| Traditionally, hydrodynamics deals with the behaviour of bodies in fluids and, in particular, with phenomena in which a force acts on a particle in a viscous solution. Very eminent scientists, such as Isaac Newton, James Clerk Maxwell, Lord Rayleigh (J. W. Strutt) and Albert Einstein, started their careers with major contributions to the science of hydrodynamics. Note that not only are the discoveries from more than one hundred years ago still highly relevant today, but also that they continue to stimulate important new developments in the field. | |
| 1731 | The science of hydrodynamics arose from the classical book by Daniel Bernoulli “Hydrodynamics”, which contained “Bernoulli law” relating pressure and velocity in an incompressible fluid, as well as a number of its consequences. The next fundamental contribution to the field was in 1879 when Sir Horace Lamb published another classical book also named “Hydrodynamics”. |
| 1821 | Botanist Robert Brown described the random, thermal motions of small plant particles suspended in water, a phenomenon that was later named Brownian motion. In 1855 Adolf E. Fick published a phenomenological description of translational diffusion and deduced the fundamental laws governing transport phenomena in solutions. In the 1990s, the method of video-enhanced microscopy was proposed for the direct observation of Brownian motion of labelled macromolecules in a membrane. |
| 1846 | J.(first name, please) L M. Poiseuille produced a theory of liquid flow in a capillary. Based on this theory, Wilhelm Ostwald invented viscometers and introduced them into physical and chemical practice. Later, the instrument was named after him. In 1962, Bruno Zimm proposed an original design for a rotational viscometer, which operates at very low velocity gradients, and has been very useful for measurements on asymmetric structures such as DNA, fibrous proteins and rod-like viruses. |
| 1856 | Sir George Stokes demonstrated that the coefficient of translational friction of a particle depends on its linear dimensions. He described the particle in terms of the radius of an equivalent sphere (the Stokes radius). The way was cleared for a direct determination of particle dimensions from hydrodynamic measurements. In 1880, Stokes analysed rotational friction and deduced an expression to relate the rotational friction coefficient of a particle to its volume. In the 1930s, François Perrin extended Stokes' formulae to ellipsoids of revolution. He also presented equations that give the three translational friction coefficients as functions of the dimensions of a general ellipsoid. |
| 1856 | James Clerk Maxwell discovered that certain liquids became birefringent when they flowed. In the 1960s and 1970s, Victor N. Tsvetkov and Roger Cerf developed a detailed method to measure flow birefringence for macromolecular solutions. The method has proven to be effective in studying flexibility and optical features of polymer and biological macromolecules that have no fixed, rigid structure. |
| 1887 | Osborne Reynolds pointed out that the ratio of inertial and viscous forces is a key feature for the characterization of any fluid movement. In the 1970s, Howard Berg and Edward Purcell applied this idea to describe the movement of different objects (from molecules to animals) in solution. It was shown that the movement of particles with molecular dimensions (10−10 000 Å) was described in terms of so-called low Reynolds numbers. It means that biological macromolecules “live” in a world without inertia. |
| 1881 | Albert Wiedemann proposed the term ‘luminescence’ to emphasise the difference between thermal equilibrium and non-equilibrium radiation emission. Among non-equilibrium processes, he studied light emission by molecules at room temperature caused by incident radiation. In 1945, Serguei Vavilov proposed a way for determining the hydrodynamic volume of a particle by using experimental data on fluorescence polarisation, a form of luminescence. |
| 1896 | John Kerr found that some solutions become birefringent under the influence of an electric field. Electric birefringence measurements became a way of obtaining information on the nature of dipole-dipole interactions and on the flexibility of macromolecules with either a natural or an induced dipole moment. The introduction of pulsed voltage to the practice of electric birefringence and the development of the basic theory of transient phenomena for macromolecules, by Henri Benoit in 1950, opened the way to study biological macromolecules in solution by electric birefringence. |
