the solvation energy is not important for small molecules (low atom density) but is extremely important for ligand binding calculations (to be explained).

Video recorded from "Water as a ferroelectric..." presentation at MIPT, November 5th, 2008

Water as a ferroelectric: anomalous properties, long range order and interactions of nano-particles in solution (in Russian)

The nature of percolation phase transition in films of hydration water around immersed bodies.

In a set of molecular dynamics calculations (MD) the percolation phase transition in water layer absorbed on a body surface was revealed at definite temperature. Below this temperature the infinite network of unbroken hydrogen bonds exists. Above it this network decays on islands. This conclusion corresponds also with measurements of conduction of moisture contained disperse materials as quartz, for example: the conductivity drops almost to zero value while heating the specimens up to definite temperature. It is known that the water conductance dominates by the “estafette” mechanism in which protons are transferred over the hydrogen bonds. The breakdown of network means the conductivity drop. These phenomena are explained in the paper in frames of early published continuous vector model of polar liquids. It is shown that the immersed bodies are surrounded by the ferroelectric film, in which the dipole moments of water molecules are ordered, arranged in one direction parallel to the interface. It is the physics behind above mentioned MD results. In addition of our previous papers the stability of this ferroelectric order is proved. The character of phase transition to the paraelectric phase is discussed and its temperature is estimated that is in agreement with MD results. Below the critical temperature the polarization vector field contains the structures as “vortex-antivortex pairs”. These pairs dissociate above this temperature that means the order breaking. The boundary conditions for the polarization vector field of molecular dipole moments are derived that is necessary to enclose the vector model equations.

Reference: accepted for publication to Journal of Structual Chemistry (Russian Journal of), 2008

Spontaneous polarization of a polar liquid next to nano-scale impurities

Numerous properties of water are determined by the hydrogen bonds between its molecules. Water does not form hydrogen bonds with hydrophobic materials, henceforth, dipole moments of its molecules are arranged mainly parallel to the interfaces with such substances. According to molecular dynamics calculations (MD) at such orientation molecules save the maximal number of hydrogen bonds: three of fourth. It is shown in this Letter that in the layer of water or ice next to surface the long-range order spontaneously forms: remaining parallel to the surface dipole moment vectors arrange in one direction. Some fraction of dipole moments form the vortex structures on the surface. At low temperatures the ordered state has small admixture of vortex-antivortex pairs. The interaction energy of vortexes in this pairs arises proportional to the distance between them. A definite temperature the phase transition takes place: pairs suffer the dissociation, the molecular dipole moments order disappears. This conclusion agrees with he results of MD calculations, in which the percolation phase transition was revealed in the hydrogen bond network of water molecules absorbed on a surface.

The spontaneous polarization of liquid induced by the immersed in it nano-size bodies (proteins, peptides, …) results in the additional long-range interaction between them that depends on their relative orientation. Polarization of liquid in this case looks like that presented in Fig.1 in agreement with MD. All mentioned MD results can not be explained in frames of standard continuous scalar theory of water. These phenomena were analyzed here in frames of continuous vector model of polar liquids applications of which looks like promising to speed the simulations of macromolecular complexes.

Reference: arXiv:cond-mat/0601129 [ps, pdf, other]

Title: Long-Range Order and Interactions of Macroscopic Objects in Polar Liquids

Authors: P.O. Fedichev, L.I. Men'shikov

Comments: 11 pages, 6 figures

Subjects: Soft Condensed Matter (cond-mat.soft); Chemical Physics (physics.chem-ph); Biomolecules (q-bio.BM)

Accepted for publication in Journal of Physical Chemistry A (Russian Journal of), 2009

Ferro-electric phase transition in a polar liquid and the nature of lambda-transition in supercooled water

Water is a major and all-important example of a strongly interacting polar liquid. Dielectric properties of water surrounding nano-scale objects pose a fundamentally important problem in physics, chemistry, structural biology and in silica drug design. The issue of temperature dependence of dielectric constant, the role of fluctuations and a possibility of a ferro-electric phase transition in a polar liquid is fairly old . It attracted a new attention when a new phase transition (so called lambda-transition) was observed in supercooled water at critical temperatures between T_{c}=228K and T_{c}=231.4K . Isothermal compressibility, density, diffusion coefficient, viscosity and static dielectric constant \epsilon and other quantities diverge as T_{c} is approached, which is signature of a second order phase transition. The singularity of \epsilon is a feature of a ferro-electric transition . However, given a complexity of interactions between water molecules, the physical picture behind this phenomenon is not entirely clear . In the phase transition is explained as a formation of a rigid network of hydrogen bonds. On the other hand a ferro-electric hypothesis was also proposed and supported by molecular-dynamics simulations (MD). For example, a ferro-electric liquid phase was observed in a model of the so called ``soft spheres'' with static dipole moments . In fact, the existence of a ferro-electric phase appears to be model independent: domains where formed both in MD calculations with hard spheres with point dipoles and with soft spheres with extended dipoles .

In the our latest publication, Ferro-electric phase transition in a polar liquid and the nature of lambda-transition in supercooled water, we develop two related approaches to calculate free energy of a polar liquid. We show that long range nature of dipole interactions between the molecules leads to para-electric state instability at sufficiently low temperatures and to a second-order phase transition. We establish the transition temperature, T_{c}, both within mean field and ring diagrams approximation and demonstrate that the ferro-electric transition is a sound physical explanation behind the experimentally observed \lambda-transition in supercooled water. Finally we discuss dielectric properties, the role of fluctuations and establish connections with earlier phenomenological models of polar liquids.

Reference: arXiv:0808.0991 [ps, pdf, other]

Title: Ferro-electric phase transition in a polar liquid and the nature of \lambda-transition in supercooled water

Authors: P.O. Fedichev, L.I. Menshikov

Comments: 4 pages, 1 eps figure

Subjects: Statistical Mechanics (cond-mat.stat-mech); Soft Condensed Matter (cond-mat.soft)