We present a scaling formula for size-dependent viscosity coefficients for proteins, polymers, and fluorescent dyes diffusing in complex liquids. The formula was used to analyze the mobilities of probes of different sizes in HeLa and Swiss 3T3 mammalian cells. This analysis unveils in the cytoplasm two length scales: (i) the correlation length ξ (approximately 5 nm in HeLa and 7 nm in Swiss 3T3 cells) and (ii) the limiting length scale that marks the crossover between nano- and macroscale viscosity (approximately 86 nm in HeLa and 30 nm in Swiss 3T3 cells). During motion, probes smaller than ξ experienced matrix viscosity: η(matrix) ≈ 2.0 mPa·s for HeLa and 0.88 mPa·s for Swiss 3T3 cells. Probes much larger than the limiting length scale experienced macroscopic viscosity, η(macro) ≈ 4.4 × 10(-2) and 2.4 × 10(-2) Pa·s for HeLa and Swiss 3T3 cells, respectively. Our results are persistent for the lengths scales from 0.14 nm to a few hundred nanometers.

We measured the viscosity of poly(ethylene glycol) (PEG 6000, 12,000, 20,000) in water using capillary electrophoresis and fluorescence correlation spectroscopy with nanoscopic probes of different diameters (from 1.7 to 114 nm). For a probe of diameter smaller than the radius of gyration of PEG (e.g. rhodamine B or lyzozyme) the measured nanoviscosity was orders of magnitude smaller than the macroviscosity. For sizes equal to (or larger than) the polymer radius of gyration, macroscopic value of viscosity was measured. A mathematical relation for macro and nanoviscosity was found as a function of PEG radius of gyration, R(g), correlation length in semi-dilute solution, xi, and probe size, R. For R < R(g), the nanoviscosity (normalized by water viscosity) is given by exp(b(R/xi)a), and for R > R(g), both nano and macroviscosity follow the same curve, exp(b(R/xi)a), where a and b are two constants close to unity. This mathematical relation was shown to equally well describe rhodamine (of size 1.7 nm) in PEG 20,000 and the macroviscosity of PEG 8,000,000, whose radius of gyration exceeds 200 nm. Additionally, for the smallest probes (rhodamine B and lysozyme) we have verified, using capillary electrophoresis and fluorescence correlation spectroscopy, that the Stokes-Einstein (SE) relation holds, providing that we use a size-dependent viscosity in the formula. The SE relation is correct even in PEG solutions of very high viscosity (three orders of magnitude larger than that of water).

In this paper we describe model calculations for the self-assembly of N,N-disubstituted melamines 1 and N-substituted cyanuric acid or 5,5-disubstituted barbituric acid derivatives 2 into linear or crinkled tapes and cyclic rosettes via cooperative hydrogen bond formation. The model description considers all possible stereoisomeric tape structures consisting of two to eight different components (270 different species in total) and one cyclic hexameric rosette structure. Furthermore, eight steric parameters (R(12)-R(28)) are included that represent the different types of steric interactions within the assemblies. Most importantly, the model calculations clearly show that the tape/rosette ratio is very sensitive to changes in parameters that directly affect the internal energy of the rosette structure. In this respect, three parameters have been characterized, i.e., the basic equilibrium constant K(0) for the bimolecular association of a melamine and cyanurate, the equilibrium constant K(r)/K(0) for the cyclization of a linear hexamer, and the parameter R(12)-a(Z)b, representing attractive or repulsive interactions between adjacent melamine and cyanurate moieties. For example, an increase in K(0) from 100 to 10,000 M(-1) ([A](0) = [B](0) = 10 mM, K(r) = 0.01 M) or in K(r) from 0.001 to 0.1 M ([A](0) = [B](0) = 10 mM, K(0) = 1000 M(-1)) raises the concentration of the rosette from <5 to approximately 90% or from approximately 10 to approximately 85%, respectively. Similarly, a change in R(12)-a(Z)b from 1.0 (no repulsive or attractive interactions) to 1.5 (slight attractive interaction) raises the rosette fraction of the mixture from 25% to 45%. In sharp contrast to this, the model calculations show that parameters that only affect the internal energy of the tapes (R(13)--R(28)) hardly change the tape/rosette ratio. For example, by changing R(13)-a(EE)a from 1.0 (no repulsive or attractive interactions) to 0.001 (maximum repulsion), the rosette fraction in the mixture changes by no more than 8%. Including all possible sterics that occur only in tapes (i.e., R(13)--R(28)), the maximum change in rosette fraction is no more than 16%. These predictions can be rationalized by considering that any change in the stability of the tapes only affects the rosette concentration by means of shifting the equilibrium between free 1 and 2 and the rosette. Since there are 270 different tapelike structures in equilibrium, this mixture represents the best buffer solution in the world. These model calculations seem to conflict with the concept of peripheral crowding as put forward by Whitesides et al., which states that bulky substituents on the periphery of the melamine (and cyanurate) components can be used to shift the tape/rosette equilibrium completely toward the rosette structure. Computer simulations (CHARMm 24.0) show that linear tapes with bulky substituents are severely distorted from planarity, while the corresponding rosette remains planar. Therefore, tapelike structures with bulky substituents are expected to have a much h...

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