| 1905 | Albert Einstein created the theory of Brownian motion. He characterized quantitatively molecular motions in simple solutions and gases. A year later he derived an equation relating the diffusion coefficient of a macromolecule in solution to its coefficient of translational friction and demonstrated that the specific viscosity of a suspension of rigid spheres is proportional to their volume fraction, and independent of their radius. In 1940 Robert Simha obtained the equation for the viscosity of a solution of ellipsoids of revolution and in 1981 Stephen Harding and Arthur Rowe obtained solution of the viscosity for three-axis ellipsoid. |
| 1920 | Theodor Svedberg invented the high-speed centrifuge, opening the epoch of analytical and preparative ultracentrifugation. He proposed the combined use of sedimentation and diffusion coefficients to obtain a direct estimate of particle molecular mass. In 1929 Otto Lamm deduced a general equation describing the behaviour of the moving boundary in the ultracentrifuge field that was later used to propose several ways for determining diffusion coefficients of macromolecules during centrifugation. A new generation of ultracentrifuges, highly automated for data collection and analysis, appeared in the 1990s and provided direct methods for the precise molecular mass determination of biological macromolecules in solution, from several hundreds to tens of millions of Daltons. |
| 1937 | Albert Tiselius used the difference in their charge on to separate macromolecules during a fractionation process, thus introducing electrophoresis into extensive biochemical practice. In the 1950s. Oliver Smithies showed that electrophoresis with molecular sieving of a gel could give much higher resolution than electrophoresis in free solution. 1975. Patrick O'Farrell developed the method of two-dimensional electrophoresis, which revolutionised protein biochemistry and is now used extensively in proteomics. |
| 1964 | Herman Z. Cummings, following the theory of Robert Pekora, published the first experimental paper on dynamic light scattering and demonstrated that the diffusion coefficients of latex particles in solution can be extracted by this method. This work confirmed the theoretical predictions of Leonid Mandel'stamm, in 1923, concerning the modulation of scattered light intensity by Brownian motion. It marked the beginning of a new trend in structural biology for the rapid and accurate determination of macromolecular diffusion coefficients from dynamic light scattering. |
| 1967–1985 | A new theoretical formalism was developed to calculate hydrodynamic properties of biological macromolecules of different shape (Victor Bloomfield, Garcia de la Torre, Stuart Allison). The hydrodynamic properties of a particle can be calculated by modelling it by a set of spheres (beads) of different radius. In alternative approaches, the particle surface is modelled as a set of small equal spheres, or as a set of panel elements. The formalism opened a way to calculate hydrodynamic characteristics for biological macromolecules of arbitrary shape. |
| 1993 | Joseph Hubbard and Jack Douglas proposed to use the mathematical similarity amongst the equations of hydrodynamics and electrostatics to perform model calculations of hydrodynamic parameters by using their electrostatic counterparts. It opened up new possibilities for accurate hydrodynamic calculations (Huan-Xiang Zhou). The main point is that electrostatic calculations are much easier to perform, than hydrodynamics calculations. |
| 1994–2000 | The spectacular progress in solving protein and nucleic acid structures to high resolution by X-ray crystallography and NMR stimulated the development of novel approaches to calculate hydrodynamic parameters from atomic-level structural details. It was shown that frictional parameters of a protein could be calculated with an accuracy of about 1−3%, from its atomic structure by including a hydration shell. |
| 2000 and now | Modern hydrodynamics is in a stage of renaissance, being one of the recognized approaches to determine size, shape, flexibility and dynamics of biological, macromolecules. Modern hydrodynamics includes many novel experimental physical methods: fluorescence photo-bleaching recovery to monitor the mobility of individual molecules within living cells, time-dependent fluorescence polarization anisotropy to calculate Brownian rotational diffusion coefficients for macromolecules, fluorescent correlation spectroscopy and localised dynamic light scattering to study the dynamical properties of macromolecules. But in spite of all these achievements we must remember that |
Because hydrodynamics is a technique developed decades ago, calculations have been performed traditionally in c. g. s. units.
The dyne is the unit of force in c. g. s. systems. 1 dyne is the force necessary to accelerate one gram mass by one centimetre per second per second.
