Dynamics in nanoworlds

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1 Dynamics in nanoworlds Interplay of energy, diffusion and friction in (sub)cellular world 1

2 NB Queste diapositive sono state preparate per il corso di Biofisica tenuto dal Dr. Attilio V. Vargiu presso il Dipartimento di Fisica nell A.A. 2014/2015 Non sostituiscono il materiale didattico consigliato a piè del programma. 2

3 References Books and other sources Biological Physics (updated 1 st ed.), Philip Nelson, Chap. 5 Biochemistry (5 th ed.), Berg et al., Chap. 4 Movies Exercise 3

4 Sorting molecules by mass/charge A way of separating molecules by their mass is to centrifuge them in order to accelerate sedimentation. Recalling Archimedes principle, force driving sedimentation of an object of mass m under the action of a acceleration gravity a s directed along z is: f = du dz = ( m V l ρ l )a s m net a s $% U = ma s Δz ( V l ρ l a s )Δz& ' Flux j of molecules under force f and diffusion D can be calculated as (analogous of Nerst-Planck formula): ( ) = m neta s c( z) j z ς D c z ( ) ς D=k B T ### j z % ( ) = D m net ' k B T a sc z & ( ) c( z) ( * ) 4

5 Sorting molecules by mass/charge Centrifuges are needed because at equilibrium (j=0) thermal agitation often keeps a relevant fraction of particles in suspension: c( z) e m neta s z k T B = e z z* z* = k B T m net a s z* scale height Example a s =g and m typical values of a globular protein (e.g. myoglobin): m 17Kg/mol (m net m/4) gives for z* a value of ~60m. In a tube of 10 cm the concentration at the top is equal to ~ 99.8 % of that at the bottom: c( 10 cm) c( 0 cm)e 0.1m 60m 99.8 c( 0 cm) The suspension never settles out equilibrium colloid or colloidal suspension. Settling occurs when: m net a s h k B T >1 "# h = z top z bottom $ % 5

6 Sorting molecules by mass/charge For a given molecule, one can act on h, T, and on a s, as done in centrifuges, where a s can be as high as 10 6 m/s 2. Equilibrium found again by imposing compensation between drift flux, due to viscosity of the medium where particles are suspended: j drift ( r) = v drift c( r) = f ς c ( r ) = m netω s ς 2 r ( ) = m netω s c r k B T 2 rd c( r) and diffusion flux due to Fick s law: ( ) = D dc ( r ) j D r dr giving the equilibrium concentration as a function of radial position: c( r) e m netω 2 s r 2 2k B T sedimentation distribution in centrifuge 6

7 Sorting molecules by mass/charge A sedimentation coefficient s can be defined which is an intrinsic property of the particle: s = v drift g = m net ς s is the time needed for a particle to reach a terminal velocity, and it is measured in svedbergs (10-13 s). s depends on friction ζ, in turn depending on viscosity η of the fluid where particles are suspended, as well as on their shape. For a spherical particle of radius R in a medium with viscosity η, friction is given by Stokes formula: ς = 6πηR η for water at room temperature is 10-3 Pa s. 7

8 Sorting molecules by mass/charge s = v drift g = m net ς ς = 6πηR 8

9 Sorting molecules by mass/charge In electrophoresis sedimentation is facilitated by means of applied electric field to a ionic solution. Drift velocity will be proportional to charge and electric field strength: v drift = f ς = qe ς As for the centrifuge, an intrinsic coefficient can be defined, the electrophoretic mobility µ e : µ e = q ς!!!!!! µ spherical particles e = q 6πηR Stokes approximation often valid also for non-spherical particles due to solvation shell of charged molecules in polar ionic solutions. With charged biomolecules mobility will depend on viscosity of medium, and on shape and charge of particles. 9

10 Sorting molecules by mass/charge Gel electrophoresis one of preferred techniques to separate and analyze molecular components of biological assemblies. Put macromolecule (DNA, RNA, protein) in a gel and switch on electric field. Add denaturing agent (e.g. SDS for proteins) to reduce effects of different folding. One SDS anion binds about two aa in proteins charge proportional to protein mass. Gel used as molecular sieve enhancing filtration of molecules by their size. Biochemistry, J. Berg et al., 5 th ed.,

11 Sorting molecules by mass/charge Separation is reflected by occurrence of different migration bands (e.g. staining with dye). Small proteins move rapidly through the gel, whereas large ones not. µe of most polypeptide chains under these conditions ~ log M (some carbohydrate-rich proteins and membrane proteins escape this rule). Biochemistry, J. Berg et al., 5th ed.,

