Upload
sen
View
39
Download
0
Tags:
Embed Size (px)
DESCRIPTION
A Simple Statistical Mechanical Model of Transport Receptor Binding in the Nuclear Pore Complex. Michael Opferman (Univ. of Pittsburgh) Rob Coalson (Dept. of Chemistry, Univ. of Pittsburgh) David Jasnow (Dept. of Physics & Astronomy, Univ. of Pgh .) - PowerPoint PPT Presentation
Citation preview
A SIMPLE STATISTICAL MECHANICAL MODEL
OF TRANSPORT RECEPTOR BINDING
IN THE NUCLEAR PORE COMPLEX
Michael Opferman (Univ. of Pittsburgh)Rob Coalson (Dept. of Chemistry, Univ. of Pittsburgh)
David Jasnow (Dept. of Physics & Astronomy, Univ. of Pgh.)Anton Zilman (Los Alamos National Lab, University of Toronto)
Nuclear Pore Complex:
NPC is a structure in the nuclear envelope which allows transport of material in and out of the nucleus (e.g. mRNA)
Walls of the NPC are lined with natively unfolded proteins called nucleoporins (“nups”)
Nups bind to transport receptors, typically karyopherins (“kaps”)
What role does binding play in transport?
Picture from: B. Fahrenkrog and U. Aebi, Nat. Rev. Mol.
Cell Biol. 4, 757 (2003). Adapted from Cryo-electron
tomography
Nuclear Pore Complex: Geometric Details
R. Lim: “Reversible Collapse Model”
Lim et al., Science 318, 640 (2007)
The Lim Experimento Nup (polymer) filaments grafted onto a nanodot
collapse in the presence of (nanoparticle) receptors...
From: Lim et al., Science 318, 640 (2007)
Our Approach1. Use a simple statistical mechanical
model (lattice gas mean field theory = MFT **) to understand the Lim experiment
Count states Minimize free energy
2. Use coarse-grained multi-particle Langevin Dynamics simulations to verify the theory and add more detail ** a la S. Alexander [J. de Phys., 1977. 38: p. 977-981] and P. de Gennes [Macromol., 1980. 13: p. 1069-1075] = “AdG”
Lattice Gas
Brush
Solution
h
Blue = Nup (Monomer)Red = Kap (Nanoparticle)
First, consider a gas of nanoparticles (“solution”) in contact with a gas of monomers mixed with nanoparticles (“brush”).
Note: v=(nanoparticle volume)/(monomer volume) = 1 here
Lattice Gas: Solution Phase
Blue = Nup (Monomer)Red = Kap (Nanoparticle)
How many ways are there to arrange NS nanoparticles on MS lattice sites? Use binomial coefficient:
!!!
SSS
SS NMN
M
0000 1ln1ln CCCCMF SS
SS MNC 0
SSF ln
Lattice Gas: Brush Phase
Blue = Nup (Monomer)Red = Kap (Nanoparticle)
How many ways are there to arrange NB nanoparticles and N monomers on MB lattice sites? Use multinomial coefficient:
!!!!
NNMNNM
BBB
BB
1ln1lnlnBB MF
BMN BB MN
Lattice Gas: (Grafted) Brush Entropy
Blue = Nup (Monomer)Red = Kap (Nanoparticle)
But these are monomers of a polymer chain, not a gas. They should have stretching entropy, not translational entropy! So replace the unphysical term.
1ln1lnln BBB MMF BMN
BB MN 1ln1ln22BB MNhF
h
Lattice Gas: Brush
Blue = Nup (Monomer)Red = Kap (Nanoparticle)
Finally, make nanoparticles “bind” to polymers by adding an “enthalpic” term to the free energy.Number of binding interactions will be (invoking “random mixing”):(Number of nanoparticles) x (Average number of monomers
neighboring each nanoparticle)
BMN BB MN
1ln1ln22BB MNhF
BM
So free energy from binding interactions will be BM
And the Total Free Energy will be:
Equilibrium ConditionsThe solution and brush can exchange nanoparticles and volume. This means that the chemical potential of nanoparticles, and the osmotic pressure must be equal in the two regions at equilibrium.
Equivalently, we can minimize a “Grand Potential”
BSBSB MNF
Note: Here [ = bulk nanoparticle concentration ]
Minimizing this function over: (1) the number of nanoparticles in the brush and (2) the volume of the brush
for fixed concentration in the solution determines the equilibrium state of the solution/brush system.
0 0( ) ; ( )S S S SC C 0C
Free Energy LandscapeHere’s what it looks like for a given , sufficiently large binding strength (χ large and negative) as you sweep through the solution concentration (C0)
Double Minimum structure – Phase Transition!Brush height suddenly collapses due to a small increase in C0
AdG MFT predicts Brush Collapse
Small binding strength: No phase transition.
Large binding strength: Discontinuity!
