1

Intermolecular interactions

and scattering

M.H.J. Koch

2

Intermolecular interactions

-Phenomena like protein folding or association depend on thebalance of

1) protein-protein interactions (intramolecular or intersubunit)

2) protein-solvent interactions3) solvent-solvent interactions

the underlying phenomena (hydrogen bonds, Van der Waals orionic 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.

3

ProteinsHydration shell

Crowding max. conc. 300-500mg/ml

IONS:Kosmotropes e.g. Na+

Chaotropes e.g. K+

OSMOLYTESe.g. free amino acidspolyhydroxy alcoholsmethylated ammoniumand sulfonium compoundsurea.

solvent

Interactions/ stability/activitymodulated by

FOLDING

Coupled equilibriaNon-contact interactions

4

Macromolecular crowding

Intermediate filament

actin

Interior of a yeast cell by D. Goodsell(from Hochachka & Somero,Biochemical Adaptation).

Crowding and excluded volume effectsstabilize proteins, but may reducespecific activity.

Microtubule Ribosome

5

Intermolecular interactions are important

1) When proteins (or e.g. colloidal particles) should crystallizeThis 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.

6

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 ifpossible back below thesupersolubility curve forgrowth.

SS

7

Intermolecular interactions

are usually difficult to quantify but it often suffices to recognizetheir 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)

8

I(s)

0

1 104

2 104

3 104

4 104

5 104

0 0.01 0.02

30�C

25�C

20�C

15�C

10�C

s = 2(sinθ)/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

9

Repulsive interactions

Repulsive interactions in a solution of BSA 5-50mg/ml

Always DECREASE the intensity at small angles

50mg/ml

5mg/ml

10

A simple case: monomer-dimer equilibrium

d d = 5nm+

Note: The scattering of the dimer is 4 timesthat of the monomerbut the number of dimers is half that of the monomers.

11

Oligomer content in protein solutions

Example: monomer and dimer of Drosophila kinesin

Kozielski, F.,et al. (2001) J. Biol. Chem. 276, 1267-1275.

12

Monomer-dimer equilibrium

as a function of

concentration

s, nm-1

0 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

0 2 4 6 8 10 12

Volume fraction

0.0

0.5

1.0

Monomer

Dimer

13

A Lennard-Jones type potential

Minimum is at 21/6σ ≈ 1.122σ

−

=

612

4dd

Vσσ

ε

is often used to explainequilibrium distances in

e.g. virus capsids (see

Zandi et al., PNAS 101,15556-15560) , although

there is nothing thatprevents the formation

of infinite structures(crystals).

repulsive attractive

σ d

14

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 suchparticles is size-dependent and the situation closer to that in

colloidal systems.

(see J. Israelachvili, Intermolecular and surface forces).

15

Intermolecular interactions� partial order

Asymmetry creates symmetry(Curie’s principle)

Finite objects

1D: fibers2D 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:

16

Regular non-periodic structures

can be described as convolution (Flip-shift-multiply-integrate)of 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 thetransforms: FT(f*g)= FT(f)·FT(g)

17

Chemical potential of the solvent in ideal solutions

...)( 323

222

2

201

011 +++−=− CACA

M

CRTVµµ

A2

A2

A2

Solute concentration

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

01V : molar volume of

the solvent

X= mole fraction

Solute-soluteInteractions

Attractive

Repulsive

A2 = A3 = 0

C2: solute concentration

18

Intermolecular interactions: Non-ideal solutions

1

22 )0,(

−

∂

Π∂

=

CM

RTCSF

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

223222 CA CA 1/M RT)/(C ++=Π

19

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

...21)0(

),0(22

2++

=MCA

ICI ideal

0=∑ iidn µ

2222022 2ln CMRTACRT +=− µµ

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

20

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

σ

Long range

repulsive (electrostatic)

Hard

sphere

Short range

attractive

21

/rs)dr1)(sin(rs)(g(r)4πρ1s),S(C0

22 −+= ∫

∞

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

>−−+−−

≤=

∞σσσσσ

σ

rdrrJdrrJ

rru

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

if)(r

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

22

2DB

2p )σ/λ0.5/(1 Lσ)/(ZJr +=

Coulomb repulsion

Protein chargehard sphere radius

Bjerrum length(0.72 nm @300K)

∑=i

2iiBD ZρL4π1/λ = 3.4/I-½

Ionic chargenumber density

In the DLVO theory the repulsive potential is:

Ionic strength

Debye length (range):

kTeL sεπε02

B 4/=

Debye length

ε0 permittivity of free space, εS=80

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

23

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

normalisationrange

q (Å-1) q (Å-1)

Niebuhr& Koch, Biophys. J. (2005)

SFI(q)

σ = 28.5Ǻ (not very sensitive 28.5-32Ǻ)da= 3Ǻdr=λD

Zp=6.5e-

24

Yukawa potential and Debye length vs [salt]

>−−+−−

≤=

∞σσσσσ

σ

rdrrJdrrJ

rru

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

if)(r

2DB

2p )σ/λ0.5/(1L σ)/(ZJr +=

25

Change of the potential with [KCl]

r/σ r/σ

26

Interactions of proteins, TMAO and urea

CH3

CH3

CH3

NOCO =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

CHO

These phenomena underlie the physiological mechanism of the TMAO/urea

balance e.g. in fishes like sharks and rays

interacts with proteins

27

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 ureaand intermolecular interactions is a propertyof the solvent system. TMAO/water is a poorer solvent for the polypeptide backbone than water, whereas urea/water is a bettersolvent. There is no need for directprotein-TMAO interactions.

28

Depth of the attractive potential

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

29

Effects of pressure on lysozyme solutions

1500bars

1bar

Kratky plot

Ortore et al. J.R. Soc. Interface (2009),

6, 8619-8634

SAXS

!

