Diffusion and Viscosity in a Crowded Environment: from Nano- to

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2006, 110, 25593-25597 Published on Web 11/22/2006

Diffusion and Viscosity in a Crowded Environment: from Nano- to Macroscale Jedrzej Szyman´ ski,† Adam Patkowski,‡ Agnieszka Wilk,‡ Piotr Garstecki,† and Robert Holyst*,†,§ Institute of Physical Chemistry PAS, Department III, Kasprzaka 44/52, 01-224 Warsaw, Poland, Institute of Physics, A. Mickiewicz UniVersity, Umultowska 85, 61-614 Poznan´ , Poland, and Cardinal Stefan Wyszyn´ ski UniVersity, Department of Mathematics and Sciences-College of Sciences, Dewajtis 5, Warsaw, Poland ReceiVed: October 11, 2006; In Final Form: October 31, 2006

Although water is the chief component of living cells, food, and personal care products, the supramolecular components make their viscosity larger than that of water by several orders of magnitude. Using fluorescence correlation spectroscopy (FCS), photon correlation spectroscopy (PCS), NMR, and rheology data, we show how the viscosity changes from the value for water at the molecular scale to the large macroviscosity. We determined the viscosity experienced by nanoprobes (of sizes from 0.28 to 190 nm) in aqueous micellar solution of hexaethylene-glycol-monododecyl-ether (in a range of concentration from 0.1% w/w to 35% w/w) and identified a clear crossover at the length scale of 17 ( 2 nm (slightly larger than persistence length of micelles) at which viscosity acquires its macroscopic value. The sharp dependence of the viscosity coefficients on the size of the probe in the nanoregime has important consequences for diffusion-limited reactions in crowded environments (e.g., living cells).

Complex fluids are materials that exhibit hierarchical structure across length scales. Consequently, properties of these heterogeneous materials depend on the length scale at which we probe them: for example, length-scale-dependent dynamics have been observed in many biological1-10 and physical systems.11-15 In an agarose gel, the coefficients of diffusion of nanoparticles depend strongly on their size and approach zero for sizes larger than the typical diameter of the pores in the polymer network.11,12 In supercooled liquids,13-15 a small probe diffusing at a temperature that is close to the glass transition experiences viscosity that is 3 orders of magnitude smaller than the macroscopic viscosity of the same liquid. In cells, the macroviscosity of the liquid constituting their interior is large because various macromolecules and lipids constitute up to 40% of their volume.16,17 Yet, at the molecular scale, small proteins move freely by diffusion in various cell compartments, including in the cell nucleus,3 cytoplasm,4-6 endoplasmic reticulum,7 and mitochondria.8 All of these observations imply that the crowded environment of cells exhibits different rheological behaviors, at the molecular- and macroscales, yet the crossover between the scales has never been elucidated. The viscosity of water-based dispersions is an important factor in the functioning of a wide variety of biological systems and industrial processes and products (biological cells, food production processes, paints, personal care products, etc.). However, the viscosity of such systems, which almost always contain supramolecular components, depends strongly on the length scale considered. The viscosity for the flow of the fluid as well as for the movement of large objects dispersed in it is usually * Corresponding author. E-mail: [email protected]. † Institute of Physical Chemistry PAS. ‡ Institute of Physics, A. Mickiewicz University. § Cardinal Stefan Wyszyn ´ ski University.

