Translational Diffusion: measuring the frictional force on the movement of a macromolecule in solution. A particle under the influence of a constant applied force will accelerate. If it interacts with the medium (solvent) the frictional force (F f ) opposing that acceleration is proportional to the velocity (u). The proportionality coefficient which relates the frictional force (F f ) to the velocity (u) is called the frictional coefficient, f. This is a function of the molecular size and shape, and also contains the viscosity of the solution, η. Note that the η here is not a property of the molecule itself, as is intrinsic viscosity [η], but is just simply the viscosity of the solution, which is usually 0.01 Poise. Hence, by measuring the speed with which a particle (protein) is moving through solution we are getting information about the limiting frictional resistance to that movement, which is contained within the frictional coefficient. It is useful here that molecules generally obey the same hydrodynamic rules that apply to macroscopic objects. In the case of macroscopic objects, such as a marble, the dependence of the frictional coefficient on molecular size and shape is known.

Translational diffusion: a sphere has the following frictional coefficient: f trans = 6πηR s. The larger the radius of a sphere, the larger the frictional coefficient, and the slower will be the velocity under a given driving force. Rotational diffusion: for a particle rotating in solution, there is also frictional resistance due to the shear force with the solvent. The hydrodynamics of this for a sphere is f rot = 8πηR s 3. Note that for rotational diffusion, the frictional coefficient increases in proportion to the volume of the sphere, whereas for translational diffusion, the dependence is proportional to the radius.

Frictional Force (F f ) applied Force (F ap ) Diffusion Frictional Resistance to Macromolecule Motion  velocity, u F f = f u f  frictional coefficient steady state - terminal velocity is reached u = (F ap / f) For a sphere: f trans = 6  R S translational motion f rot = 8  R S 3 rotational motion Measure molecular motion  f  R s  molecular size and shape information

Diffusion is a purely statistical (or entropic) concept. It represents the net flow of matter from a region of high chemical potential (or concentration) to a region of low chemical potential. Diffusion can be considered as 1) the net displacement of a molecule after a large number of small random steps: random walk model. Alternatively, 2) it can also be looked at in terms of the net flow of matter under the influence of a gradient in the chemical potential, which can be formally treated as a hypothetical "force" field. Both approaches provide useful physical insight into how the diffusion of macromolecules can be utilized to yield molecular information. We will derive the concepts in terms of 1-dimension, and then generalize to 3-dimensions.

Translational diffusion Measured classically by observing the rate of “spreading” of the material: flux across a boundary Two equations relate the rate of change of concentration of particles as a function of position and time: Fick’s first law Fick’s second law Phenomenological Equations

A  area of reference plane n 1 n 2 J = flux  moles (net) area time l Flux of particles depends on the concentration gradient Fick’s first law (concentration) Define the diffusion coefficient: D (cm 2 /sec) Fick’s 1 st law: J = -D dn dx

Fick’s 2 nd law: = D d 2 n dx 2 dn dt area  A dn (J 1 - J 2 ) A dt dx A = dx J1J1 J2J2 Change in concentration requires a difference in concentration gradient Fick’s second law volume change in # particles per unit time Constant gradient: same amount leaves as enters the box Gradient higher on left: more enters the box than leaves

Translational Diffusion Solve Fick’s Laws - differential equations ( relate concentration, position, time) (define boundary and initial conditions) diffusion in 1 - dimension All (N) particles are at x = 0 at t = c(x,t) = (  D t) 1/2 e  solution N2N2 -x 2 / 4 D t Note: = average value of x 2 = p(x) dx (x 2 ) where p(x) is the probability of the molecules being at position x = 2 D t mean square displacement +  -  Material spreads (Gaussian)

Mean square displacement - in 3-dimensions isotropic diffusion D x = D y = D z = 2Dt l 2 = 6Dt where l 2 = + + D = cm 2 /sec NOTE: Mean displacement   time l 2 6t average displacement from the starting point = (6Dt) 1/2

Values of Diffusion Coefficients 1. Diffusion in a gas phase: D  1 cm 2 /sec (will depend on length of λ, the mean free path) 2. Diffusion within a solid matrix: D ≈ cm 2 /sec or much smaller 3. Diffusion of a small molecule in solution, such as sucrose in water at 25° C: D ≈ cm 2 /sec (takes 3 days to go ~ 4 cm) 4. Diffusion of a macromolecule in solution, such as serum albumin (BSA, 70,000 mol weight) in water at 20° C: D ≈ 6 · cm 2 /sec D  cm 2 /sec (3 days  1 cm) 5. Very large molecules such as DNA diffuse so slowly that the measurements cannot even be made. NOT a useful technique