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In order to construct reasonable physical models for flow systems involving biological particles, it is necessary to make a number of simplifications. In this Section it is assumed that the flow is laminar and, further, that it is sufficiently “slow” that inertial effects need not be considered in the equations of motion, which describe the movement of particles relative to fluid (solvent). The approximation is justified, since biological systems of interest consist of very small particles, and even though the particles move rapidly with respect to the container wall, as, for example, in the viscosity and flow birefringent methods, they still move slowly with respect to the fluid surrounding them.
We consider an object moving with some velocity through a fluid of specific density and viscosity. The Reynolds number is a dimensionless parameter, which determines the relative importance of inertial and viscous effects
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The ratio as a significant intrinsic number to characterise a system was proposed more than one hundred years ago by Reynolds. It depends on the relative size of terms that describe inertial and viscous forces, respectively. When the Reynolds number is low, viscous forces dominate. If it is high, inertial forces dominate.
We calculate the Reynolds number for a virus 500 Å (5×10−6 cm) in diameter moving in water with a speed of order 10−3 cm s−1. Considering that ρ = 1 g cm−3 and η = 10−2 g cm−1 s−1 we obtained a Reynolds number of 5×10−7, i.e. the Reynolds number for the virus is negligibly small. A small Reynolds number means that molecule of virus will stop immediately when force disappears. Of course virus is still subject to Brownian motion, so in reality it does not stop.
Calculations show that for large biological complexes including bacteria in water the Reynolds number is also very small (Comment D1.1). So all biological macromolecules from small proteins to bacteria live in a world without inertia where viscous forces predominate (Comment D1.2).
Comment D1.1 Different objects in water
Bacteria
E. Purcell was the first to calculate Reynolds number for bacteria and fish (Purcell, 1977 Am. J. Phys., 45, 3–11). He considered that a bacterium is 10–4 cm in diameter and swims with a velocity of order 2×10−3 cm/s. Considering that ρ= 1 g/cm3 and η0= 10−2 g/cm s, he obtained a Reynolds number of 10−5, i.e. very small. The bacterium therefore lives in a world without inertia.
Fish
The same calculation for a fish with l = 10 cm, moving with velocity ∼100 cm/s in water yields a Reynolds number of about 105. This case in hydrodynamics is called hydrodynamics at high Reynolds number. The fish lives in a water medium with inertia.
Whale
For a whale l = 10 m (1000 cm) moving with velocity 3.6 km/h (100 cm/s) in water the Reynolds number is about 109. The whale swims in water with very large inertia.
Comment D1.2 Definition of the movement at very low Reynolds number
The best definition of the movement at very low Reynolds number belongs to E. Purcell: “What You are doing at the moment is entirely determined by the forces that are exerted on You at the moment, and by nothing in the past.” (Purcell, 1977, Am. J. Phys., 45, 3–11).
In hydrodynamic experiments, a biological macromolecule moves together with a certain amount of bound solvent, thus defining the concept of a hydrated particle as a core of particle material and an envelope of bound water. Fig. D1.1 shows that hydration is manifested as increased size or volume of the core particle. The hydrated volume, Vhyd, is larger than the “dry” volume, Vanh, which can be obtained from molecular mass, M, and the partial specific volume, υ¯ of the protein
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Fig. D1.1. Schematic presentation of a rigid hydrated particle. Hydration influences the overall shape of the protein in the sense of smoothing out some structural details such as pockets or cavities (Garcia de La Torre, 2001, Biophys. Chem., 93, 159–170)
Protein hydration, δ (g/g), expresses the ratio of the mass of the bound water to that of the protein
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Comment D1.3 Uniform expansion for compact and elongated particles
For typical value for globular protein in water if ρ = 1 cm3/g, v = 0.73 g/cm3 and δ = 0.3 g/g, then h = 0.12, which corresponds to 12% in linear dimensions and 41% in volume.
For very elongated particle with dimensions 20 Å in diameter and 200 Å in length hydration leads to reasonable increasing in diameter to approximately 22 Å. The same value of hydration applied to a particle length would increase 280 Å, i.e. 40 Å at each end. Evidently that this result is not realistic because leads to abnormal hydrodynamic solution properties.
