Percolation transition of hydration water

Percolation transition of hydration water is intrinsically a site-bond percolation problem. At some temperature, percolation transition occurs upon increase in the surface coverage C, which is analogue of the occupancy variable p. At low coverages, only finite clusters are present in the system, whereas there is an infinite cluster above the percolation threshold. In Fig. 66, typical arrangement of water molecules, adsorbed at hydrophilic plane, is shown for three surface coverages. Visual inspection does not allow determination of the percolation threshold. This can be done by the analysis of various cluster properties for a system of a given dimensionality [396]. As hydration water is not a strict 2D system, the reliable estimation of a percolation threshold assumes an independent use of several criteria. 

The cluster size distribution ns is an occurrence frequency of water clusters of sizes S. Right at the percolation threshold, the cluster size distribution obeys the universal power law  

When approaching the percolation threshold via increase of the surface coverage, the cluster size distribution undergoes qualitative changes. At low surface coverage, most of the water molecules belong to small clusters and ns shows a rapid exponential decay with increasing S. Upon increas- 

Spanning probability i is a probability that system percolates, i.e. contains an infinite cluster [396]. In an infinite system, R = I above and R = 0 below the percolation threshold. In a finite system of linear dimension L, the probability of a spanning cluster to be present in the system 

At the percolation threshold, the largest cluster has a specific structure, which may be characterized by the fractal dimension that is universal for all systems of a given dimensionality. The universal value of the fractal dimension of the infinite cluster in 2D systems df = 1.896 [396]. No clusters with the fractal dimension lower than can be infinite in the 2D space. The fractal dimension df of a cluster can be evaluated by fitting the cumulative radial distribution of its molecules 

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Oleinikova A, Brovchenko I. Percolation transition of hydration water in bio-systems. Mol. Phys. 2006 104 3841-3855. Oleinikova A, Brovchenko I. Percolating networks and liquid-liquid transitions in supercooled water. J. Phys. Cond. Matter 2006 18 S2247-S2259. 

Above, we have considered the percolation transition of hydration water at idealized smooth surfaces. Real surfaces are structured at nanoscopic level and may be strongly heterogeneous. This can suppress the critical temperature of the layering transition and/or causes smearing... 

In experiments, the percolation transition of hydration water can be detected by conductivity and dielectric measurements. Formation of a condensed hydrogen-bonded network of water should provide a media for the charge transport (proton or ions) and should change qualitatively the dielectric properties of the system. Sharp stepwise increase of the conductivity of the system with increasing water content at some threshold hydration level may directly indicate the appearance of an infinite hydrogen-bonded water network via a percolation transition. The dielectric response is also expected to increase drastically at the percolation threshold. Note, however, that the strongly attractive sites on the surface, which immobilize water molecules, may complicate interpretation of the results. As these effects occur on the surfaces, their experimental observation is possible first in the system with high surface/volume ratio (in various porous media). 

The detailed studies of the percolation transition of hydration water in model biosystems (Section 7.1) makes it possibile to consider various physical properties of these systems below and above the percolation threshold. The total MSD (( )) of water at the surfaces of rigid and flexible lysozyme molecules continuously increases upon hydration (Fig. 112). Similar behavior was observed in the simulation studies of water near the surface of differently hydrated plastocyanin [640, 641]. 

Analysis of the properties of biosystems at various hydrations in the vicinity of the percolation transition of hydration water should clarify which particular property changes in a drastic way at the percolation threshold and makes possible the biological function. Rotational motions of water molecules change qualitatively near the percolation threshold. 

Nevertheless, a two-peak distribution of S ax is manifested in pronounced shoulders or as an almost flat P(S,aax) at T = 320 K. The latter temperature may be considered a midpoint of a temperature-induced percolation transition of hydration water. Distribution P( S max) makes possible an estimation of some minimal size required for the largest cluster to be spanning. One of the possible choice of is being equally distant to both peaks of P(-S max)- Integration of P(S,aax) for S ax > A ax yields an estimation of the spanning probability R. For a given temperature, the number of water molecules in the hydration shell fluctuates... 

The temperature evolution of the cluster size distribution ns allows estimation of the true quasi-2D percolation transition of hydration water. 

Figure 135 Phase diagram of surface water in fully hydrated systems (upper panel). Fragile to strong transition of the hydration water [243] and anomaly in thermophysical properties [108] that indicate a continuous transition from tetrahedrally ordered to orientationally disodered water [45] are shown by open and closed circles, respectively. The line of percolation transition of hydration water in the case of full hydration is shown schematically by solid lines based on the results of Ref [566]. Location of the percolation transitions in low-hydrated systems is shown schematically by dashed and dot-dashed lines (lower panel). Reprinted, with permission, from [612].

A. Oleinikova, I. Brovchenko, A. Geiger, Percolation transition of hydration water at hydrophilic surfaces, Physica A 364 (2006) 1-12. 

It turned out, however, that the behavior of hydration water near neutral DNA depends on how its surface was neutralized. Properties of hydration water were found to be similar in the cases when the neutralizing charge was uniformly distributed over the whole system including DNA and water and between all DNA atoms only. In both cases, water undergoes a normal percolation transition with increasing F. With ions removed, the percolation threshold of hydration water is shifted by AT 4 toward... 

Formation of the hydrogen-bonded water networks may affect conductivity of a system in a drastic way, as these networks provide the paths for the conduction of protons, ions, or other charges in the system. So, the qualitative changes in the conductivity may be expected at hydrations, close to the percolation transition of water. Surface conductivity of quartz increases relatively slowly with increasing hydration level until the completion of the adsorbed water monolayer, but much faster at higher hydrations [582]. The hydration dependence of the dielectric losses of hydrated collagen... 

