Solution structure and dynamics of biopolymers

Transcription

1 Solution structure and dynamics of biopolymers Atomic-detail vs. low resolution structure Information available at different scales Mobility of macromolecules in solution Brownian motion, random walk, diffusion Transport coefficients: friction factor Stokes-Einstein relation for spherical particles Relations for other shapes Shape estimation from comparing low-angle scattering with hydrodynamic properties Light scattering Instrumentation Static scattering Guinier plot Radius of gyration, molecular mass Dynamic scattering Diffusion equation in Fourier space Autocorrelation function Examples Data analysis Methods for the study of biopolymer dynamics Analytical ultracentrifugation Fluorescence correlation spectroscopy

2 Dynamics of the genome time and length scales

3 Atomic-detail vs. low resolution structure Crystallography very detailed information about atomic-level chemistry and physics requires preparation and immobilization of sample in non-native state Solution structural methods sample can be maintained in close-to-native state information about the dynamics can be obtained But: information from any single method is very limited we must combine different techniques and compare the results with models Crystallography Solution structure

4 Information available at different scales unknown biopolymer point object solid sphere sphere with mass distribution elongated shape flexible object atomic detail M, v sedimentation equilibrium solution scattering molecular mass partial specific volume R h hydrodynamic radius R g radius of gyration sedimentation, diffusion R h = k BT 6phD solution scattering electron microscopy axial ratio combining information: M, v, R h, R g rotational diffusion fluorescence, NMR more complex shapes modelling dynamic light scattering fluorescence correlation spectroscopy crystallography, NMR

5 Transport of biomolecules in solution (and in the living cell!) Most transport processes in living cells are random. Many of these random processes come from thermal motion. What is heat? The random motion of molecules at temperatures above 0K. Each molecule contains the energy E = 1/2 k B T for each degree of freedom (translation, rotation, etc) k B = J K -1 is Boltzmann s constant N A = is Avogadro s number, the number of particles in one mol Thus, each degree of freedom contains the energy E = 1/2 N A k B T = 1/2 RT per mol R = N A k B = J mol -1 K -1 is the universal gas constant At room temperature (T=293 K), RT = 2.5 kj//mol

6 Brownian motion The random motion can be described by the time law: r 2 = 2n D Dt where D is the diffusion coefficient of the molecule and n D the number of spatial dimensions, i.e. <r 2 > = 2 Dt for 1-dimensional, <r 2 > = 4 Dt for 2-dimensional, <r 2 > = 6 Dt for 3-dimensional random motion Thus, to move 10 times as far, we need 100 times longer. example: typical protein, radius r = 2 nm, D = m 2 s -1 = 130 μm 2 s -1 traversing E.coli cell (r = 1 μm) takes t = <r 2 > / 6D = about 1 ms traversing eukaryotic cell (r = 20 μm) takes about 400 ms diffusing the length of a nerve axon (r = 1 m) would take about 10 9 s = 30 years!! Consequence: active transport plays no important role on short length scales but is absolutely necessary for distances > about 1 mm (in biology, not in geology)

7 Diffusion c(x) A x x+dx Diffusive flux through the plane A: (number of particles n per unit time, per surface A) Net flux in x-direction = (flux from left to right) (flux from right to left) j net ( x) = D ( ) - c( x + dx) ( c x ) dx = -D c x j(x) = 1 A n A µc(x) Ê Á Ë dn dt ˆ x Fick s first law of diffusion j x ( ) = -D c x = -grad c

8 Diffusion c(x) A x x+dx Concentration change in volume element dv = A dx c t = ( ) - n ( x + dx) n x dv ( ( ) - j(x + dx) ) = A j x A dx = - j 2 x = D c x 2 Fick s second law of diffusion in 3 dimensions: c t = D Ê 2 c x + 2 c 2 y + 2 c ˆ Á = D 2 c Ë 2 z 2

9 Diffusion and friction are related force F velocity v f is friction coefficient For a sphere with radius R, f = 6πηR (Stokes equation) Einstein showed that F = - f v D = k B T f (i.e. the diffusion coefficient of a molecule is inversely proportional to its size) Thus, if the molecular mass is M and the partial specific volume is v, the radius of a globular protein would be R h = 3V 3 = 3v M 3 4p 4p and its diffusion coefficient D = k BT 4p 3 6ph 3v M Thus, D M -1/3 (difference between monomer and dimer only about 25%!)

10 Diffusion for elongated objects measured f is often larger than that expected for a sphere (f0) Perrin developed relationships for f of ellipsoidal objects of same volume as a sphere f/f0 is also called Perrin factor or form factor prolate = cigar oblate = discus For cylinders, another relationship has been derived (Garcia de la Torre, 1977) f = 6pha ( ) + g ln a b a = long, b = short half-axis γ = 0.3 for a/b = 1, 0.5 for a/b large

11 Molecular shape from diffusion and molecular mass Example: protein with M = , v = 0.75 cm 3 /g, volume per molecule: V = m 3 3 = m 2 s -1 D = k BT 6ph 4p 3v M but measured only m 2 s -1 two possible reasons: hydration: Volume is (6.8/5.0) 3 = 2.5 times larger due to associated water elongation, f/f 0 = 1.36, axial ratio a/b = 6 (from previous plot) for you to choose! THL: it is hard to distinguish between monomers and dimers in hydrodynamics molecular mass cannot be determined from hydrodynamic experiments (i.e. gel filtration, electrophoresis)