Measuring the translational diffusion constant The classical method of measuring the translational diffusion constant is to observe the broadening of a boundary that is initially prepared by layering the protein solution on a solution without protein. In practice, this is not done. The preferred method for a purified protein is to use dynamic or quasi-elastic light scattering to get a value of the diffusion coefficient. The method measures the fluctuations in local concentrations of the protein within solution, and this depends on how fast the protein is randomly diffusing in the solution. There are commercial instruments to perform this measurement. Note that this does not yield a molecular weight but rather a Stokes radius. However, it is more common to use mass transport techniques to measure the Stokes radius of macromolecules. As we saw with osmotic pressure and with intrinsic viscosity, it is often necessary to make a series of diffusion measurements at different concentrations of protein and then extrapolate to infinite dilution. When this is done a superscript “0” is added: D o. The next slide shows results obtained for serum albumin.

Translational Diffusion Classical technique  boundary spread (mg/mL) x (distance) BSA * values of D have NOT been corrected for pure water at 20 o C DoDo C (mg/mL) x D Extrapolate to infinite dilution (designated by D o ) startend

Another way to measure the value of the Diffusion Coefficient is by Fluorescence Correlation Spectroscopy (FCS) By measuring fluctuations in fluorescence, the residence time of a fluorescent molecule within a very small measuring volume (1 femtoliter, L) is determined. This is related to the Diffusion Coefficient molecules moving into and out of the measuring volume: fast (left) vs slow (right) excitation emitted photons

Fluorescence Correlation Spectroscopy: FCS the time-dependence of the fluorescence is expressed as an autocorrelation function, G(  ), the is the average value of the product of the fluorescence intensity at time t versus the intensity at a short time, , later. If the values fluctuate faster than time  then the product will be zero. deviation from the average intensity

An example of FCS: simulated autocorrelation functions of a free fluorescence ligand and the same ligand bound to a protein free ligand (small, fast diffusion) bound ligand on slow moving protein 1:1 mix of free/bound ligand G(  ) goes to zero at long times

Relating D to Molecular Properties Treat Diffusion as material flow due to “force” J = flux = c V concentration V = (F ap / f) F ap = - d  dx Velocity “Force” t o t1t1  =  o + kT ln c F ap = - = - kT = V f d  d(ln c) dx J = c V = - = - D c k T d(ln c) dc f dx dx Fick’s 1 st law 1 dc c dx - (or thermodynamic activity) Net “velocity” of Diffusing material D = k T f

D = kT 6  R S Stokes-Einstein Equation Relating D to molecular properties Rate of mass flux is inversely proportional to the frictional drag on the diffusing particle D = kT f k = Boltzman constant f = frictional coefficient But f = 6  R S for a sphere or radius R s  = viscosity of solution This expression appears to be valid for large objects such as marbles, and for macromolecules, and even for small molecules, at least to the extent that D·η is a constant for a molecule of fixed radius as the viscosity is changed.

Stokes Radius obtained from Diffusion kT Assumes a spherical shape 6  R S Stokes Radius kT 6  D 1 You need additional information to judge whether the particle is really spherical -A highly asymmetric particle behaves like a larger sphere - higher frictional coefficient (f) 2 Deviations from the assumption of an anhydrous sphere (R min ) are due to either a) hydration b) asymmetry 3 Stokes radius from different techniques need not be identical D = R S = 1. Measure D 2. Calculate R s

kT kT f 6  R S f = 6  R s Define: f min = 6  R min so: R s f R min f min Hydrodynamic theory defines the shape dependence of f, frictional coefficient D = = = b a (a / b) f / f min this allows one to estimate effects due to molecular asymmetry R min vol = [V 2 ] =  R 3 min 4343 MNMN Experimental: Theoretical: 1. Measure D and get R s 2. Compare R s to R min Interpreting the meaning of the Stokes Radius

Shape factor for translational diffusion frictional coefficient of ellipsoids for a prolate ellipsoid frictional coefficient shape factor viscosity shape factor much larger effect of shape on viscosity than on diffusion (Cantor + Schimmel) oblate prolate f/f min

