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

Intermolecular interactions and scattering M.H.J. Koch 1

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 2 than solutions.

Proteins Hydration shell solvent FOLDING Interactions/ stability/activity modulated by Coupled equilibria Non-contact interactions OSMOLYTES IONS: e.g. free amino acids Kosmotropes e.g. Na + polyhydroxy alcohols Chaotropes e.g. K + methylated ammonium Crowding max. conc. and sulfonium compounds 3 300-500mg/ml urea.

Macromolecular crowding 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. Intermediate filament actin Ribosome Microtubule 4

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. 5

Intermolecular forces and crystal growth 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. Adjustable parameters e.g.: pH, concentration of precipitant, ionic strength, concentration of detergent, amphiphile, SS surfactants…… Factors affecting crystallization: purity, T, P. vibrations, viscosity and dielectric constant of solvent, chemical modification, pI…… (see Chayen, Curr. Opin. Struct. Biol. 2004, 6 14:577)

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) 7

Attractive Interactions always INCREASE the intensity at small angles 4 5 10 30 � C 25 � C 4 4 10 20 � C 15 � C I(s) 4 3 10 10 � C T Example: Temperature induced 4 aggregation in a solution of 2 10 γ -crystallins c=160 mg/ml in 50mM Phosphate pH 7.0 4 1 10 0 A. Tardieu et al., LMCP (Paris) 8 0 0.01 0.02 -1 θ)/λ θ)/λ s = 2(sin θ)/λ θ)/λ A

Repulsive interactions Always DECREASE the intensity at small angles 5mg/ml 50mg/ml Repulsive interactions in a solution of BSA 5-50mg/ml 9

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. 10

Oligomer content in protein solutions Example: monomer and dimer of Drosophila kinesin ���������������� ����������� ��������������� ��� ������������ 11

lg I, relative Monomer-dimer equilibrium (1) -1 as a function of concentration (2) -2 -3 (3) Volume fraction -4 (4) 1.0 Monomer -5 (5) Dimer -6 (6) 0.5 -7 (7) -8 (8) 0.0 -9 0 2 4 6 8 10 12 c, mg/ml 0 1 12 2 s, nm -1

A Lennard-Jones type potential is often used to explain σ d equilibrium distances in e.g. virus capsids (see Zandi et al., PNAS 101, 15556-15560) , although there is nothing that prevents the formation of infinite structures Minimum is at 2 1/6 σ ≈ 1.122 σ (crystals).   12 6     σ σ 4       = ε − V       d d   13 repulsive attractive

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). 14

Intermolecular interactions � partial order Finite objects Asymmetry creates symmetry (Curie’s principle) Infinite 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. 15

Regular non-periodic structures can be described as convolution (Flip-shift-multiply-integrate) of a motif with an array of δ -functions e.g. chain molecules: * = concentrated solutions: * = semi-crystalline materials: ∗ = x x The Fourier transform of a convolution is the product of the 16 transforms: FT(f*g)= FT(f)·FT(g)

Chemical potential of the solvent in ideal solutions Solute-solute Interactions A 2 Attractive A 2 A 2 = A 3 = 0 Solute concentration A 2 Repulsive X= mole fraction 0 : molar volume of V C 1 0 0 2 3 ( 2 ...) the solvent µ − µ = − + + + RTV A C A C 1 1 1 2 2 3 2 M 2 C 2 : solute concentration 17 See: van Holde, Johnson & Cho, Principles of physical biochemistry

Intermolecular interactions: Non-ideal solutions * Pseudo-lattice * solute = solution Convolution: L(C 2 ,s) X F(0,s) = F(C 2 ,s) solution FT SF(C 2 ,s) X I(0,s) = I(C 2 ,s) solution 1 −     ∂ Π RT 2 /(C RT) 1/M A C A C     Π = + + ( , 0 ) = SF C   2 2 2 3 2 2    ∂  M C 2 Osmotic pressure 1/SF(C 2 ,0) =1+2MA 2 C 2 18

Virial coefficient ∑ 0 n µ = i d Using the Gibbs-Duhem equation with n i the i number of moles of component i one can show that 0 ln 2 µ − µ = + RT C A RT M C 2 2 2 2 2 2 in the case of moderate interactions, the intensity at the origin varies with concentration of the solute according to : ( 0 ) I ( 0 , ) ideal = I C 2 1 2 ... + + A MC 2 2 where A 2 is the second virial coefficient which represents pair interactions and I(0) ideal is ∝ to C 2 . A 2 is evaluated by performing experiments at various concentrations c. A 2 is ∝ to the slope of C 2 /I(0,C 2 ) vs C 2 . (e.g. in light scattering). 19

The DLVO (Derjaguin, Landau, Verwey, Overbeek) potential Long range repulsive (electrostatic) σ Short range Hard attractive sphere 20

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 ∞ ∫ 2 S(C , s) 1 4 π (g(r) 1)(sin(rs) /rs)dr ρ = + − 2 0 Pair distribution function: g(r) = exp [-u(r)/k B T + h(r) - c(r)] The total (h(r) = g(r)-1) and direct c(r) correlation functions are related to g(r) by the hypernetted approximation The pair potential is: Hard sphere (Ø= σ ) potential ∞  if ≤ σ ( � r )  = u r ( / ) exp[ ( ) / ] ( / ) exp[ ( ) / ] if  σ − − σ + σ − − σ > σ J r r d J r r d r a a r r 21 attractive and repulsive Yukawa potentials d =range

In the DLVO theory the repulsive potential is: Coulomb repulsion Debye length 2 2 J r (Z / σ ) L /(1 0.5 σ / λ ) = + p B D Bjerrum length (0.72 nm @300K) Protein charge 2 / 4 = πε 0 ε L e kT hard sphere radius B s ε 0 permittivity of free space, ε S =80 ∑ 2 λ 1/ 4 π L ρ Z Debye length (range): = = 3.4/I -½ D B i i i Ionic charge number density Ionic strength Note: If the ionic strength increases, λ D decreases (increased screening) 22

Intermolecular interactions: Lysozyme-KCl 100 I(q) 1.4 scattering intensities KCl series in water SF KCl salt series 1.2 1.0 normalisation range 10 0.8 SF (s) 0 mM 0.6 5 mM 0 mM 10 mM 5 mM q (Å -1 ) 20 mM 0.4 q (Å -1 ) 10mM 50 mM 20mM 1 100 mM 50mM 100mM 250 mM 0.2 250mM simulated SFs 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.02 0.04 0.06 0.08 0.10 0.12 σ = 28.5 Ǻ (not very sensitive 28.5-32 Ǻ ) d a = 3 Ǻ d r = λ D Z p =6.5e - 23 Niebuhr& Koch, Biophys. J. (2005)