10.1021/jp0666784 CCC: $33.50

much larger than that of pure water, whereas the viscosity that determines the movement of very small objects in such complex fluids is often close to that of pure water. Despite its great importance, there has heretofore been no systematic study of the transition from this “microviscosity” (or nanoviscosity) for small objects to the “macroviscosity” for larger objects and the overall fluid flow. The macroscopic coefficient of viscosity can be derived theoretically only for simple fluids. On the basis of the works of Maxwell, Boltzmann, Chapman, and Enskog, it is possible18 to connect the coefficient of viscosity to the molecular theory of gases. Recently, Dzugutov19 was able to relate the viscosity of a liquid η ) CkT exp(-S/k)/(Fd3) to the excess entropy S per particle (over ideal gas) and the Enskog frequency of collisions, F, between molecules of diameter d (k is the Boltzmann constant, T is temperature, and C is a universal constant). His theory is a step further from the long-known Hildebrand-Batchinski free-volume theory of viscosity and in the spirit of Adam-Gibbs theory.19 For structured fluids, the problem has been met only with partial solutions and only for selected systems. Einstein demonstrated that the viscosity (η) of a dilute suspension of colloidal particles to scale as η ) η0(1 + 2.5φ) where φ is the concentration of spheres and η0 is the viscosity of the solvent. The solution serves as a limiting case for the theory of the viscosities of polymer solutions and suspensions.20 Concentrated suspensions are, in practice, described by one of two phenomenological formulas: either the Krieger-Dougherty algebraic relation21 for viscosity ηφ ) (1 - φ/φmax)-λ, where λ is proportional to intrinsic viscosity multiplied by the maximum packing fraction φmax, or the Mooney exponential formula22 η/η0 ) exp[Aφ(1 - Bφ)], which contains two fitting parameters describing the attraction between © 2006 American Chemical Society

25594 J. Phys. Chem. B, Vol. 110, No. 51, 2006 suspending objects (A) and their excluded volume (B). A firstprinciple theory exists only for a dense suspension of hard spheres and is based on nonequilibrium averaging over the configuration of spheres23 in the fluid. In theory, the macroviscosity increases algebraically with the concentration of spheres and the viscosity is related to the static structure factor of the fluid. Experiments using capillary electrophoresis, sedimentation, or light scattering (with a limited number of nanoprobes and limited concentrations of polymers in solutions)24-28 suggest that probes of size d experience a viscosity η/η0 ) exp(adφ), where a is a constant. Interestingly, this exponential dependence of viscosity on concentration is not typically observed for macroviscosity. Currently, the most commonly used techniques1 for studying the micro-rheological properties of fluids involves monitoring the thermal motion of colloidal probes.1,2,29-31 For example, Mason and Weitz29 applied the generalized EinsteinStokes relationship to study the viscoelastic properties of actin-F networks;30 Granick et al. used fluorescence correlation spectroscopy and surface force measurements to study the flow and viscosity at the microscale simultaneously.32 De Gennes and Pincus in private communication suggested that the crossover between nano- and macroviscosity occurs at a lengthscale, d, that is much larger than the correlation length, ζ, in a polymer solution.25 ζ corresponds to the size of the channels in the polymer network, decreases with concentration, and is independent of the molecular mass of polymers. The authors observed a crossover in the sedimentation of 4-nmdiameter tracers28 in a solution of hydrogenated polyisoprene (polyethylene-propylene or PEP). In this paper, we use fluorescence correlation spectroscopy (FCS), photon correlation spectroscopy (PCS), and data from nuclear magnetic resonance (NMR) and rheology measurements to demonstrate an unambiguous, sharp crossover from nano- to macroviscosity in surfactant solutions spanning 2 orders of magnitude in concentration (0.1% to 35% by weight of hexaethylene glycol monododecyl ether, C12E6 in water, w/w) and three orders in the diameters, d, of the probes [ranging from 0.28 nm for water to 190 nm for fluorescent polystyrene spheres (Figure 1)]. The size of the water channels in the solution changes from 150 nm for 0.1% of C12E6 to 3 nm for 35% of C12E6 (Figure 2). We point out the importance of the fluctuation-dissipation theorem (FDT) (i.e., the relation between the hydrodynamic drag and the diffusion coefficient of tracers) in the determination of viscosity. In principle, one could check FDT comparing the viscosity obtained from FCS measurements to electrophoretic mobility of the charge particle33-35 assuming that the net charge of the tracers does not change in the crowded environment in comparison to the pure buffer. FCS36,37 makes it possible to characterize the coefficients of diffusion of fluorescent nanoprobes. The experiments involve a probe diffusing across the laser-illuminated focal volume of a confocal microscope. The distribution of the intensity (I) of the laser light in the focal volume is often approximated37 as a three-dimensional Gaussian: I(x,y,z) ) I0 exp(-2(x2 + y2)/F2 - 2z2/P2), where F is the cross-sectional length in the x-y plane, and P is the height of the illuminated element of volume. We recorded the intensity S(t) of fluorescence emitted from this volume as a function of time for fluorescent probes, shown in Figure 1. This intensity fluctuated as single fluorescently labeled molecules, and particles diffused in and out of the focal volume (i.e., 10-15 L). The signal S(t) allowed us to extract the distribution of residence times (τ) of the tracers in the focal volume by analyzing the autocorrelation function of S(t): g(∆t)