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Fig. D1.2. Illustration of δ value as a uniform expansion in globular proteins (Garcia de La Torre, 2001, Biophys. Chem., 93, 159–170)
The second interpretation of the δ value is based on the assumption that the anhydrous core is coated by a bound water shell which has a constant thickness t, measured in the direction normal to the protein surface (Fig. D1.3). The hydration shell is considered as an intrinsic property of all proteins (Ch. D3).
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Fig. D1.3. Illustration of interpretation of δ value as thick uniform hydration shell (Garcia de La Torre, 2001, Biophys. Chem., 93, 159–170)
Early interpretations of hydrodynamic data led to hydration levels that varied widely from protein to protein. For example, hydration values deduced from diffusion coefficients and intrinsic viscosity are in a broad range, from 0.1 to more than 1 g of water per g of protein. These results were obtained by modelling proteins as spheres or ellipsoids, for which the translation friction and intrinsic viscosity are known analytically.
The use of models obtained from detailed protein structures for the calculation of translation, rotational friction and intrinsic viscosity leads to a much smaller hydration range, from 0.3−0.4 g of water per g of protein, corresponding to a less than single molecular layer in the hydration shell. The development of this type of calculation (Ch. D3) allows hydrodynamic measurements to join other techniques such as NMR, infrared spectroscopy, calorimetry, and small angle X-ray and neutron scattering to provide a unified picture of protein hydration.
The following picture has emerged (Fig. D1.3). A protein is hydrated at a definite level, corresponding to a 1.2-Å-thick hydration shell in average. The local density of water in the hydration shell is about 10% higher than that of bulk water. If in a hydrodynamic experiment a hydration value that differs greatly from usual levels were required to fit the data, it would be a likely indication that the hydrodynamic equivalent model used is wrong (Comment D1.4).
Comment D1.4 Estimation of hydration
The estimation of hydration from hydrodynamic properties of protein is sensitive to several types of errors because the extent of hydration is determined as “the small difference of two great values' between hydrated and dry volumes (Eq. D1.4). The main source of uncertainty in the estimation of hydration from hydrodynamic parameters is experimental errors in the data of hydrodynamic and other solution properties. Many of the tabulated data for common proteins are up to 50 years old, and it is evident that, for a quantitative, more accurate evaluation of hydration more precise data are required.
Because of their net negative charge in solution, more water is associated with RNA or DNA molecules leading to hydration values of about 0.6±0.2 g/g. For glycosylated proteins and for carbohydrates, hydration values are also larger (0.5 g/g) owing to the generally higher affinity for water of these glycopolymers.
In classical hydrodynamic theory two extreme solvent–particle interaction cases are usually considered called “slip” and “stick” boundary conditions (Comment D1.5). In the “slip” approximation, there is no interaction between solvent and particle and the solvent slips over the particle surface (Fig. D1.4a). The other extreme is represented by the “stick” approximation, in which the first solvent layer sticks to the particle surface and moves with it (Fig. D1.4b).
Comment D1.5 Mathematical definitions of stick and slip boundary conditions
Mathematical definitions of stick and slip boundary conditions are particularly complex. Interested readers can find them in the specialised literature (Hu and Zwanzig, 1974, J. Chem. Phys., 60, 4354–4357).
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Fig. D1.4. Hydrodynamic slip (a) and stick (b) boundary conditions.
It is important to note that, the value of the constant in equation connecting measured and calculated hydrodynamic values depends on the boundary conditions (see below, equations D1.1 and D1.2, D1.4, and D1.5 and D1.6).
It is generally accepted that for small protein molecules (< 2000 Da) “slip” conditions are more applicable, whereas for the large Brownian particles (> 5000 Da) “stick” boundary conditions hold (see discussion in Ch. D3).
Experiments in hydrodynamics can be divided into four groups. In the first, we find experiments that measure the equilibrium velocity of the particles. Translational diffusion is observed when the effective force arises from particle concentration gradients in the solution. When the acting force is gravitational (either under natural gravity or through ultracentrifugation), the phenomenon is called sedimentation. If the acting force is electrical in nature, the phenomenon is called electrophoresis.
The second group includes experiments in which the rate of particle rotation under the action of a pair of forces (a torque) is determined. If a velocity gradient in the solvent plays the role of an orienting force, the phenomenon is known as