McGuffee & Elcock (2010) Diffusion, crowding & protein - - PowerPoint PPT Presentation

McGuffee & Elcock (2010) Diffusion, crowding & protein stability in a

Macromolecular Crowding

Molecules and Complexes: E. coli Census

• Nproteins ~ 3x10^6
• Nribosomes ~ 20,000
• Nlipids ~ 2x10^7

Spacing for a molecule of given concentration c:

d = c−1 3

Protein Spacing in E. coli

• Estimate
• Average protein-protein distance ~ 10 nm
• 1 mM protein in vitro distance ~ 100 nm

Radius of protein ~ 2nm 1 um

Linked Polymer Network Architecture

Gram positive bacterial wall (atomic resolution) Collagen fibrils Basement membrane Actin web in a miving cell Cellulose network

• f a plant cell

Ligament collagen bundle Axon bundle Intenstinal microvilli

• Effect on binding and interaction
• Difference between cells and dilute

solutions

Ten Commandments of Enzymology, Amended

Kornberg (2003) Tr. Biochem. Sci.

• 1. Thou shalt rely on enzymology to resolve

and reconstitute biologic events

• 2. Trust the universality of biochemistry and

the power of microbiology (Escherichia sapiens)

• 3. Not believe something just because you can

explain it

• 4. Not waste clean thinking on dirty enzymes
• 5. Not waste clean enzymes on dirty substrates
• 6. Use genetics and genomics
• 7. Be aware that cells are molecularly crowded
• 8. Depend on viruses to open windows
• 9. Remain mindful of the power of radioactive

tracers

• 10. Employ enzymes as unique reagents

Arthur Kornberg Nobel Prize for Chemistry (1959) Discovery of DNA polymerase (now known as DNA polymerase I)

Be aware that cells are molecularly crowded

Fish skin keratocyte

Fish Keratocyte Leading Edge

Membrane stripped platinum coated E.M. Front edge: Ordered, Branched Middle zone: Randomly

• verlaid

filaments

Cellular Effects of Crowding

• Equilibrium binding
• Diffusive processes

7.5% 5% 2.5% 0% PEG concentration

ATPase Rate

T4 DNA polymerase clamp-loader proteins

Experimental Measures

• f Diffusivity

BODIPY-FI sizes

Verkman (2002) TiBS

Method 1 Method 2 FCS Time resolved Fluorescence Anisotropy Method 3

Cellular Diffusion Coefficients

Verkman (2002) TiBS

Single Molecule Imaging Membrane Proteins

GFP-Lck Lck Tyrosine kinase Jurkat T-cells TIRF microscopy Anti-T cell receptor Abs stimulate clustering

Douglass & Vale (2005)

Trapped Long distance

Membrane Microdomains

T-cell Signalling domains in CD2-enriched signalling domains Signalling activated by antibody-patch on coverslip

CD2 Lck Merge

Lipid Rafts

GPI-anchored proteins Raft-associated lipids

Mayor Lab, NCBS Bangalore

Effective Diffusion

<d^2> = < [d(t)-d(t+δt)]^2 >, For different values of δt. Plotting d^2 vs δt gives us a profile that can be fitted by <d^2>=2*D*t^α α = 1 for normal diffusion

Bacher, Reichenzeller, Athale et al. (2004) Klopfstein et al. (2002) Cell

Raft targeted Lck-10 GFP Lck GFP

TIRF: At the Surface

Evanescent wave illumination with limited range

Nikon Instruments

Lattice Model of Crowding

Ωv = reaction volume L=ligand no. C=Crowding

• molec. no.

Ligand-Receptor Binding pbound = 1 1+ Ω− L − C L

( )eβΔε L

L << Ω When C increases, pbound increases

ΔεL = εL

b −εL sol

Crowding Changes L-R binding Probability

7.5% 5% 2.5% 0% PEG concentration Binding constant PEG dependent)

Dissociation Rate

Kd = 1 v eβΔe

pbound = L

[ ] Kd

( )

n

1+ L

[ ] Kd

( )

n

v=volume of single lattice site Δε = binding energy

PEG as Crowdant

T4 atpase data, PEG size (12 kDa) << Protein size (164 kDa) Ω large boxes r small boxes in each large box

Crowdant Smaller than Ligand

Binding

Where, volume fraction of the crowding molecules in solution Assuming L << Ω And (N+r)!/N! ~ Nr

φC = C rΩ

pbound = 1 1+ Ω L (1− φC )reBΔε L

Dissociation Constant

Volume fraction dependent dissociation constant Kd

Kd φC

( )

Kd φC = 0

( )

= 1− φC

( )

r

Kd = 1 v eβΔe

Factors Affecting Crowding

Osmotic Pressure and Crowding

Osmotic pressure due to excess Hemoglobin p = pressure v=volume of single box in lattice [H]=concentration of Hb molecules=H/Ωv

p = − kBT v ln(1−[H]v)

Crowding and Osmotic Pressure

Free parameter is v V=5.8 nm Hard sphere gas model Experiment Lattice gas

Hard Sphere Gas Model

Boltzmann 1899 Where x = 4V[H] V=volume of hard sphere

p = kBT[H](1+ x + 0.625x 2 + 0.287x 3 + 0.11x 4)

Next

• Crowded polymers and ordering
• Cytoskeleton
• Motors
• Cells in tissues
• FRAP data
• Paper presentations

2011-03-23

Macromolecular Crowding

• 10-100% of fluid volume of cytoplasm lies

within 1 molecular diameter of the surface

• f fibrous and membraneous structures
• Pores
• Reactant X and pore size comparable

Sieving Effect

Verkman (2002) TiBS

Volume Exclusion

Minton (2001) J. Biol. Chem.

