Intermolecular interactions and scattering M.H.J. Koch 1 - - PowerPoint PPT Presentation

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Intermolecular interactions and scattering

M.H.J. Koch

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Intermolecular interactions

• Phenomena like protein folding or association depend on the

balance of 1) protein-protein interactions (intramolecular or intersubunit) 2) protein-solvent interactions 3) solvent-solvent interactions the underlying phenomena (hydrogen bonds, Van der Waals or ionic interactions etc) are the same.

• The intermolecular interactions can be neglected in ideal

solutions, but these tend to be far remote from any real physiological or practical situation.

• Many systems of interest consist of fibers, or lipid systems

which may align and/or form gels (physical or chemical gels), colloidal suspensions, or even anisotropic systems rather than solutions.

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Proteins

Hydration shell Crowding max. conc. 300-500mg/ml IONS: Kosmotropes e.g. Na+ Chaotropes e.g. K+ OSMOLYTES e.g. free amino acids polyhydroxy alcohols methylated ammonium and sulfonium compounds urea. solvent Interactions/ stability/activity modulated by

FOLDING

Coupled equilibria Non-contact interactions

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Macromolecular crowding

Intermediate filament

actin

Interior of a yeast cell by D. Goodsell (from Hochachka & Somero, Biochemical Adaptation). Crowding and excluded volume effects stabilize proteins, but may reduce specific activity.

Microtubule Ribosome

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Intermolecular interactions are important

1) When proteins (or e.g. colloidal particles) should crystallize This is mainly a problem in protein crystallography. The interactions must be strong enough to induce crystallization and weak enough to avoid massive aggregation Proteins rarely crystallize inside cells (see e.g. Doye &Poon,

• Curr. Opin.Colloid Interface Sci. 2006, 11,40).

2) When proteins (or colloidal particles) should NOT crystallize The surface of proteins seem to have evolved to avoid crystallization in the crowded environment of the cell. Nanotechnology requires to produce particles with a finite size.

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Intermolecular forces and crystal growth

Adjustable parameters e.g.: pH, concentration of precipitant, ionic strength, concentration of detergent, amphiphile, surfactants…… Factors affecting crystallization: purity, T, P. vibrations, viscosity and dielectric constant of solvent, chemical modification, pI…… (see Chayen, Curr. Opin. Struct. Biol. 2004, 14:577)

The aim is to bring the macromolecules in a suitable state of supersaturation for nucleation and if possible back below the supersolubility curve for growth.

SS

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Intermolecular interactions

are usually difficult to quantify but it often suffices to recognize their signature in the scattering patterns to understand what happens.

• 1. Solutions of globular proteins (temperature, concentration,

salt, osmolytes, pressure)

• 2. Interactions of fibers
• 3. Interactions of lipids and proteins
• 4. In vivo these forces are associated with important PHASE

TRANSITIONS (e.g. chromatin condensation)

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I(s)

1 10

4

2 10

4

3 10

4

4 10

4

5 10

4

0.01 0.02 30C 25C 20C 15C 10C

s = 2(sin θ)/λ θ)/λ θ)/λ θ)/λ A

• 1
• A. Tardieu et al., LMCP (Paris)

Attractive Interactions

always INCREASE the intensity at small angles

Example: Temperature induced aggregation in a solution of γ-crystallins c=160 mg/ml in 50mM Phosphate pH 7.0

T

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Repulsive interactions

Repulsive interactions in a solution of BSA 5-50mg/ml Always DECREASE the intensity at small angles 50mg/ml 5mg/ml

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A simple case: monomer-dimer equilibrium

d

d = 5nm

+

Note:

The scattering of the dimer is 4 times that of the monomer but the number of dimers is half that of the monomers.