Letters

Figure 1. Fluorescent dye, proteins, and core-shell particles [core quantum dot CdSe/ZnS nanocrystals or particles of polysterene derivatized with fluorescent dyes (FS1 and FS3) and silica derivatized with fluorescein FS2] used as nanoprobes. Their hydrodynamic diameters, d, were determined from their diffusion in water using photon correlation spectroscopy. All of the proteins were labeled with the fluorescent dye TAMRA (d ) 1.7 nm). Additionally, we used the NMR data for water (diameter 0.28 nm).42

Figure 2. (a) Cartoon representation of the micellar solution of C12E6 and (b) the size of the water channels (average distance between elongated micelles) as a function of the surfactant C12E6 weight percent. This is a rough estimate based on the hexagonal geometry. However, irrespective of the precise distribution of micelles we roughly get the same size, L, of the water channels.41

) ∫ dt S(t) S(t+∆t). We calculated the coefficient for diffusion as D ) F2/(4τ). Figure 1 illustrates the nanoprobes we used in our study. We determined their hydrodynamic diameters from their diffusion coefficients in water, using PCS, and verified the obtained values with the data in the Protein Data Bank.38 We labeled all of the proteins with TAMRA [i.e., 5(6)-carboxytetramethylrhodamine N-hydroxysuccinimide ester]: a single fluorescent dye has a hydrodynamic diameter (d) of 1.7 nm. In addition to five proteins ranging from d ) 3.8 (lysozyme) to d ) 13.8 nm (appoferritin), we also studied the core-shell particles: CdSe/ ZnS core quantum dots covered with nonfunctional shells (d ) 25 nm), polystyrene spheres derivatized with fluorescent dyes (d ) 60 and 190 nm), and silica shells derivatized with fluorescein (d ) 114 nm).