12 Mixing and viscosity From Einstein and Stokes relations, inverse proportionality between diffusion coefficient of spherical object and viscosity of medium: D = k B T ς η 1 Greater the viscosity, slower will be diffusion Explains why a small blob of thick fluid can apparently mix and reappear if a slow mixing movement is followed by its reverse. Biological Physics, P. Nelson, 1 st ed. updated,

13 Mixing and viscosity In laminar flow conditions (low Reynolds number) with thick fluids, motion of stirring rod or one cylinder relatively to a second concentric one only cause mixture of peripheral layers of molecules, due to sliding of fluid layers one over another. Diffusion randomize molecules over relatively long times, thus fluid layers can slide back if reverse movement is applied, reassembling blob. The blob has never fully mixed due to the viscosity of the fluid. The opposite occurs with turbulent flows (e.g. milk on coffee). Biological Physics, P. Nelson, 1 st ed. updated,

14 In laminar flow viscous force due to shear motions with (low) v 0 among fluid planes of area A and separated by distance dx given by: f frict = ηadv dx Newtonian fluid, planar geometry Inertia vs. friction Biological Physics, P. Nelson, 1 st ed. updated, 2008 Though no intrinsic definition of large and small viscosity, for isotropic Newtonian fluids viscous regime takes place when ratio f frict /f crit is small, the viscous critical force f crit being: f crit = η 2 ρ m f frict /f crit > 1 inertial forces dominates over friction (turbulent flow: large density mass or low viscosity of fluid). f frict /f crit < 1 friction quickly damps out inertial effects (laminar flow). 14

15 Inertia vs. friction Biological Physics, P. Nelson, 1 st ed. updated, 2008 Newtonian fluid has not intrinsic scale length! Size of forces involved in the process relative to a critical value discriminates between laminar and turbulent flows. f crit of water in range of nn. In molecular world forces have values ~pn friction dominates molecular processes occurring in aqueous solvents! Flows are laminar! 15

16 Reynolds number Reynolds number corresponds to the ratio between inertial and frictional forces in the dynamics of a fluid: R = f inertial f frict = vlρ m η L typical linear dimension of object moving relatively to the fluid. R small (< 1000) friction dominates, laminar flow. R large (> 3000) inertial effects dominate, turbulence. Experiments by Reynolds shown that distinction based on R is generally applicable whenever the object can be characterized by a length scale L. R related to the concept of critical force: R f frict f crit 16

17 Reynolds number Relation demonstrated considering a spherical object immersed in a fluid under laminar flow conditions. Biological Physics, P. Nelson, 1 st ed. updated, 2008 Newton law for small volume of fluid involves pressure and viscous forces: f inertial = mdv dt = f p + f frict Small volume of fluid accelerates nearly sphere: direction changes in Δt R/v. Acceleration comparable to v in size dv/dt v 2 /R. f inertial = mdv dt = ρ m l 3 v 2 R 17

18 Reynolds number Inertial term must be compared to frictional force: f frict = ηl 2 dv dx Net force due to pushing by upper layer and pulling to lower one, approximated by derivative times length of volume fluid: f frict l df dx = ηl 3 d 2 v dx 2 ηl 3 v R 2 Dividing the modulus of this quantity by f frict gives the Reynolds number definition: R = f inertial f frict = vrρ m η A 30m long whale swimming in water at 10 m/s has R~ A 1µm thick bacterium swimming at 30 µm/s has R~3 10-5! 18

19 Reversibility of flow At low Reynolds number, flow is reversible 19

20 Swimming of bacteria Reversibility means that anything swimming by repeated flapping motions can t get anywhere. If it moves forward in one stroke, the other stroke will bring it right back to where it started! At low Reynolds number, body and paddles of bacterium move with friction coefficients ζ 1 and ζ 2 respectively. Δx = u Δt Δx' = u'δt ' Net movement: Δx Δx' =? To estimate u (drift velocity of the body) and Δx due to paddle movement with velocity v, apply reaction law drag forces: f body = f paddle u( ζ 0 +ζ 1 ) = vζ 1 ( ) Δx = vζ 1 ( ζ 0 +ζ 1 ) u = vζ 1 ζ 0 +ζ 1 ( )Δt 20

21 Swimming of bacteria Reversibility means that anything swimming by repeated flapping motions can t get anywhere. If it moves forward in one stroke, the other stroke will bring it right back to where it started! At low Reynolds number, body and paddles of bacterium move with friction coefficients ζ 1 and ζ 2 respectively. Δx = u Δt Δx' = u'δt ' Net movement: Δx Δx' =? u and Δx due to back movement with velocity v are: u' = v'ζ 1 ( ζ 0 +ζ 1 ) Δx' = ( v'ζ 1 ( ζ 0 +ζ 1 ))Δt ' as the paddles return to their initial position, it must be: vδt = v'δt ' Δx' = ( vζ 1 ( ζ 0 +ζ 1 ))Δt = Δx 21

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