BME Engineering Molecular Cell Biology. Basics of the Diffusion Theory. The Cytoskeleton (I)

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1 BME Engineering Molecular Cell Biology Lecture 07: Basics of the Diffusion Theory The Cytoskeleton (I) BME Lecture 07, September 20,

2 Outline Diffusion: microscopic theory Diffusion: macroscopic theory A method to determine the diffusion coefficient An overview of the cytoskeleton 2

3 Outline Diffusion: microscopic theory Diffusion: macroscopic theory A method to determine the diffusion coefficient An overview of the cytoskeleton 3

4 Introduction Cellular lar molecules les are subject to thermal force due to collisions with water and other molecules. The resulting motion and energy are called thermal motion and thermal energy. 4

5 Movement of a Free Molecule (I) The average kinetic energy of a particle of mass m and velocity v is 1 2 kt mvx 2 2 Boltzmann constant= J/K t K = t C Joule = 1 N m = 1 kg m 2 / s 2 where k is Boltzmann's constant and T is absolute temperature (Einstein 1905). Principle of equipartition of energy 1 2 3kT mv 2 2 5

6 Movement of a Free Molecule (II) Molecular mass of GFP is 27 kda. One atomic mass unit (Da) is g. So the mass of one GFP molecule is g. At 27 degree C, kt is g cm 2 /sec 2. 2 kt v x cm/sec m 6

7 1D Random Walk in Solution (I) Assumptions: (1) A particle i has equal probabilities to walk to the left and to the right. (2) Particle movement at consecutive time points are independent. (3) Movement of different particles are independent. (4) Each particle moves at a average step size of δ=v x τ i 1 x n x n i N N 1 1 xn xin xin 1 N i 1 N i1 N 1 x i n1 x n1 N i 1 7

8 1D Random Walk in Solution (II) Property 1: The mean position of a particle (or an ensemble of particles) undergoing random walk remains at the origin. i 1 x n x n i N N 1 1 xn xin xin 1 N i 1 N i1 N 1 x i n 1 x n 1 N i 1 8

9 1D Random Walk in Solution (III) Property 2: The mean square displacement of a particle undergoing random walk increases linearly w.r.t. time. 1 1 x n x n x n x n N N i i 12 i 1 N i1 N i1 2 2 x n 1 t x n n 2Dt r n x n y n 4Dt r n x n y n z n 6Dt 9

10 1D Random Walk in Solution (IV) Property 3: The displacement of a particle follows a normal distribution. p k;n n! 1 1 k k! n k!2 2 nk p k 1 e 2 2 k 2 n n where and x n k n k k n x n 2 k n x n k k nn n n n n n x 1 4Dt 2 p x e where n 2 Dt 4 Dt 10

11 Application of the Microscopic Theory (I) Object Distance diffused 1 μm 100 μm 1mm 1m K ms 2.5s s s (7 hrs) (8 yrs) Protein 5ms 50s s (6 days) s (150 yrs) Organelle 1s 10 4 s 10 8 s s (3 hrs) (3 yrs) (31710 yers) K+: Radius = 0.1nm, viscosity = 1mPa s -1 ;T=25 C; D=2000 μm 2 /sec Protein: Radius = 3nm, viscosity = mPa s -1 ; T = 37; D = 100 μm 2 /sec Organelle: Radis = 500nm, viscosity = mPa s -1 ; T = 25 C; D = 0.5 μm 2 /sec 11

12 Application of the Microscopic Theory (II) Diffusion with external flow x 2 > acement < uare displa Mean squ H. Qian, M. P. Sheetz, E. L. Elson, Single particle tracking: analysis of diffusion and flow in two-dimensional systems, Biophysical Journal, 60(4): , Pure diffusion Diffusion in a cage t 12

13 Outline Diffusion: microscopic theory Diffusion: macroscopic theory A method to determine the diffusion coefficient An overview of the cytoskeleton 13

14 Macroscopic Theory of Diffusion (I) Fick's first equation: net flux is proportional to the spatial gradient of the concentration function. 1 2 N x N x 1 Fx lim 0 2 N x N x / A 2 1 Nx Nx lim 0 2 A A 1 lim D Cx Cx 0 C D x 14

15 Macroscopic Theory of Diffusion (II) Fick's second equation 1 Ct Ct FxxFxxA A Ct Ct Fxx Fxx A A 1 Fx x Fx x 2 C F x C D 2 t x x The time rate of change in concentration is proportional to the curvature of the concentration ti function. 15

