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**Intrinsic Viscosity of Macromolecular Solutions**

Viscometry: Intrinsic Viscosity of Macromolecular Solutions

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**Annalen der Physik Band 19, 1906, 289-306:**

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**Annalen der Physik Band 34, 1911, 591-592:**

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**Viscosity of biomolecules**

Why viscometry? Simple, straightforward technique for assaying Solution conformation of biomolecules & volume/ solvent association Molecular weight of biomolecules Flexibility “ “

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**Types of Viscometer: “U-tube” (Ostwald or Ubbelohde)**

“Cone & Plate” (Couette) Diagram from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 154 Ostwald Viscometer

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**Types of Viscometer: “U-tube” (Ostwald or Ubbelohde)**

“Cone & Plate” (Couette) Extended Ostwald Viscometer

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**Types of Viscometer: “U-tube” (Ostwald or Ubbelohde)**

“Cone & Plate” (Couette) “U-tubes” simplest, allows very slow “creeping” shear experiments. “Cone and plate” allows variation of viscosity with shear rate to be studied. There is a third type known as a “Pressure Imbalance” differential viscometer – based on measurement of the pressure drop between a solution flowing through a capillary and solvent flowing through a capillary. We will refer to this later. Diagram from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 154 Couette-type Viscometer

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**Anton-Paar AMVn Rolling Ball viscometer**

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**Auto-timer Coolant system Density meter Water bath + 0.01oC Solution**

Temperature control is essential for accurate measurement of solution viscosity.

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**h = t/(dv/dy) Definition of viscosity: At 20.0oC, h(water) ~ 0.01P**

For normal (Newtonian) flow behaviour: t = (F/A) = h . (dv/dy) h = t/(dv/dy) units: (dyn/cm2)/sec-1 = dyn.sec.cm = POISE (P) At 20.0oC, h(water) ~ 0.01P Viscosity: a measure of the resistance of a fluid to flow. Units. When dealing with macromolecular solutions we usually prefer the c.g.s. (centimetre gram second) system of units rather than SI, because on a laboratory scale we are not dealing with kilogrames of cubic metre quantities. The “S.I” unit of viscosity is (N/m2)/s-1 or Pa. s. Diagram from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 142

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**h = t/(dv/dy) Definition of viscosity: At 20.0oC, h(water) ~ 0.01P**

For normal (Newtonian) flow behaviour: t = (F/A) = h . (dv/dy) viscosity shear rate h = t/(dv/dy) units: (dyn/cm2)/sec-1 = dyn.sec.cm = POISE (P) shear stress At 20.0oC, h(water) ~ 0.01P Viscosity: a measure of the resistance of a fluid to flow. Units. When dealing with macromolecular solutions we usually prefer the c.g.s. (centimetre gram second) system of units rather than SI, because on a laboratory scale we are not dealing with kilogrames of cubic metre quantities. The “S.I” unit of viscosity is (N/m2)/s-1 or Pa. s. Diagram from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 142

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**Viscosity of biomolecular solutions:**

A dissolved macromolecule will INCREASE the viscosity of a solution because it disrupts the streamlines of the flow:

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We define the relative viscosity hr as the ratio of the viscosity of the solution containing the macromolecule, h, to that of the pure solvent in the absence of macromolecule, ho: hr = h/ho no units For a U-tube viscometer, hr = (t/to). (r/ro) r = density of solution ro= density of solvent. With the density correction, out the density correction hr is known as the dynamic viscosity, without it, it is given the symbol h’r and is known as the kinematic viscosity. Since, for a given concentration polysaccharide solutions have a much higher relative viscosity than proteins, we can do experiments at low enough concentrations so that a density correction is not necessary, i.e. h’r ~ hr. Unfortunately many polysaccharide solutions show a strong shear dependence of hr: in these cases U-tube viscometers may not be suitable.

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**The first step is to define the reduced viscosity hred = (hr – 1)/c **

The relative viscosity depends (at a given temp.) on the concentration of macromolecule, the shape of the macromolecule & the volume it occupies. If we are going to use viscosity to infer on the shape and volume of the macromolecule we need to eliminate the concentration contribution. The first step is to define the reduced viscosity hred = (hr – 1)/c If c is in g/ml, units of hred are ml/g (hr – 1) is sometimes referred to as the specific viscosity and hred as the reduced specific viscosity.