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Oligomer content in protein solutions

Example: monomer and dimer of Drosophila kinesin

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Monomer-dimer equilibrium as a function of concentration

s, nm-1 1 2 lg I, relative

• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

(1) (2) (3) (4) (5) (6) (7) (8)

c, mg/ml

2 4 6 8 10 12

Volume fraction

0.0 0.5 1.0

Monomer Dimer

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A Lennard-Jones type potential

Minimum is at 21/6σ ≈ 1.122σ

              −       =

6 12

4 d d V σ σ ε

is often used to explain equilibrium distances in e.g. virus capsids (see Zandi et al., PNAS 101, 15556-15560) , although there is nothing that prevents the formation

• f infinite structures

(crystals). repulsive attractive σ d

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The limitations of Lennard-Jones potentials

Arise from the fact that it is isotropic. Its lowest energy minimum with a large number of atoms corresponds to hexagonal close packing and at higher temperatures cubic close packing and then liquid. Proteins are anisotropic and are much larger than atoms for which the Lennard-Jones is valid. The potential between such particles is size-dependent and the situation closer to that in colloidal systems. (see J. Israelachvili, Intermolecular and surface forces).

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Intermolecular interactions partial order

Asymmetry creates symmetry (Curie’s principle) Finite objects 1D: fibers 2D crystals 3D crystals Objects made by repetition of a motif can be described as the convolution of the motif with an array of δ-functions. Infinite objects:

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Regular non-periodic structures

can be described as convolution (Flip-shift-multiply-integrate)

• f a motif with an array of δ-functions e.g.

concentrated solutions: semi-crystalline materials:

*

=

*

=

∗

= x x chain molecules: The Fourier transform of a convolution is the product of the transforms: FT(f*g)= FT(f)·FT(g)

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Chemical potential of the solvent in ideal solutions

...) (

3 2 3 2 2 2 2 2 1 1 1

+ + + − = − C A C A M C RTV µ µ

A2 A2 A2 Solute concentration

See: van Holde, Johnson & Cho, Principles of physical biochemistry

1

V

: molar volume of the solvent X= mole fraction

Solute-solute Interactions Attractive Repulsive A2 = A3 = 0 C2: solute concentration

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Intermolecular interactions: Non-ideal solutions

1 2 2

) , (

−

        ∂ Π ∂       = C M RT C SF

Osmotic pressure

1/SF(C2,0) =1+2MA2C2

Pseudo-lattice * solute = solution L(C2,s) X F(0,s) = F(C2,s)solution SF(C2,s) X I(0,s) = I(C2,s)solution Convolution:

* FT

2 2 3 2 2 2

C A C A 1/M RT) /(C + + = Π

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Using the Gibbs-Duhem equation with ni the number of moles of component i one can show that where A2 is the second virial coefficient which represents pair interactions and I(0)ideal is ∝ to C2. A2 is evaluated by performing experiments at various concentrations c. A2 is ∝ to the slope of C2/I(0,C2) vs C2. (e.g. in light scattering).

Virial coefficient

... 2 1 ) ( ) , (

2 2 2

+ + = MC A I C I

ideal

=

∑

i id

n µ

2 2 2 2 2 2

2 ln C M RT A C RT + = − µ µ

in the case of moderate interactions, the intensity at the origin varies with concentration of the solute according to :

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The DLVO (Derjaguin, Landau, Verwey, Overbeek) potential

σ

Long range repulsive (electrostatic) Hard sphere Short range attractive

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/rs)dr 1)(sin(rs) (g(r) 4π ρ 1 s) , S(C

2 2

− + =

∫

∞

g(r) = exp [-u(r)/kBT + h(r) - c(r)] Pair distribution function: The total (h(r) = g(r)-1) and direct c(r) correlation functions are related to g(r) by the hypernetted approximation

   > − − + − − ≤ =

∞

σ σ σ σ σ σ r d r r J d r r J r r u

r r a a

if ] / ) ( exp[ ) / ( ] / ) ( exp[ ) / ( if ) (

attractive and repulsive Yukawa potentials

Model based on the DLVO theory

(Tardieu et al. (1999) J. Crystal growth 196, 193-203 and Malfois et al. (1996) J. Chem. Phys. 105, 3290-3300 and Hard sphere (Ø=σ) potential The pair potential is: d =range

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2 D B 2 p

) σ/λ 0.5 /(1 L σ) / (Z Jr + =

Coulomb repulsion Protein charge hard sphere radius Bjerrum length(0.72 nm @300K)

∑

=

i 2 i i B D

Z ρ L 4π 1/ λ

= 3.4/I-½ Ionic charge number density

In the DLVO theory the repulsive potential is:

