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 Approach

1. 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:

!!!

!

NNMNN

M

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 NvMN

vM

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 - B

z2

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

ula

tions

MW

C t

heory 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