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A Simple Statistical Mechanical Model of Transport Receptor Binding in the Nuclear Pore Complex

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 .)

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A Simple Statistical Mechanical Model of Transport Receptor Binding in the Nuclear Pore Complex

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  1. 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)

  2. 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

  3. Nuclear Pore Complex: Geometric Details

  4. R. Lim: “Reversible Collapse Model” Lim et al., Science 318, 640 (2007)

  5. The Lim Experiment • Nup (polymer) filaments grafted onto a nanodot collapse in the presence of (nanoparticle) receptors... From: Lim et al., Science 318, 640 (2007)

  6. Our Approach • Use a simple statistical mechanical model (lattice gas mean field theory = MFT **) to understand the Lim experiment • Count states • Minimize free energy • 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”

  7. Lattice Gas First, consider a gas of nanoparticles (“solution”) in contact with a gas of monomers mixed with nanoparticles (“brush”). Solution Brush h Blue = Nup (Monomer) Red = Kap (Nanoparticle) Note: v=(nanoparticle volume)/(monomer volume) = 1 here

  8. Lattice Gas: Solution Phase How many ways are there to arrange NSnanoparticles on MS lattice sites? Use binomial coefficient: Blue = Nup (Monomer) Red = Kap (Nanoparticle)

  9. Lattice Gas: Brush Phase How many ways are there to arrange NBnanoparticles and N monomers on MB lattice sites? Use multinomial coefficient: Blue = Nup (Monomer) Red = Kap (Nanoparticle)

  10. Lattice Gas: (Grafted) Brush Entropy But these are monomers of a polymer chain, not a gas. They should have stretching entropy, not translational entropy! So replace the unphysical term. h Blue = Nup (Monomer) Red = Kap (Nanoparticle)

  11. Lattice Gas: Brush 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) So free energy from binding interactions will be And the Total Free Energy will be: Blue = Nup (Monomer) Red = Kap (Nanoparticle)

  12. Equilibrium Conditions The 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” 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.

  13. Free Energy Landscape Here’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

  14. AdG MFT predicts Brush Collapse Small binding strength: No phase transition. Large binding strength: Discontinuity!

  15. 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”

  16. Simulation Snapshot Solution White = Polymer Beads (Nups) Red = Transport Receptors (Kaps) Top: Reservoir of Red particles Bottom: Hard wall to which polymers are grafted Sides: Periodic boundary conditions C0 = (# of red) (volume) Brush h Grafting Sites

  17. 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)

  18. Continuous Polymer Compression for weakly attractive kap-nup interactions Homogeneous Extended Collapsing Homogeneous Collapsed Increasing C0

  19. Attempted Phase Separation for strongly attractive kap-nup interactions Homogeneous Extended Inhomogeneous Homogeneous Collapsed Increasing C0

  20. Lattice Gas Mean Field Theory for Large Nanoparticles (v>1) When nanoparticles are larger than monomers, place the larger particles first so that the number of available “super-lattice” sites is easily calculated. Solution MS/v supersites Brush NB red N blue Msites M/v supersites h Blue = Nup (Monomer) Red = Kap (Nanoparticle)

  21. 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

  22. 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.

  23. A better level of theory is provided by … Milner-Witten-Cates (MWC) / ZhulinaMean Field Theory of a Plane-Grafted Polymer Brush: 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

  24. Illustration of MWC theory inversion procedure: at every distance z from the grafting surface, there is a unique value of monomer density Ψ : A A - Bz2

  25. Langevin simulation data vs. MWC theory for v=1,20,100. ** V=1 Overall, the agreement between Langevin 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.) V=20 A few conclusions: No true “phase transition” (discontinuity in h vs. c) even for v=1. The collapse transition is sharper for smaller v. V=100 ** MGO, RDC, DJ and AZ, Langmuir, in press.

  26. Spatial distribution of monomers, ψ(z), and nanoparticles, Φ(z), for v=1, a=4: Comparison of Langevin simulations to MWC theory. Simulations Red = Φ Blue = ψ MWC theory Increasing nanoparticle concentration, c → extended state collapse regime collapsed state

  27. New results from Lim et al. ** on a nup-based brush grafted to a flat surface with attractive kap proteins in solution: 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.] ** Schoch, R.L., L.E. Kapinos, and R.Y. Lim , PNAS 2012. 109: p. 16911–16916.

  28. Potential Nanotechnology Application: Tunable nano-valves (for separations applications): Control via temperature: Control via solution pH: Yameen, B., M. Ali, R. Neumann, W. Ensinger, W. Knoll, and O. Azzaroni, Small, 2009. 5: p. 1287-1291. Iwata, H., I. Hirata, and Y. Ikada, Macromol., 1998. 31: p. 3671-3678. Our variation on this theme: Control via nanoparticle concentration

  29. 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

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