Letters As the structured liquid, we used a solution of the nonionic surfactant C12E6 (hexaethylene glycol monododecyl ether) in phosphate buffer (0.02 M, pH ) 7, I ) 0.154 M). The surfactant is fully miscible in water up to 38% (w/w) by weight of C12E6, and for temperatures between 0 and 50 °C. We confirmed that the replacement of water with phosphate buffer does not change the phase behavior (transition temperature) of C12E6 solutions. At very low concentrations [below 0.1% (w/w)] and at low temperatures (below 20 °C,) we expected spherical micelles in the system. At higher concentrations, the micelles grow in length and eventually form a network39 (see Figure 2) at concentrations close to the isotropic hexagonal phase [the transition occurs at 38% (w/w)]. X-ray scattering40 experiments performed on the hexagonal phase (above 38%) indicated that the diameter of the columns in this phase was 44 Å. We used this data to estimate the average diameter, L, of the water channels (i.e., diameter of a sphere inscribed between the columnar micelles distributed on the hexagonal lattice).41 At 0.1% of C12E6 (w/w) L ≈ 150 nm and at 35% it is slightly larger than 3 nm. This estimate does not depend strongly on the particular geometry of micellar solution (see Figure 10 in ref 41). We chose the size of the nanoprobes (Figure 1) to fully cover this spectrum of dimensions of the water channels. Diffusion in structured fluids is often anomalous;29,30 the displacement of the tracers does not scale with the square root of time. Our results give only normal diffusion for the tracers that we used.41 At all concentrations, the TAMRA-labeled proteins yielded two components of diffusion (Figure 3): one characterized by a small diffusion coefficient (slow mode) and one with a large diffusion coefficient (fast mode). At low concentration of surfactants (below 4% w/w), the slow diffusion amplitude was very low.41 The colloidal particles (quantum dots, polystyrene beads, and fluorescent silica particles) produced only one diffusion coefficient. We attribute the slow mode to the motion of the tracer-micelle complexes (the hydrophobic region of the dye molecule is buried into the hydrophobic cores of the micelles). This assertion is consistent with NMR data for the diffusion of micelles in the solution.42 The amplitude of the slow component is small: it grows with concentration of C12E6, but even for 35% C12E6 (w/w) it was 10 times smaller than the amplitude for the signal from the fast diffusion of tracers alone. The coefficient of viscosity, η, was determined using the Smoluchowski-Einstein formula η ) kT/3πDd, where k is the Boltzmann constant, T is temperature in Kelvin, and d is the diameter of the probe. This formula can only be applied if the movement of the nanoprobes satisfies the fluctuation-dissipation theorem (FDT).43,44 FDT states that in thermal equilibrium the response of the system to external perturbation is the same as its response to spontaneous fluctuations, or in other words, externally induced perturbations are not faster than the relaxation of the medium. Quantitatively, FDT relates the diffusion coefficient, D, of a sphere of diameter d to that the hydrodynamic drag, 3πηd, experienced by a sphere moving in a viscous fluid under the influence of a constant force. Here D arises from fluctuations, whereas the drag is a result of external perturbation. Intuitively, we expected that the viscosity experienced by the probes would reach the macroscopic value once the size of the tracers exceeded the typical size of the water channel (for polymer solutions it would correspond roughly to the correlation length).28 Thus, we expected that for each of the probes there would be a crossover to macroviscosity at a different concentration of surfactant. Our results contradict this expectation. We performed measurements at 25 °C and at a concentration range of surfactant from 0.1% to 35%. We used the measurements of

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Figure 3. (a) An example of the signal S(t) (number fluctuations of the fluorescence photons) obtained in the FCS experiment for TAMRA dye molecules. (b) The autocorrelation function (ACF) g(∆t) derived from S(t). The time decay of the ACF carries the information about the time of diffusion (τ) through the focal volume. (c) The diffusion coefficients determined for TAMRA dye molecules in surfactant solution together with the NMR data for the diffusion of micelles in the solution. Slow diffusion of dye molecules is consistent with the NMR42 data, demonstrating that dye molecules are attached to the micelles. The fast diffusion is a free diffusion of dye molecules and in all cases constituted 90% of the signal.

the residence time of the nanoprobes, τ0, in pure buffer as a reference for the viscosity in the surfactant solutions, η, according to the formula

η/η0 ) τ/τ0

(1)

where η0 is the bulk viscosity of the buffer (η0 ) 0.917 mPas at 25 °C, measured using a glass Cannon type viscometer and a Parr densimeter). In Figure 4, we show the measured dependence of the coefficients of viscosity on the concentration of C12E6. We find that all of these coefficients follow an exponential dependence on the concentration of surfactant, φ

η/η0)exp(aφ),

(2)

where a is independent of φ and is a function of the diameter of the diffusing object. The macroscopic viscosity ηmacro also satisfies eq 2 as expected from the growth of micelles as a function of concentration.45-47 We note that in systems in which micelles do not grow in length with concentration the viscosity grows algebraically: this characteristic is common for polymer systems.48 To our surprise, we found that for all tracers of d < 20 nm the proportionality constant in the exponent is a linear function of the hydrodynamic diameter of the tracer (a ∝ d). This linear

25596 J. Phys. Chem. B, Vol. 110, No. 51, 2006

Letters

dependence is also consistent with the diffusion of molecules of water42 (with the hydrodynamic diameter dwater ) 0.28 nm) in the micellar network, as illustrated in Figure 4. The linear dependence (a ∝ d) crosses the value of the proportionality constant for the macroviscosity at a length scale of 17 ( 2 nm (Figure 4); all of the colloidal probes we examined (25 nm, 60 nm, 114 nm, 190 nm) had a macroscopic coefficient of viscosity (Figure 4). Thus, the viscosity, η, saturates at its macroscopic value, ηmacro, once the diameter of the probe exceeds the critical length scale LP. This length scale cannot be linked to the diameters of the water channels in the structured fluid we studied because these change between ∼3 to ∼150 nm in the range of concentrations of C12E6 that we tested. The only other relevant length scale of our micellar system is the persistence length, LP ≈ 11 nm, of the elongated micelles of C12E6, which coincides closely with the determined crossover.49 On the basis of the above observations, we describe the coefficients of viscosity of the tracers (Figure 4) with the following formula