Interpreting Diffusion Experiments: does a reasonable amount of hydration explain the measured value of D? Protein M D o 20,W x10 -7 cm 2 /s R S (Å) ( diffusion) RNAse 13, (R min =17Å) Collagen 345, (R min =59Å) protein Maximum solvation Maximum asymmetry RNAse  H2O = 0.35 a/b = 3.4 Collagen  H2O = 218 a/b = 300 R S (V p +  H2O ) 1/3 R min V p solve for  H2O = Volume per gram of anhydrous protein

What is the Diffusion Coefficient of a Protein in the bacterial cytoplasm? Are proteins freely mobile? Some proteins will be tethered Some proteins will interact transiently with others and appear to move slowly Free diffusion will be slower due to “crowding” effect excluded volume effect at high concentration of protein

Measuring the Diffusion of Proteins in the Cytoplasm of E. coli Fluorescence Recovery After Photobleaching (FRAP) 1. Express a protein that is fluorescent: green fluorescent protein, GFP. 2. Use a laser to “photo-bleach” the fluorescent protein in part of a single bacterial cell. This permanently destroys the fluorescence from proteins in the target area. 3. Measure the intensity of fluorescence as the protein diffuses into the region which was photo-bleached. Ready.. Aim... Fire! Diffusion of protein into the spot t0t0 t1t1 t2t2 E. coli cell

Single cell, expressing GFP Bleach cell center with a laser, t 0 t = 0.37 sec after flash t = 1.8 sec after flash J. Bacteriology (1999) 181, Diffusion of the Green Fluorescent Protein inside E. coli 4 µm one can observe the molecules diffusing back into the bleached area

Diffusion of the Green Fluorescent Protein inside E. coli Results: D = 7.7 µm 2 /sec (7.7 x cm 2 /sec) this is 11-fold less than the diffusion coefficient in water = 87 µm 2 /sec Slow translational diffusion is due to the crowding resulting from the very high protein concentration in the bacterial cytoplasm ( mg/ml) J. Bacteriology (1999) 181,

Mass Transport Techniques Measure the steady state velocity of hydrodynamic particles under the influence of an applied force 1 Sedimentation velocity 2 Electrophoresis 3 Gel filtration chromatography F ret retardation force F ap applied force

(A) Sedimentation velocity F ap = centrifugal force F ret = frictional drag measure: velocity = sedimentation centrifugal acceleration coefficient S (B) Electrophoresis F ap = electrical force F ret = frictional drag (through gel) measure: velocity = electrophoretic electric field mobility U

(C) Gel filtration chromatography F ap = solvent flow F ret = partitioning into stationary phase particles small molecules partition into beads - This retards their progress down the column Large particles are excluded from the bead interior -elute faster Measure elution volume, V e, or  V o = elution volume of totally excluded molecules V I = elution volume of totally included molecules measure particle velocity solvent velocity solvent flow V o V e V I elution volume V e - V o V I - V o  = beads -stationary phase matrix

F ap = centrifugal force F ret = frictional drag measure: velocity = sedimentation centrifugal acceleration coefficient S Sedimentation velocity F ret retardation force F ap applied force

Sedimentation Velocity F ap =  2 r (m h -  h  )....  = circular velocity radians / sec r centrifuge mass of particle corrected for buoyancy Note: terms for bound water drop out of equation in steady state: F ap = F ret = f(velocity) measure F ap =  2 r M (1 - V 2  ) N m h = (1 +  H2O ) ;  h = (V 2 + V H2O  H2O ) M N S = = velocity M (1 - V 2  )  2 r Nf Substitute:

Sedimentation Coefficient: depends on three molecular variables: M, V 2, and f units : seconds 1 Svedberg = seconds mol. wt inverse density of particle shape dependence S = M (1 - V 2  ) Nf f = 6  R s

infinite dilution S o 20,w 20 o C in water (correct for viscosity and temperature from conditions of actual measurement) S = M (1 - V 2  ) Nf Sedimentation value depends on solution conditions:  and  S-values are usually reported for “standard conditions” f = 6  R s

Types of Centrifuges used to measure the S-value 1Analytical Ultracentrifuge (monitor the distribution of material by absorption or dispersion) as a function of time –Method of choice, but requires specialized equipment –Beckman “Optima” centrifuge –small sample, but must be pure - optical detection used to determine sedimentation velocity  S (frontal analysis (moving boundary method) - not zonal method) 2Preparative Ultracentrifuge –common instrumentation –sedimentation coefficient obtained by a “zonal method”  requires a density gradient to stabilize against turbulence / convection  obtaining S usually requires comparison to a set of standards of known S value