Ionic strength Debye length (range):

kT e L

s

ε πε0

2 B

4 / =

Debye length

ε0 permittivity of free space, εS=80

Note: If the ionic strength increases, λD decreases (increased screening)

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Intermolecular interactions: Lysozyme-KCl

KCl series in water

0.00 0.02 0.04 0.06 0.08 0.10 0.12

SF (s)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0 mM 5 mM 10mM 20mM 50mM 100mM 250mM simulated SFs

scattering intensities KCl salt series

0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 10 100 0 mM 5 mM 10 mM 20 mM 50 mM 100 mM 250 mM

normalisation range

q (Å-1) q (Å-1) Niebuhr& Koch, Biophys. J. (2005) SF I(q)

σ = 28.5Ǻ (not very sensitive 28.5-32Ǻ) da= 3Ǻ dr=λD Zp=6.5e-

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Yukawa potential and Debye length vs [salt]

   > − − + − − ≤ =

∞

σ σ σ σ σ σ r d r r J d r r J r r u

r r a a

if ] / ) ( exp[ ) / ( ] / ) ( exp[ ) / ( if ) (

2 D B 2 p

) σ/λ 0.5 /(1 L σ) / (Z Jr + =

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Change of the potential with [KCl]

r/σ r/σ

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Interactions of proteins, TMAO and urea

CH3 CH3 CH3 N O C O = NH2 NH2 UREA denatures proteins TMAO stabilizes proteins

• does NOT interact with proteins
• counteracts the effects of urea if [urea]/[TMAO] = 2
• protects against heat and pressure denaturation
• induces folding
• increases attractive interactions between proteins
• decreases Km (i.e. increases affinity for the substrate)

Note that TMAO is isosteric with tert-butanol (TBA) CH3 CH3 CH3 C HO These phenomena underlie the physiological mechanism of the TMAO/urea balance e.g. in fishes like sharks and rays interacts with proteins

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Lysozyme-KCl with TMAO or urea

s(Å-1)

KCL+TMAO KCl only KCl+urea The effect of TMAO on protein stability, folding, crystallization, counteraction of urea and intermolecular interactions is a property

• f the solvent system. TMAO/water is a

poorer solvent for the polypeptide backbone than water, whereas urea/water is a better

• solvent. There is no need for direct

protein-TMAO interactions.

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Depth of the attractive potential

The repulsive potential is the same in the three cases and depends only on [KCl].

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Effects of pressure on lysozyme solutions

1500bars 1bar Kratky plot Ortore et al. J.R. Soc. Interface (2009), 6, 8619-8634

SAXS

!

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Effect of pressure on lysozyme solutions

Ortore et al. J.R. Soc. Interface (2009), 6, 8619-8634 1000

Pressure (bar)

1bar 1500bars

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The potential changes with pressure

Ortore et al. J.R. Soc. Interface (2009), 6, 8619-8634 ρ/ρ0 Ζ(ε) Pressure (bars) Ja(KBT) da(Ǻ)

• hydr. shell/bulk

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Note that:

Temperature, pressure, pH, osmolytes, mainly change the properties of the solvent, not of the macromolecules. The DLVO and similar potentials can explain a few general effects but not yet phenomena like the Hofmeister series, which describes the effect of different salts on protein stability and solubility. Anions have large effects than cations The mechanism of the Hofmeister series remains unclear. but does not seem to result from changes in general water structure, instead more (specific interactions between ions and proteins and ions and the hydration shell?)

m guanidiniu Ca Mg Li Na K NH : Cations SCN CLO I ClO Br NO Cl Ac HPO SO F : Anions

2 2 4 4 3 3 2 4 2 4

> > > > > > > > > > > > > > > ≈

+ + + + + + − − − − − − − − − −

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The human eye

The cornea is transparent The sclera is opaque LENS

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The lens

Is the most concentrated protein (cristallins) solution in the body (300 mg/ml), yet it does not scatter light! This property results from the short range order arrangement of the cristallins. The central part of the lens is older than we are! With age, under the influence of radiation or in certain diseases like diabetes the lens becomes opaque as a result of cross-linking due to the Maillard reaction and the formation of larger aggregates.