Simulations Langevin Dynamics
Overdamped regime, Implicit solvent, Coarse-grained
Lennard-Jones Repulsion between all particles
Lennard-Jones Attraction to represent binding FENE springs to connect polymer strands Polymers grafted in a square array to the
“floor” Periodic boundary conditions on “walls”
Simulation SnapshotWhite = Polymer Beads (Nups)Red = Transport Receptors (Kaps)
Top: Reservoir of Red particles
Bottom: Hard wall to which polymers are grafted
Sides: Periodic boundary conditions
Solution
Brush
Grafting Sites
h
C0 = (# of red) (volume)
Comparing MFT to BD Simulation
Vertical Drop: “Phase Transition!”
M. Opferman, R.D. Coalson, D. Jasnow and A. Zilman, http://arxiv.org/abs/1110.6419, 2011 and Phys. Rev. E 86 , 031806 (2012)
Continuous Polymer Compression for weakly attractive kap-nup interactions
Increasing C0
Homogeneous Extended
Homogeneous Collapsed
Collapsing
Attempted Phase Separation for strongly attractive kap-nup interactions
Increasing C0
Homogeneous Extended
Homogeneous Collapsed
Inhomogeneous
Lattice Gas Mean Field Theory for Large Nanoparticles (v>1)
Brush
NB redN blueM sitesM/v supersites
Solution
MS/v supersites
h
Blue = Nup (Monomer)Red = Kap (Nanoparticle)
When nanoparticles are larger than monomers, place the larger particles first so that the number of available “super-lattice” sites is easily calculated.
!!
!
SSS
SS NvMN
vM
!!
!1
BB NvMNvM
SSF ln
21ln BF
!!
!2 NvNMN
vNM
B
B
Large Nanoparticles: Predictions of AdG MFT
v>1 shares many qualitative similarities with the v=1 case, including the decrease in brush height when more nanoparticles are bound and the phase transition between an extended and collapsed state when the binding strength is sufficiently high.
v=10 v=1
Comparison of MFT vs. BD simulations for v=10.Note: BD simulations for v=10 were performed with spherical nanoparticles having spherically homogeneous attraction nup (polymer) monomers.
Milner-Witten-Cates (MWC) / Zhulina Mean Field Theory of a Plane-Grafted Polymer Brush:
2 ( )A Bz z
Here: z =distance from grafting plane
= monomer (polymer bead) density (volume fraction)
= function derived from the brush free energy function above (sans polymer chain stretching energy term)
A,B = positive constants dependent on polymer chain length and grafting density
( )z
( )
A better level of theory is provided by …
0 0.2 0.4 0.6 0.8 1
0
1
2
3
psi
mu(
psi)
AA - Bz2
Illustration of MWC theory inversion procedure: at every distance z from the grafting surface, there is a unique value of monomer density Ψ :
Langevin simulation data vs. MWC theory for v=1,20,100. **
Overall, the agreement betweenLangevin simulations and MWC is quite reasonable (good?) over the entire range v=1-100.
(Quantitative agreement degrades as v increases, but all qualitative features are faithfully reproduced.)
A few conclusions:
1) No true “phase transition” (discontinuity in h vs. c) even for v=1.
2) The collapse transition is sharper for smaller v.
V=1
V=20
V=100
** MGO, RDC, DJ and AZ, Langmuir, in press.
Spatial distribution of monomers, ψ(z), and nanoparticles, Φ(z), for v=1, a=4: Comparison of Langevin simulations to MWC theory.
Increasing nanoparticle concentration, c →
Sim
ulat
ions
MW
C th
eory Red = Φ Blue = ψ
extended state collapse regime collapsed state
New results from Lim et al. ** on a nup-based brush grafted to a flat surface with attractive kap proteins in solution:
** Schoch, R.L., L.E. Kapinos, and R.Y. Lim , PNAS 2012. 109: p. 16911–16916.
Δd = change in brush height from its value when there are no nanoparticles (here, “kaps”) in solution, and ρkapβ1 is the number of nanoparticles inside the brush per unit surface area. [N.B.: ρkapβ1 increases monotonically with bulk nanoparticle concentration, which is indicated in parentheses in the figure.]
Potential Nanotechnology Application: Tunable nano-valves (for separations applications):
Our variation on this theme: Control via nanoparticle concentration
Control via solution pH:
Iwata, H., I. Hirata, and Y. Ikada, Macromol., 1998. 31: p. 3671-3678.
Control via temperature:
Yameen, B., M. Ali, R. Neumann, W. Ensinger, W. Knoll, and O. Azzaroni, Small, 2009. 5: p. 1287-1291.
Conclusions We developed a simple theory capable of
explaining the collapse of a polymer brush when exposed to binding particles
Depending on the binding strength, collapse may be quite sharp over a small
nanoparticle concentration range.
Next steps: I) Add more realism. E.g.: discrete binding
sites on the (large) nanoparticles, cylindrical geometry, range of polymer grafting densities and nanoparticle sizes.
II) Applications to both biology (NPCs) and materials science (controlling the morphology of a polymer brush) are envisaged.
$$: NSF