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**The Intrinsic Viscosity [h]**

The next step is to eliminate non-ideality effects deriving from exclusion volume, backflow and charge effects. By analogy with osmotic pressure, we measure hred at a series of concentrations and extrapolate to zero concentration: [h] = Limc⃗0 (hred) Units of [h] are ml/g

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**Form of the Concentration Extrapolation 2 main forms **

Huggins equation: hred = [h] (1 + KH[h]c) Kraemer equation: (lnhr)/c = [h] (1 - KK[h]c) KH (no units): HUGGINS CONSTANT KK (no units): KRAEMER CONSTANT ln(hr)/c is known as the “inherent viscosity” hinh (ml/g). The plot is from K. Jumel, PhD Thesis, University of Nottingham, 1994, and is for irradiated (10kGy) guar in phosphate chloride buffer (pH=6.8, I=0.10). The “common” intercept gives [h]. The slopes are KH[h]2 for the Huggins line (data: circles) and Kk[h]2 for the Kraemer (triangles).

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**A variant of the Huggins equation is: hred = [h] (1 + kh.c) kh: ml/g **

and another important relation is the SOLOMON-CIUTA relation, essentially a combination of the Huggins and Kraemer lines: [h] ~ (1/c) . [2 (hr – 1) – 2 ln(hr) ] 1/2 The Solomon-Ciuta equation permits the approximate evaluation of [h] without a concentration extrapolation. kh: viscosity concentration dependence coefficient. The Solomon-Ciuta or “Solomon Göttesman” equation is particularly useful for a new type of viscometer called a “differential viscometer” which is based on measuring the pressure difference between solution flowing through a capillary and solvent flowing through a capillary:

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**Differential Pressure Viscometer:**

Pi DP hr = 1 + {(4DP).(Pi -2DP)} These “pressure imbalance” viscometers are particularly useful when combined on-line with a column chromatography system: since a concentration extrapolation is difficult in such set-ups the Solomon-Ciuta equation is commonly employed.

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**Intrinsic Viscosity and its relation to macromolecular properties**

[h] so found depends on the shape, flexibility and degree of (time-averaged) water-binding, and for non-spherical particles the molecular weight: .

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M (g/mol) [h] (ml/g) Glucose Myoglobin Ovalbumin Hemoglobin Soya-bean 11S Tomato bushy stunt x virus Fibrinogen Myosin Alginate GLOBULAR RODS, COILS

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**Intrinsic Viscosity and Protein Shape and Hydration**

[h] = n . vs (1) n: Simha-Saito function (function of shape & flexibility) vs: swollen specific volume, ml/g (function of H2O interaction) n: Einstein value of 2.5 for rigid spheres >2.5 for other shapes vs: volume of “hydrated” or “swollen” macromolecule per unit anhydrous mass = v + (d/ro) = v . Sw d: “hydration” (g H2O/g protein) v: partial specific volume (anhydrous volume per unit anhydrous mass) Hydration of proteins and other macromolecules is a complicated dynamic process. We are dealing here with a “time averaged” value. Sw = the “swelling ratio” (volume of swollen protein/volume of anhydrous protein)

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So, 3 forms of Eqn. (1): [h] = n . vs or [h] = n . {v + (d/ro)} or [h] = n . v . Sw For proteins, v ~ 0.73ml/g, vs ~ 1ml/g, Sw ~ 1.4, {For polysacchs, v ~ 0.6ml/g, vs>>1ml/g, Sw >>1}

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**Getting a shape from the viscosity n parameter**

SIMPLE ELLIPSOIDS OF REVOLUTION: axial ratio: a/b n: Simha-Saito ratio. Prolate ellipsoid (rugby ball or cigar shape): 2 equal short axes and one long axis. Oblate ellipsoid (disc, smartie or M&M shape): 2 equal large axes and one short axis. Both shapes defined by the axial ratio a/b where a>b (a=b for a sphere). The prolate ellipsoid is more applicable than the oblate (with a few exceptions, e.g. the Fc fragment of an antibody). No macromolecular surface is of course an ellipsoid but it is an approximation. Computer programmes such as ELLIPS1 can be used for this type of modelling. For more general modelling with all three axes of the ellipsoid unequal, programmes ELLIPS2,3 and 4 can be used, but this requires the use of additional data (such as sedimentation and light scattering)m as well as intrinsic viscosity. All these programmes are downloadable from: Diagram on the right is from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 34 Computer program ELLIPS1 downloadable from