12 Modeling hydrodynamic friction factors System of several subunits shows hydrodynamic interaction: force on subunit i induces a solvent flow velocity at j <T> = rotationally averaged Oseen tensor r i i j T = (zero order approximation) Kirkwood approximation for a molecule of N spherical subunits of radius R: Assumptions: j Dv i = T F j subunits small, all same radius 1 6phr ij I f = 1 + R N N 6phR ÂÂ i j π i rotationally averaged hydrodynamic interaction (no orientation in the flow) works approximately (but imagine a hollow object with a small sphere inside!) Better algorithms exist (HYDRO, HYDROPRO Garcia de la Torre) r ij -1

13 How do we measure diffusion coefficients? Diffusion in a viscous fluid has nothing to do with mass (this should be clear by now) particle in a viscous medium, force f: for typical protein, τ = 10 ps, thereafter v is constant = v 0 = F/f Measure transport properties Forced motion: F = m dv dt -fv; v t sedimentation coefficient (analytical ultracentrifugation) electrophoretic mobility gel filtration Random motion: classical diffusion measurements (boooooooring...) dynamic light scattering fluorescence correlation spectroscopy Ê ( ) = v 0 Á 1 - e - Á Ë t t ˆ ; t = m f

14 Fluorescence correlation spectroscopy (FCS) (our first single molecule technique) FCS can be used to determine: Diffusion coefficients and local concentration of fluorescent molecules Association of biomolecules at very low concentrations Mobility of fluorescent probes inside biological structures

15 Fluctuations of intensive properties of small systems Example: 1nM solution of rhodamine Size [mm] Volume [l] # of particles N N/N [%] e e e e Ê DN ˆ Ë N 2 Ê = Dc ˆ Ë c 2 = 1 N ; N = c N Lv M M = c Ê Á Ë ˆ Dc 2 c NL v

16 Autocorrelation analysis The autocorrelation function characterizes the temporal fluctuations of a measured quantity: Fluorescence signal I(t) τ Calculating the normalized autocorrelation function (ACF) G ( t ) = di ( t )di ( t +t ) I( t ) 2 Normalized autocorrelation fuction t ln(τ ) Fitting appropriate model functions to the autocorrelation function yields the diffusion coefficient the concentration of several species with different hydrodynamic properties

17 Fluorescence autocorrelation function for Gaussian beam profile Detected fluorescence intensity profile in confocal geometry: Microscope lens I( x, y, z) = I e 0 È -2Í Í Î ( x 2 + y 2 ) 2 w 0 + z2 z 0 2 beam waist w 0 z 0 Fluorescence intensity autocorrelation function of particles diffusing through the beam: G ( 2 ) t Ê Á Ë ( ) =1 + 1 N 1 + 4Dt w 0 2 ˆ Ê 1 + 4Dt ˆ Á 2 Ë z 0 = Ê N 1 + t ˆ Ë Á t D 1 + t -1 2 Ê ˆ Ë Á k 2 t D -1 2 κ = z 0 /w 0 ( structure factor ) τ D : diffusion time N: number of particles in focus (Aragon & Pecora, J. Chem. Phys. 64 (1976) 1791)

18 Cross correlation analysis Fluorescence signals I g (t), I r (t) τ Calculating the normalized cross correlation function (CCF) G ( t ) = di g( t)di r ( t +t ) I g ( t)i r ( t) Normalized cross correlation fuction t ln(τ ) Fitting appropriate model functions to the cross correlation function yields the diffusion coefficient the fraction of double labelled species

19 Fluorescence Fluctuation Microscope (Malte Wachsmuth, Michael Tewes) European Patent No (2001) cytoplasm nucleus cell nucleus laser filters dichroic mirrors AT-1 expressing EGFP scan lens detectors pinhole dichroic mirror rotating scan mirrors Characteristics: Confocal FCS module with two avalanche photodiode detection channels Operating modes: FCS, cross correlation, photon count histogram, singlemolecule FRET Integrated galvanometer scanner allows positioning of FCS focus spot and enables imaging in intensity and correlation mode

20 Modular FCS device - system components Olympus IX-70 inverted microscope 60X1.2W water immersion lens with cover slide correction x-y translation stage, focus drive (z), 50 nm resolution, 0.5 µm precision Scanner two-mirror galvanometer scanner (GSI Lumonics), Olympus scan lens positioning accuracy 25 nm FCS module box with filters and dichroic mirrors, fiber coupler Avalanche photodiode detectors (EG&G SPCM AQ-14) QE 630 nm, dead time = 30 ns Laser Omnichrome 50 YB Ar-Kr, / 568 nm pulsed Nd-vanadate (4 ps pulses) He/Cd, nm Correlator ALV-5000 correlator board Fast data acquisition board for time correlated single photon counting Control and data analysis software: own development

21 Modular FCS device - characteristics 1000 N LUMPlanFl 60x /0.9 W UPlanApo 60x /1.2 W 0.1 Sensibility tested with rhodamin 6G Detection limit < 10 pm Confocal volume: 0.26 fl (60X/1.2W) 1.21 fl (60X/0.9W) Structure factor κ = 5 (w 0 = 0.46 µm, z 0 = 2.3 µm) Diffusion time [ms] E-5 1E-4 1E c[nm] LUMPlanFl 60x /0.9 W UPlanApo 60x /1.2 W E-5 1E-4 1E c[nm]