30

Effect of pressure on lysozyme solutions

Ortore et al. J.R. Soc. Interface (2009),

6, 8619-8634

0 1000

Pressure (bar)

1bar

1500bars

31

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

32

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 effectsbut not yet phenomena like the Hofmeister series, which

describes the effect of different salts on protein stability andsolubility.

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?)

mguanidiniuCaMgLiNaKNH:Cations

SCNCLOIClOBrNOClAcHPOSOF :Anions22

4

43324

24

>>>>>>

>>>>>>>>>≈++++++

−−−−−−−−−−

33

The human eye

The cornea is transparent

The sclera is opaque

LENS

34

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 thecristallins.

The central part of the lens is older than we are!

With age, under the influence of radiation or in certain diseases likediabetes the lens becomes opaque as a result of cross-linking dueto the Maillard reaction and the formation of larger aggregates.

35

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.

36

Mixed solventsStrong 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

37

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

DNA is a fibre!

Slope � Rx=1.0 nm � ø 2.8 nm

38

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

Calf thymus DNA, long andPolydisperse (5mg DNA/ml)

150 bp monodisperse DNA

d

The results depend only on the

centre-to-centre distance between fibersand are a measure of the repulsive force.

39

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

on [DNA]1/2 as foreseen for the semidilute (C> 1rod/Length3)regime by polyelectrolyte theory.

dssrsrsSCsrg )]2/()2sin()[1)(()/4(1)(0

2 πππ ∫∞

−+=g(r)

r(nm)

40

ISIS: a spallation neutron source

moderator

Neutron beam

T: Target: Tantalum-cladded Tungsten platesRFQ: radiofrequency quadrupole

DLT: drift tube linac (linear accelerator)

41

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 ≈ 6Ǻ20 ms!

Spectrum of a reactor source drops off like this

TOF: Time of flight

42

SANDALS: Small Angle Neutron Diffractometer

for Liquids and Amorphous Samples – ISIS (UK)

633 scintillators ZnS+PMDetectors

0.75 – 4 mFinal trajectory

11 mIncidenttrajectory

Liquid methane @ 110KModerator

0.1 – 50 Ǻ-1Q-range

0.05 - 4.95 ǺIncidentwavelength

Incident neutrons

scintillator andphotomultiplier

43

Studying interactions in solution

CH3

CH3

CH3

NOCO =NH2

NH2

OH

H

Use isotopic substitution H�D

and make first and second order differences

44

The ideal case requires 7 samples

HH7

HDHD6

DD1Solute-solvent

(2nd order differences)

DH5

DHD4

DD1Solute-solute

(first order differences)

HD3

HDD2

DD1Solvent-

solvent

(first order differences)

Solvent and

exchangeable H

Solute

non-exchangeable H

45

WANS

drQr

QrrgrQS ∫

∞

−=−0

2 sin)1)((41)( αβαβ πρ

∫=2

1

2)(4r

r

drrrgcn αββαβ ρπ

∑∑ ∑∑ ==

−+=

≠ α

αNNNNcQSbbccbcNQI and / where]1)([2)( 2

ααα α αβ

αββαβααα

∫ −+= dQQRQQSrN

Vg sin)1)((

21 αβαβ

π

Rewrite Debye equation in terms of atomic fractions:

Partial structure factor

Partial pair distributionfunction (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 theCOORDINATION NUMBER

46

EXAMPLE: The TMAO-TMAO correlations can be obtained viathe 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

47

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

( ) )(exp2

1

QRQ i NSiN

i

≡⋅∑=

TMAO-urea 1:2

XRD

Simulations N:N+X:

48

Empirical Potential Structure Refinement (EPSR)see A.K. Soper Phys. Rev. B. (2005) 72: 104204

jijijir

qq

rrU

ijiref

βα

βα

βαβα

βα

βα

βαβα

πε

σσε

0, ,

6

,,, 4

421

12

+

−

= ∑ ∑≠

( )αβαβαβ

βα αβ

µβα

/ 2

22

2

intra dww

rCU

i

ii == ∑∑≠

−

+== ∑

σσπρσσσ

rr

nrprpCkTrU

n

nnn

i

i

EP

iexp

)!2(41

),( where),()(3

)()(,1

QSwQF j

Nj

iji ∑=

=

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 thesimulations

After the simulation with Uref has equilibrated the EP guides the atomic and molecular moves to obtain the best fit to the experimental data.

49

Are the results plausible ? TMAO vs TBA

TMAO:nO-Ow: 2.5

nO-Hw: 2.5

TBA:nO-Ow: 2.2

nO-Hw: 1.3R-O

H

H-OH

Me3N�O

H-OH

OH

H-OH

H-OH

H

50

Clustering of urea

urea-water

51

Clustering of urea

6.7M urea

urea-urea

At 2M urea the clusters contain at most 40molecules, at 4M urea around 70 but at

6.7M urea most urea is in large clusters

of 600-650 molecules

52

Weak TMAO-urea interaction

1M 0.1a][TMAO][ure

urea][TMAOK −≅

−= which is of the same order as the

protein site-urea interaction

53

- The structure of water is hardly changed by osmolytes,butTMAO seems to strengthen H-bonds and urea to weaken them.

- There is a weak direct TMAO-urea interaction which is of thesame 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.

54

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

NwaterNurea

Uurea

Uwater

Nwater

UTMAO

NTMAONwater

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

55

Bound or simply present?

PR

O

TE

IN

urea

water

TMAO

This zone is enriched in ureabut excluded volume for TMAO

For low K-values mass action is no longer valid!

56

Conclusions

Understanding of protein stability and interactions is stillvery 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

ones.

For routine measurements of A2 use light scattering.