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Cataract

Age (years) 1/s nm-1 Cataract can be easily detected by light scattering or fluorescence

Suarez, G. et al. (1993) J. Biol. Chem. 268 (24) 17716-21.

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Mixed solvents

Strong temperature-dependent X-ray scattering is observed with e.g. Trifluoro-ethanol (TFE) and -propanol (TFP) or hexafluoro-2-propanol (HFP) in conditions commonly used in NMR work on peptides. This is due to:

• Formation of clathrate hydrate-like aggregates of

alcohol with water

• Further heterogeneity of the solution due to

immiscibility of the two components.

Kuprin, S. et al. (1995) BBRC 217 1151-6

Iwasaki, K. & Fujiyama, T.(1976)

• J. Phys. Chem. 81, 1908-1912.

HFP

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DNA: scattering intensity I(s)= F2(s)·SF(s)

DNA is a fibre!

Slope Rx=1.0 nm ø 2.8 nm

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Divide by I(s)150 mM NaCl = F2(s) SF(s)

Calf thymus DNA, long and Polydisperse (5mg DNA/ml)

150 bp monodisperse DNA d The results depend only on the centre-to-centre distance between fibers and are a measure of the repulsive force.

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Pair distribution function (Zernike-Prins eq.)

Log([DNA]) Position of max ~ C1/2

g(r) gives a measure of the probability of finding the centre of a DNA rod at a distance r from any given rod. The position of the maximum does not depend much on [salt] but

• n [DNA]1/2 as foreseen for the semidilute (C> 1rod/Length3)

regime by polyelectrolyte theory.

ds sr sr s S C s r g )] 2 /( ) 2 sin( )[ 1 ) ( ( ) / 4 ( 1 ) (

2

π π π

∫

∞

− + =

g(r) r(nm)

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ISIS: a spallation neutron source

moderator

Neutron beam

T: Target: Tantalum-cladded Tungsten plates RFQ: radiofrequency quadrupole DLT: drift tube linac (linear accelerator)

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Spallation sources:moderator

On short pulse sources, the moderator must be thin in order not to degrade the pulse width too much large epithermal component. The example is for a methane moderator at ISIS TS1 with 12m flight path Cutoff due to the frequency of the source (50 Hz) λmax ≈ 6Ǻ 20 ms!

Spectrum of a reactor source drops off like this

TOF: Time of flight

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SANDALS: Small Angle Neutron Diffractometer

for Liquids and Amorphous Samples – ISIS (UK)

633 scintillators ZnS+PM Detectors 0.75 – 4 m Final trajectory 11 m Incident trajectory Liquid methane @ 110K Moderator 0.1 – 50 Ǻ-1 Q-range 0.05 - 4.95 Ǻ Incident wavelength

Incident neutrons scintillator and photomultiplier

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Studying interactions in solution

CH3 CH3 CH3 N O C O = NH2 NH2 O H H Use isotopic substitution HD and make first and second order differences

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The ideal case requires 7 samples

H H 7 HD HD 6 D D 1 Solute-solvent (2nd order differences) D H 5 D HD 4 D D 1 Solute-solute (first order differences) H D 3 HD D 2 D D 1 Solvent- solvent (first order differences) Solvent and exchangeable H Solute non-exchangeable H

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WANS

dr Qr Qr r g r Q S

∫

∞

− = −

2

sin ) 1 ) ( ( 4 1 ) (

αβ αβ

πρ

∫

=

2 1

2

) ( 4

r r

dr r r g c n

αβ β αβ

ρ π

∑ ∑ ∑∑

= =       − + =

≠ α α

N N N N c Q S b b c c b c N Q I and / where ] 1 ) ( [ 2 ) (

2 α α α α α β αβ β α β α α α

∫

− + = dQ QR Q Q S rN V g sin ) 1 ) ( ( 2 1

αβ αβ

π

Rewrite Debye equation in terms of atomic fractions:

Partial structure factor Partial pair distribution function (PPDF)

gαβ(r) is related to the probability of finding a site of type β at a distance r from a site of type α located at the origin. For a concentration cβ, the average number of atoms of type β in a shell extending from r1 to r2 surrounding the central α-atom is: For the first shell this is the COORDINATION NUMBER