η/ηmacro ) exp[R(d - Lp)φ] for d < LP η/ηmacro ) 1 for d g LP

(3)

with the exponential form consistent with earlier observations.25,26 In eq 3, R is a constant with units of inverse length, [R ) 0.84 ( 0.08 (1/nm) for φ expressed as the weight fraction φ ∈ (0,1)]. In general, we expect that LP corresponds to the characteristic correlation length-scale at which stress is transmitted in the medium. R is an unknown function of the structure of the medium; that is, in general it may depend on LP. We stress the importance of the fluctuation dissipation theorem (FDT) in observing the transition from nano- to macroscopic rheological behavior. For a probe moving in a structured fluid there are two important time scales: a typical time, tP , needed for the probe to change the local structure of the fluid and the time, tR, required for relaxation of this phenomenon. If tR , tP , then the structured fluid relaxes rapidly during the motion of the object, FDT is satisfied, and the crossover, the saturation at the macroscopic value of viscosity coefficient, is observed. Both times are easily computed for probes larger than the size of a pore in our system. A probe with d ) 25 nm (Figure 1) and diffusion coefficient D ) 10-12 m2/s (from Figure 4) which moves across a pore of size L ) 10 nm (Figure 2) in time tP ) L2/D ) 100 µs, affects the fluid at a distance of characteristic length LP ) 17 nm. The relaxation time of the disturbance is associated with the collective diffusion coefficient in the micellar solution DC ≈ 10-10 m2/s, which is determined by PCS and rheological measurements.50 This time tR ) L2P/DC ≈ 3 µs is much smaller than tP ) 100 µs. In fact, the relaxation in micellar systems can be even faster than our estimate as pointed out by Cates and Candau51 because micelles break and recombine with characteristic time tB, and the full relaxation time in micellar solution is given by xtBtR. In the system we have studied, on the time scale of its motion, the probe feels the time average of the local structure of the fluid, the FDT applies, and we are able to observe a clear crossover in rheological properties. The same is not true for solutions of stiff polymers, semidilute networks of F-actin (with a persistence length 1000 times the diameter of the actin fiber),52 solutions of fd virus (of size 800 nm and persistence length 2300 nm),53 or in solutions of supramolecular polymers that are hydrogen bonded (of persistence length well exceeding 100 nm).54 In these cases several characteristics are observed, including different diffusion coefficients at short and long time

Figure 4. (a) Viscosity, η, relative to the viscosity of the buffer, η0, (eq 1) for all of the probes shown in Figure 1. Additionally, we show the data for water42 (d ) 0.28 nm) obtained from diffusion measured with NMR. The macroviscosity (MACRO)45 of the solution is shown as a function of the weight percent of surfactant C12E6. By macroviscosity, we mean the value of viscosity obtained in a rheometer in zero shear limit.45 The fit to the data is given by eq 2. (b) Measured values of ln(η/ηmacro)/φ as a function of the diameter, d, of the probes (Figure 1). φ ∈ (0,1) is the weight fraction of surfactant. The line crosses 0 at the crossover length scale equal to 17 nm (see also L given by Figure 2). This length scale is slightly larger than the persistence length of the elongated micelles at room temperature (11 nm)49 and does not depend on the concentration of surfactant. Persistence length corresponds roughly to the length over which the micelle is stiff. The slope of the line gives the value of the parameter characterizing the surfactant medium (eq 3), R ) 0.84 (nm-1). Full black circles indicate the data we measured, whereas empty circles are data for water by Yethiraj et al.42 The inset shows data for all probes from 0.28 to 190 nm.