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**Getting a shape from the viscosity n parameter**

n: Simha-Saito ratio. Prolate ellipsoid (rugby ball or cigar shape): 2 equal short axes and one long axis. Oblate ellipsoid (disc, smartie or M&M shape): 2 equal large axes and one short axis. Both shapes defined by the axial ratio a/b where a>b (a=b for a sphere). The prolate ellipsoid is more applicable than the oblate (with a few exceptions, e.g. the Fc fragment of an antibody). No macromolecular surface is of course an ellipsoid but it is an approximation. Computer programmes such as ELLIPS1 can be used for this type of modelling. For more general modelling with all three axes of the ellipsoid unequal, programmes ELLIPS2,3 and 4 can be used, but this requires the use of additional data (such as sedimentation and light scattering)m as well as intrinsic viscosity. All these programmes are downloadable from: Diagram on the right is from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 34 Computer program ELLIPS2 downloadable from

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**IgE IgG1 For more complicated shapes: BEAD & SHELL MODELS**

IgE IgG1 This type of modelling is particularly useful for non-simple shaped molecules such as antibodies. Despite the fact the viscosity and other shape factors can only be calculated exactly for ellipsoids, excellent approximate numerical solutions are now available permitting the calculation of n for any shape to better than a few percent. Programmes such as HYDRO and SOLPRO developed by J. Garcia de la Torre and colleagues can be downloaded from:

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**GENERAL CONFORMATIONS**

The three extremes of macromolecular conformation (COMPACT SPHERE, RIGID ROD, RANDOM COIL) are conveniently represented at the corners of a triangle, known as the HAUG TRIANGLE: Conformation of a given globular protein generally has conformation between that of a sphere and a rod. Conformation of a given polysaccharide, DNA or mRNA generally has a conformation between that of a random coil and a rod. Conformation of a denatured protein perhaps more between a sphere and a random coil. The Haug triangle is described in O.Smidsrød & L. Andresen (1979) Bipolymerkjemi, Tapir Press, Trondheim, Norway.

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**Mark-Houwink-Kuhn-Sakurada equation [h] = K’.Ma**

Each extreme has its own characteristic dependence of [h] on M. Mark-Houwink-Kuhn-Sakurada equation [h] = K’.Ma Analagous power law relations exist for sedimentation, diffusion and Rg (classical light scattering) so20,w= K”.Mb; Do20,w = K’”.M-e; Rg = K””.Mc; By determining a (or b, e or c) for a homologous series of a biomolecule, we can pinpoint the conformation type a: MHKS “power law” viscosity coefficient. b,c,e corresponding power law coefficients so20,w: sedimentation coefficient (sec) corrected to standard solvent conditions (density and viscosity of water at 20.0oC) Do20,w: translational diffusion coefficient (cm2/sec) corrected to standard solvent conditions (temperature of 20.0oC and viscosity of water at 20.0oC) Rg: radius of gyration (cm)

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[h] = K’.Ma a = 0 a = a = 1.8 Globular proteins, a~0.0, polysaccharide, a ~ 0.5 – 1.3 Sphere: n = 2.5 (i.e. constant, independent of M). Thus [h] independent of M so long as vs ~ constant. Rod: n approx (axial ratio)1.8. Since axial ratio for a rod of uniform diameter M, n M1.8 hence [h] M1.8. Coils: range of values reflects also the effects of intra-chain excluded volume and draining phenomena. More complicated power law type of expressions exist for modelling flexibilites of linear and branched types of polymer – in terms of Kratky-Porod worm-like coil theory, further developed by researchers such as Yamakawa, Fujii and Bohdanecky. The simplest parameter reflecting flexibility is the Persistence Length, Lp.

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**The intrinsic viscosity is ideal for monitoring conformation change:**

Denaturation of ribonuclease [h] (ml/g) Denaturation of ribonuclease at pH The process is reversible. Diagram from Van-Holde (1971): Physical Biochemistry, Prentice Hall, Englewood-Cliffs, USA, page 149 T(oC)

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**The intrinsic viscosity is also ideal for monitoring stability:**

Storage of chitosan (used in nasal drug delivery) Fee et al, 2006

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**Demonstration of H-bonding in DNA**

Creeth, J.M., Gulland J.M. & Jordan, D.O. (1947) J. Chem. Soc

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J.Michael Creeth,

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**Follow up reference sources:**

Serydyuk, I.N., Zaccai, N.R. and Zaccai, J. (2006) Methods in Molecular Biophysics, Cambridge, Chapter D9 Harding, S.E. (1997) "The intrinsic viscosity of biological macromolecules. Progress in measurement, interpretation and application to structure in dilute solution" Prog. Biophys. Mol. Biol 68, Tombs, M.P. and Harding, S.E. (1997) An Introduction to Polysaccharide Biotechnology, Taylor & Francis, ISBN The 1997 review expands in considerable details what is in these slides and notes. Chapter 1 in the 1997 book gives a simple review of the usefulness of viscosity and other solution methods for characterising polysaccharides

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