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EXAMPLE: The TMAO-TMAO correlations can be obtained via the g(r) of the methyl hydrogen/deuterium sites.

methyl hydrogens methyl deuteriums remaining part of TMAO Heavy water: 1.25 M TMAO 1.25 M d-TMAO D2O 2.5 M TMAO 2.5M d-TMAO D2O

EXTRACTING THE PPDF

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Structure factors

Poorer fit due to inelasticity correction T + d-U in D2O d-T + d-U (D2O) ½T+½d-T+d-U (H2O/D2O) d-T+½ U+ ½d-U (H2O/D2O) ½(d-T+T+d-U+U) (H2O/D2O) T+U (H2O) d-T+U (H2O) Experimental

( )

) ( exp

2 1

Q R Q

i

NS i

N i

≡ ⋅

∑

=

TMAO-urea 1:2

XRD Simulations N: N+X:

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Empirical Potential Structure Refinement (EPSR)

see A.K. Soper Phys. Rev. B. (2005) 72: 104204

j i j i j i

r q q r r U

i j i ref β α β α β α β α β α β α β α β α

πε σ σ ε

, , 6 , , ,

4 4 2 1

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+                           −         = ∑

∑

≠

( )

αβ αβ αβ β α αβ

µ β α / 2

2 2 2 intra

d w w r C U

i i i

= = ∑∑

≠

     −       + = =

∑

σ σ πρσ σ σ r r n r p r p C kT r U

n n n n i i EP

i

exp )! 2 ( 4 1 ) , ( where ) , ( ) (

3

) ( ) (

, 1

Q S w Q F

j N j ij i

∑

=

=

Total potential= reference potential + empirical potential (EP)

Estimated from the experimental data Fit all separate data sets by Monte Carlo calculations involving intra- and intermolecular translations, rotations Standard form used to start the simulations After the simulation with Uref has equilibrated the EP guides the atomic and molecular moves to obtain the best fit to the experimental data.

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Are the results plausible ? TMAO vs TBA

TMAO: nO-Ow: 2.5 nO-Hw: 2.5 TBA: nO-Ow: 2.2 nO-Hw: 1.3

R-O H H-OH Me3NO H-OH O H H-OH H-OH H

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Clustering of urea

urea-water

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Clustering of urea

6.7M urea urea-urea At 2M urea the clusters contain at most 40 molecules, at 4M urea around 70 but at 6.7M urea most urea is in large clusters

• f 600-650 molecules

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Weak TMAO-urea interaction

1

M 0.1 a] [TMAO][ure urea] [TMAO K

−

≅ − = which is of the same order as the protein site-urea interaction

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• The structure of water is hardly changed by osmolytes,but

TMAO seems to strengthen H-bonds and urea to weaken them.

• There is a weak direct TMAO-urea interaction which is of the

same order as the protein site-urea interaction (K =0.1 M-1)

• The results of all molecular dynamics calculations in the

literature are incompatible with the neutron and X-ray scattering curves.

• Urea and TMAO affect the mobility of the fast fraction of

water in opposite ways (Rezus & Bakker, 2009). This can of course not be detected in an elastic neutron scattering experiment.

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What matters is the quality of the solvent

In a good solvent

• Rg increases
• protein - solvent interactions increase
• protein - protein interactions decrease
• solvent - solvent interactions decrease

Uwater Nwater Nurea Uurea Uwater Nwater UTMAO NTMAO Nwater Uwater UTMAO NTMAO poor TMAO neutral H2O good urea Structured proteins Unstructured proteins

See: D.W. Bolen & G.D. Rose (2008) Ann. Rev. Biochem. 77, 339-362

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Bound or simply present?

P R O T E I N urea water TMAO This zone is enriched in urea but excluded volume for TMAO For low K-values mass action is no longer valid!

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Conclusions

Understanding of protein stability and interactions is still very rudimentary. Scattering techniques offer a bridge between thermodynamics, molecular dynamics and spectroscopic methods but there are no simple answers with just one technique. The study of interactions is an active research area with many papers appearing in physics journals rather than biochemical

• nes.

For routine measurements of A2 use light scattering.