scales, anomalous diffusion and lack of clear crossover between the nano- and macroscale. In such systems, the diffusing tracers do not probe the average structure of the fluid; they instead probe the local nonequilibrium configurations of the structured liquid. We used probes of sizes ranging from 0.28 to 190 nm, covering almost 3 orders of magnitude in their diameters, and tested their diffusion in solutions of surfactants that range in concentrations over 2 orders of magnitude (corresponding to almost 2 orders of magnitude in the diameter of the pores in the network). For the first time we have demonstrated a clear and sharp crossover between the nano- and macroscale for viscosity using probes of sizes changing over 3 orders of magnitude. The Dzugutov phenomenological formula19 for simple liquids, which relates the self-diffusion coefficient to the configurational entropy for simple liquids, points out the importance of probing the equilibrium ensemble, or time average, of the structure of the fluid with a probe. The FDT is a necessary condition that must be taken into consideration for determining the viscosity via monitoring of the Brownian motion

Letters of the tracers and for clearly observing the crossover between nano- and macroscale rheological behavior. In spite of the importance of microrheological properties in biology and in industrial applications, our understanding of the flow of complex materials is limited. No single theory derives, nor provides an explanation for, the values of the coefficient of viscosity of structured fluids. In this paper we have demonstrated a complete set of experimental measurements that (i) clearly show the change in the coefficient of viscosity with the scale at which it is probed, (ii) clearly show a linear dependence of the logarithm of the coefficient of viscosity on the diameter of the nanoprobe, and (iii) allow the identification of the length scale at which the coefficient of viscosity saturates at its macroscopic value. In living cells, macromolecules and lipids form a crowded environment with concentrations up to several hundred grams per liter. They can occupy up to 40% of the available volume. From our work, it follows that in such an environment the protein diffusion coefficients decrease exponentially with their size and this very fact should have tremendous impact on the reaction constants of diffusion-limited processes in cells. For example, an estimate of the ratio of diffusion coefficients for lysozyme and appoferritin on the basis of the Einstein-Stokes relation for simple liquids yields a 3-fold decrease (from LYS to APO), whereas the correct estimate for a crowded environment (through eq 3) yields a factor of 100. This suggests that size can have a critical impact on the kinetics of biochemical reactions in cells and that, for diffusion-limited reactions, there might be critical differences between the reaction constants determined in vitro (in buffer) and reaction constants in vivo. Further applications of the current work are possible in nanoand microelectrophoresis, control of nanoflows, nanofluidics in analytical systems, and in controlled release of macromolecules in cosmetics and pharmaceutical applications. Acknowledgment. This work was supported by the grant (2006-2008) from the budget of the Ministry of Education and Science and Unilever grant PS-2005-069. P.G. acknowledges financial support of the Foundation for Polish Science. We thank Professor George Whitesides, Professor Robert Pecora, and Professor Douglas Weibel for critical reading of the manuscript. References and Notes (1) Waigh, T. A. Rep. Prog. Phys. 2005, 68, 685. (2) Shin, J. H.; Gardel, M. L.; Mahadevan, P.; Weitz, D. A. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 9636. (3) Phair, R. D.; Mistell, T. Nature 2000, 404, 604. (4) Weiss, M.; Elsner, M.; Kartberg, F.; Nilsson, T. Biophys. J. 2004, 87, 3518. (5) Elowitz, M. B.; Surette, M. G.; Wolf, P. E.; Stock, J. B.; Leibler, S.; J. Bacteriol. 1999, 181, 197. (6) Swaminathan, R.; Hoang, C. P.; Verkman, A. S. Biophys. J. 1997, 72, 1900. (7) Dayel, M. J.; Horn, E. F. Y.; Verkman, A. S. Biophys. J. 1999, 76, 2843. (8) Partikian, A.; O ¨ lveczky, B.; Swaminathan, R.; Li, Y.; Verkman, A. S. J. Cell Biol. 1998, 140, 821. (9) Kao, H. P.; Abney, J. P.; Verkman, A. S. J. Cell Biol. 1993, 120, 175.

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