The conductivity exponent of about 1.23 indicates 2D character of the percolation transition. Similar values of the conductivity exponent were obtained for the hydration dependence of the conductivity of embryo and endosperm of maize seeds [595, 596], where the percolation threshold is /t = 0.082 and 0.127 g/g, respectively. In hydrated bakers yeast, protonic conductivity evidences 2D percolation transition of water at h = 0.163 g/g, and the value of the conductivity exponent is about 1.08 [597]. In this system, increase in conductivity due to 3D water percolation is observed at essentially higher hydration level h= 1.47 g/g, where conductivity exponent is about 1.94, i.e., close to the 3D value t = 2.0. Conductivity measurements of Anemia cysts at various hydrations show strong increase in conductivity starting from the threshold hydration h = 0.35 g/g [598] (see Fig. 97). The conductivity exponent in this system is 1.635, which is in between the values expected for 2D and 3D systems. DC conductivity of lichens, evaluated from the dielectric studies at frequencies between 100 Hz and 1 MHz [599], shows strong enhancement at some hydration level. Fit of the conductance-hydration dependence to equation (24) gave the following parameters he = 0.0990 g/g, t = 1.46 for Himantormia lugubris and he = 0.0926 g/g, t = 1.18 for Cladonia... 

The above experimental studies of the conductivity of hydrated biosystems directly evidence the formation of a spanning network of hydration water via percolation transition. The charge transfer itself may play a crucial role in biofunction [607]. In most of the cases, described above, the percolation transition of water occurs at the hydration level, where various forms of biological activity develop in a stepwise manner (see Section 6). In particular, the following biological processes starts close... 

Below we show how the appearance of spanning water networks may be detected in computer simulations. In particular, a percolation transition of water upon hydration was studied by simulations in model lysozyme powders and on the surface of a single lysozyme molecule. In protein crystals, increase in hydration of a biomolecular surface may be achieved by applying pressure. In some hydration range, pressurization leads to the formation of spanning water networks enveloping the surface of each biomolecule. Finally, the formation of the spanning water network is shown for the DNA molecule at various conformations and for different forms of DNA. 

In low-humidity tetragonal crystal with the partial density of lysozyme of about 0.80 g/cm, approximately 120 water molecules are in the first hydration shell of lysozyme molecule. In order to explore a wide range of hydration level up to monolayer coverage (about 300 water molecules), partial density of lysozyme in powder should be < 0.80 g/cm. In Ref. [401], two models for protein powder were studied densely packed powder with the density of dry protein 0.66 g/cm and loosely packed powder with a density 0.44 g/cm. In loosely packed powder, the percolation transition of water was noticeably (by a factor of two) shifted to higher hydration levels compared with experiment. The fractal dimension of the water network at the percolation threshold as well as other properties evidenced that the percolation transition of water in this model was not two dimensional. The spanning water network consists of the 2D sheets at the protein surface as well as of the 3D water domains, formed due to the capiUaiy condensation of water in hydrophilic cavities. The latter effect causes essential distortion of various distribution functions of water clusters in loosely packed powder. Therefore, below we present an overview of the results obtained for the densely packed model powder. 

Spanning probability R, defined as a probability to observe a water cluster that crosses the model system at least in one dimension, shows sigmoid dependence on the mass fraction C of water (Fig. 98, upper panel). At ambient temperature (T = 300 K), the inflection point of this dependence corresponding to R = 50% is located at about C = 0.122. This hydration level is close to that where the mean cluster size Smean passes through a maximum (Fig. 98, middle panel). Fractal dimension of the largest water cluster achieves the value at C 0.155 (Fig. 98, lower panel). Summarizing, the percolation transition of water may be attributed to the hydration level C 0.155. The cluster size distribution ns supports this conclusion [401]. 

Figure 98 2D percolation transition of water in the hydrated densely packed powder of lysozyme at two temperatures. Spanning probability R (upper panel), mean cluster size 5 mean (middle panel), and fractal dimension of the largest cluster d (lower panel) are shown as functions of a mass fraction of water C. The dashed lines are guides for eyes only. Vertical hnes indicate the 2D percolation threshold. Reprinted, with permission, from [401]. 

About 2.0 water-water H-bonds are necessary to create a spanning network of hydration water around B- and A-DNA molecules at ambient temperature [621]. The lower value of h in comparison with lysozyme systems obviously reflects the trend of water clustering toward three dimensionality. Indeed, a 2D percolation threshold of water in binary mixtures close to ambient temperature occurs when h 1-80 [100]. Note that 3D percolation transition in neat supercritical water occurs when 1-78 [24]. 

So, experimental and simulation studies show the formation of a spanning network of hydration water in various biosystems with increasing water content via the percolation transition. Analysis of the various properties of water clusters allows locaUzation of the percolation threshold and characterization of the properties of the largest (spanning) water cluster. In the next section, we consider how increasing hydration level and appearance of a spanning water network affect the properties of hydrated biosystems. 

Effect of hydration on the properties of biosystems was extensively studied both experimentally and by computer simulations. We have already considered how biological activity and conformational dynamics of hydrated biomolecules (Section 6) as well as conductivity of biosystems (Section 7.1) develop upon hydration. Now we analyze some other physical properties of hydrated biosystems (first, their dynamical properties) in relation to the percolation transition of water. Typical biomolecular surface is characterized by heterogeneity (presence of strongly hydrophilic and strongly hydrophobic groups), roughness, and finite size (closed surface of a single biomolecule). These features determine several steps in the process of hydration of biomolecules. 

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