Excluded volume Available volume

Minton (2001) J. Biol. Chem.

Relative Sizes

Volume Available

Effective and actual concentrations vtot=total volume va,i=volume available to species i

γ i ≡ ai ci

( ) = vtot va,i

( )

Haemoglobin Concentration

Normal RBC Concentration ~ 300gm/L

2011-03-29

Forces due to Volume Exclusion

Large particle near surface Two large particles in solution Two rod-like molecules in solution Depletion forces Volume available to smaller molecules increases

Origin of Depletion Forces

R= radius of large disk r = radius of small disk Surface 2D geometry Find area available to small disks, as a function of distance z between large disk and surface

Excluded Volume Interactions

R= radius of large disk r = radius of small disk Surface 2D geometry z=distance between large disk and surface

Area available to small disks Entropy increases

Free Energy Change

No conventional forces- van der Walls, electrostatics, etc. Free energy change by change in entropy is: Vbox = volume of box, Vex=excluded volume, v=volume

• f unit cell, N=no. of SMALL MOLECULE particles

Gex = −NkBT ln Vbox −Vex v       + NkBT ln Vbox v      

Free Energy

If Vex << Vbox, approximate ln(1+x) ≈ ln(x) If 2 large particles overlap excluded volumes, Vex increases entropy of small particle ~ ideal gas (osmotic) pressure of small particles in box

Gex = NkBT Vex Vbox

NkBT Vbpx

Depeletion Force

Volume and Force

Total excluded volume Volume of spherical cone Volume cone Overlap Depletion Force

Vex = 2⋅ 4π 3 R + r

( )

3 −Voverlap

Vsphericalcone = 2π 3 R + r

( )

2 ⋅ R + r − D 2

( )

Vcone = π 3 D 2 R + r

( )

2 − D 2

( )

2

[ ]

Voverlap = 2π 3 R + r + D 2

( )

2 2R + 2r + D 2

( )

Fdepletion = −∂Gex ∂D = −pπ R + r

( )

2 − D2

4      

p = nkBT , n=N/Vbox, and distance 2R<D<2(R+r)

Depletion Force Measurement

2 beads, R=625 nm, Move in line Depletion agent- Phage λ DNA r=500 nm

DNA conc. DNA conc. DNA conc.

p(D) ∝e−βGex (D)

Gex(D) = pVoverlap

Entropic Ordering

Rods: Filamentous viruses Spheres

Volume Exclusion

Mutual exclusion v=volume occupied N=number of macromolecules Ω=total number of boxes N=no. of macromolecules In absence of excluded volume Zex(N) = Ω! N! Ω− N

( )!

Znex(N) = ΩN N!

Free Energy for Excluded Volume

Free energy Using stirling’s approximation, and assuming Ω >> N, and

G = −kBT lnZ

ΔGex = Gex − Gnex = −kBT ln Zex Znex

1− N Ω

( )

Ω ≈ e−N

Zex Znex ≈ 1− N Ω      

N

ΔGex = −NkBT ln 1− N Ω       ≈ kBT N 2 Ω

Polymers, Crowding and RW Model

• Random walk model ignores self-avoidance
• Size of macro-molecule like DNA ~

N=number of segments, a=persistence length

Na2

Competing effects:

• Entropy makes chain compact
• Self-avoidance swells the chain

Random Walk Polymer

Free energy R=radius of polymer SRW(R)=random walk entropy of chain of length R Entropy from probability distribution P(R;N)

G(R) = −TSRW (R) + Gex(R)

SRW (R) = kBT lnP(R;N) + const = −kB 3R2 2Na2 + const

Excluded volume

• Assume polymer to be gas with hard cylinders
• f length a and diameter d
• Mutual orientation angle θ decides excluded

volume v = 2da2 sinθ

• For d<<a
• Averaging sin(θ) over all orientations gives

estimate for excluded volume

• Free energy
• Volume fraction of N hard cylinders

πa2d 2

Gex = kBTNφ

φ(R) = N 1 2 πa2d 1 2 πR3 = N 3a2d 8R3

Free Energy Difference

• Since
• Flory’s estimate of free energy of polymer
• Size of chain

ΔGex ≈ kBT N 2 Ω = kBTNφ Gex(R) = kBTN 2 3a2d 8R3 G flory(R) = kBT 3R2 2Na2 + kBTN 2 3a2d 8R3

Rflory = 3 8 a4d      

1/ 5

N 3/ 5

RW vs. SAW

• Power scaling RW (N1/2), SAW (N3/5)
• Short polymers RW still valid

For DNA d~2nm, a=100nm For N<<16(d/a)=40,000, G self avoid < RW entropy L=Na=40,000x100 nm (kuhn length) ~ 16 µm

Gex = kBT 3 8 d a N

1 2

SRW = 3 2 kBT

Protein Folding & Crowding

Chaperones

Diffusion in Crowded Environs

Diffusion in Crowded Environments

• For single time step t=τ
• After time t, steps N=t/τ, so
• Diffusion coefficient

For a random walker

x 2 τ = a2 ⋅ pright + a2 ⋅ pleft + 0⋅ pstay = a2(1− φ) x 2 t = t τ x 2 τ = a2 τ (1− φ)t

D = D0(1− φ)

D0 = a2 2τ

FITC-Aldolase diffusing

Aldolase BSA Ovalbumin

Self-Diffusion and Crowding

NEXT

• Cytoskeleton and motors
• Cells
• Development