16 Outline Diffusion: microscopic theory Diffusion: macroscopic theory A method to determine the diffusion coefficient An overview of the cytoskeleton 16

17 A Method to Measure Diffusion Coefficient Einstein-Smoluchowski Relation v d 1 1 Fx a 2 2 m 2 2m 2 2 Fx 2m mvx kt f 2 vd D D kt D f: viscous drag coefficient f Stokes' relation: the viscous drag coefficient of a sphere moving in an unbounded fluid f 6 r : viscousity r: radius 17

18 An example of D calculation Calculation of diffusion coefficient D kt 6r k= J/k= N m/k T = =0.8904mPa s= N m -2 s r= 500nm=0 0.5μm D=0.5 m 2 /s 18

19 References Howard Berg, Random Walks in Biology, Princeton University Press, Jonathon Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associated,

20 Outline Diffusion: microscopic theory Diffusion: macroscopic theory A method to determine the diffusion coefficient An overview of the cytoskeleton 20

21 The Cytoskeleton is Highly Dynamic 21

22 The Cytoskeleton (I) Three classes of filaments - actin: stress fiber; cell cortex; filopodium - microtubule: centrosome - intermediate filaments Red: actin Green: microtubule 22

23 The Cytoskeleton (II) Intermediate filaments Spatial organization of cytoskeletal filaments is dependent on many factors, e.g. Green: vimentin IF Red: microtubule -cell type - cell states (cycle) - cell activities Orange: keratin IF Green: desmosome 23

24 The cytoskeleton plays a critical role in many basic cellular l functions, e.g. The Cytoskeleton (III) - structural organization & support - shape control - intracellular transport - force and motion generation -signaling g integration Highly dynamic and adaptive 24

25 Overview of Cytoskeletal Filaments Shape Diameter Subunits Polarized actin cable ~6 nm actin monomer microtubule tube ~25nm tubulin heterodimer intermediate filament rope ~10nm Various dimers yes yes no 25

26 Organization of Actin with a Cell 26

27 Actin Structure and Function Each actin subunit is a globular monomer. One ATP binding site per monomer. Functions - Cell migration -Cell shape - Used as tracks for myosin for short distance transport 27

28 Basics Terms of Chemical Reaction Kinetics A reversible bimolecular binding reaction A + B AB Rate of association = k + [A][B] Rate of disassociation = k - [AB] At equilibrium k + [A][B] = k - [AB] 28

29 Actin Nucleation and Nucleotide Hydrolysis Actin polymerizes and depolymerizes substantially faster at tthe plus end d(barbed b end) d)than at the minus end (pointed end). 29

30 Actin Accessory Proteins (I) More than 60 families identified so far. Functions - Monomer binding - Nucleation - Filament capping - Filament severing - Filament side-binding and supporting - Filament crosslinking -Signaling g adapter Functional overlap and collaboration between actin- binding proteins 30

31 Questions? 31

32 Actin Accessory Proteins (II) Monomer binding proteins - profilin: to bind actin monomer and accelerate elongation - thymosin: to bind and lock actin monomer - ADF/cofilin: to bind and destabilize ADP-actin filaments 32

33 Actin Accessory Proteins (III) Actin nucleation - Formins: to initiate unbranched actin filaments - Arp2/3: to bind the side of actin and initiate branching 33

34 Actin Accessory Proteins (IV) Actin capping protein - Blocks subunit addition and disassociation Actin severing protein Three families of proteins perform both functions - Gelsolin - Fragmin-severin - ADF/cofilin 34

35 Actin Accessory Proteins (V) Actin side-binding proteins tropomyosin, nebulin, caldesmon Actin crosslinking gprotein - -actinin - filamin - spectrin -ERM 35

36 Actin Adapter Protein WASP & VASP 36

37 Actin Regulation GTPase: Molecule switch; Family of proteins that are activated by GTP binding and inactivated by GTP hydrolysis and phosphate dissociation. Rho GTPase: cdc42: its activation triggers actin polymerization and bundling at filopodia. Rho: its activation promotes actin bundling. Rac: its activation promotes polymerization at the cell periphery. 37

38 Rac on Actin Organization 38

39 Summary: actin Relatively soft (quantification in following lectures). Often form bundles; mechanical strength comes mostly from bundling and crosslinking. Mostly function to withstand tension rather than compression. Relatively stable and easy to work with (biochemically). 39

40 Summary: actin accessory proteins Different proteins have distinct functions. Proteins with multiple functional domains can have multiple functions. Some of them are essential. Most of the proteins have functional overlap. 40