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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)

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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

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Lim et al., Science 318, 640 (2007)

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

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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”

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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

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Blue = Nup (Monomer) Red = Kap (Nanoparticle) How many ways are there to arrange N S nanoparticles on M S lattice sites? Use binomial coefficient:

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Blue = Nup (Monomer) Red = Kap (Nanoparticle) How many ways are there to arrange N B nanoparticles and N monomers on M B lattice sites? Use multinomial coefficient:

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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. h

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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) So free energy from binding interactions will be And the Total Free Energy will be:

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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.

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Here’s what it looks like for a given, sufficiently large binding strength ( χ large and negative) as you sweep through the solution concentration (C 0 ) Double Minimum structure – Phase Transition! Brush height suddenly collapses due to a small increase in C 0

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Small binding strength: No phase transition. Large binding strength: Discontinuity!

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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”

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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 Solution Brush Grafting Sites h C 0 = (# of red) (volume)

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Vertical Drop: “Phase Transition!” M. Opferman, R.D. Coalson, D. Jasnow and A. Zilman, http://arxiv.org/abs/1110.6419http://arxiv.org/abs/1110.6419, 2011 and Phys. Rev. E 86, 031806 (2012)

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Increasing C 0 Homogeneous Extended Homogeneous Collapsed Collapsing

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Increasing C 0 Homogeneous Extended Homogeneous Collapsed Inhomogeneous

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Brush N B red N blue M sites M/v supersites Solution M S /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.

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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=10v=1

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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.

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Milner-Witten-Cates (MWC) / Zhulina Mean 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 A better level of theory is provided by …

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A A - Bz 2 Illustration of MWC theory inversion procedure : at every distance z from the grafting surface, there is a unique value of monomer density Ψ :

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Langevin simulation data vs. MWC theory for v=1,20,100. ** 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.) 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.

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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 → Simulations MWC theory Red = Φ Blue = ψ extended statecollapse regimecollapsed state

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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. Δ d = change in brush height from its value when there are no nanoparticles (here, “kaps”) in solution, and ρ kap β 1 is the number of nanoparticles inside the brush per unit surface area. [N.B.: ρ kap β 1 increases monotonically with bulk nanoparticle concentration, which is indicated in parentheses in the figure.]

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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.

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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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