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MIT 3 052 - ELASTICITY OF SINGLE POLYMER CHAINS

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REVIEW : LECTURE 18 NANOMECHANICS AND BIOCOMPATIBILITY : PROTEIN-BIOMATERIAL INTERACTIONS 2 SUMMARY : THEORETICAL MODELS FOR SINGLE POLYMER CHAIN ELASTICITY VARIOUS MATHEMATICAL FORMS FOR THE (INEXTENSIBLE) FREELY JOINTED CHAIN (FJC) MODEL COMPARISON OF VARIOUS MATHEMATICAL FORMS FOR THE INEXTENSIBLE FREELY JOINTED CHAIN (FJC) MODEL EXTENSIBLE FREELY JOINTED CHAIN (FJC) MODEL WORM LIKE CHAIN (WLC) MODEL (*Kratky-Porod Model) APPENDIX : THEORETICAL MODELS FOR THE ELASTICITY OF SINGLE POLYMER CHAINS : FULL CITATIONS OF ORIGINAL REFERENCES3.052 Nanomechanics of Materials and Biomaterials Thursday 04/26/07 Prof. C. Ortiz, MIT-DMSE I LECTURE 19: ELASTICITY OF SINGLE POLYMER CHAINS : THEORETICAL FORMULATIONS Outline : REVIEW LECTURE #18 : NANOMECHANICS AND BIOCOMPATIBILITY 2 ........................................... 2 SUMMARY : THEORETICAL MODELS FOR SINGLE POLYMER ELASTICITY...................................... 3 VARIOUS MATHEMATICAL FORMS FOR THE INEXTENSIBLE FREELY JOINTED CHAIN (FJC) ....... 4 Graphical Comparison ................................................................................................................... 5 EXTENSIBLE FJC ..................................................................................................................................... 6 WORM LIKE CHAIN (WLC) MODEL ......................................................................................................... 7 APPENDIX : FULL CITATIONS ................................................................................................................. 8 Objectives: To understand the theoretical formulations of single macromolecule elasticity Readings: Course Reader Document 31, CR Documents 32-39 are the original theoretical papers for reference, English translations of CR 33 and 36 are available on Stellar Multimedia : Podcast : Elasticity of fibronectin; Abu-Lail, et al. Matrix Biology 2006 25 175 13.052 Nanomechanics of Materials and Biomaterials Thursday 04/26/07 Prof. C. Ortiz, MIT-DMSE REVIEW : LECTURE 18 NANOMECHANICS AND BIOCOMPATIBILITY : PROTEIN-BIOMATERIAL INTERACTIONS 2 -Two examples of biomaterials : vascular graft and endotracheal tube (materials, design issues, relation to nanomechanics) -Kinetics of protein adsorption; contributions to diffusion; ideal and activated (Szleifer model-CR 29,30); initial and secondary stage of protein adsorption -Modes of protein adsorption : (I.) adsorption of proteins to the top boundary of the polymer brush (II.) local compression of the polymer brush by a strongly adsorbed protein (III.) protein interpenetration into the brush followed by the non-covalent complexation of the protein and polymer chain (IV.) adsorption of proteins to the underlying biomaterial surface via interpenetration with little disturbance of the polymer brush -Use of steric repulsion (conformational entropy) to inhibit protein adsorption (Halperin model for polymer brushes-posted on stellar- Polymer brush is a layer of polymers attached with one end to a surface whereby the distance between neighboring chains, s<Rg where Rg is the radius of gyration of an isolated chain; this condition causes extension of the chains away from the surface) Ubrush(s, Lo )Ubare→ VDW, hydrophobic, etc.repulsive barriersecondary adsorptionprimary adsorptionUbrush(s, Lo )Ubare→ VDW, hydrophobic, etc.repulsive barriersecondary adsorptionprimary adsorption(Halperin, Langmuir 1999) For a protein interacting with a planar surface : eff bare brush=+U (z) U (z) U (z) U*=activation barrier determining rate of primary adsorption kads= adsorption rate constant Kramers rate theory: ads-*expkT⎛⎞≈⎜⎟⎝⎠0UDkαL D= diffusion constant α= width of barrier at U*-kT Lo= uncompressed height of polymer brush -Polyethylene oxide (PEO, PEG) - hydrophilic and water-soluble at RT, forms an extensive H-bonding network; intramolecular H- bond bridges between -O- groups and HOH→ large excluded volume, locally (7/2) helical supramolecular structure (tgt axial repeat = 0.278 nm), high flexibility, molecular mobility, low van der Waals attraction, neutral. However: poor mechanical stability, protein adhesion reported under certain conditions (long implant times), maintains some hydrophobic character. 23.052 Nanomechanics of Materials and Biomaterials Thursday 04/26/07 Prof. C. Ortiz, MIT-DMSE SUMMARY : THEORETICAL MODELS FOR SINGLE POLYMER CHAIN ELASTICITY Freely-Jointed Chain (FJC)(Kuhn and Grün, 1942 James and Guth, 1943)ExtensibleFreely-Jointed Chain(Smith, et. al, 1996)Worm-Like Chain (WLC)(Kratky and Porod, 1943Fixman and Kovac, 1973Bustamante, et. al 1994)ExtensibleWorm-Like Chain (Odijk, 1995)FFr≈FFr≈FFrf≈(a, n)(a, n, ksegment)(p, n)(p, n, ksegment)MODEL SCHEMATIC FORMULASFr≈F()BB2contourBB-13k T 3k T=na aL1aExact Formula : ) = na coth x - where: x = (2)xkTkTLangevin Expansion : = (3)a= Inverse Langevin F⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠f(r) r = r (1)fr(ff(r)naββGaussian:Non- Gaussian :rL35 7-1Bcontour-1Btotalunction9 297 1539= 3 ...5 na 175 na 875 nakTHigh Stretch Approximation : = 1- (4)aLNon - Gaussian : kT=a⎛⎞⎛ ⎞⎛ ⎞⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞++ + +⎜⎟⎜ ⎟⎜ ⎟⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟⎜⎟⎜ ⎟⎜ ⎟⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠⎝⎠⎝ ⎠⎝ ⎠⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎜⎟⎝⎠narf(r)f(r)Lrr r rrLtotal contourB2contourcontourB ; L = L + nExact : Numerical SolutionkT 1 1Interpolation Formula : =pL441LkTInterpolation Formula : =pL⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎜⎟⎛⎞⎜⎟+−⎜⎟⎜⎟⎛⎞⎝⎠⎜⎟−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎜⎟⎝⎠segmentfkrf(r)rrf(r)2totaltotaltotal contour11; 441LL= L + n⎛⎞⎜⎟⎜⎟+−⎜⎟⎛⎞⎜⎟−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎜⎟⎜⎟⎝⎠segmentrfkfffffffFreely-Jointed Chain (FJC)(Kuhn and Grün, 1942 James and Guth, 1943)ExtensibleFreely-Jointed Chain(Smith, et. al, 1996)Worm-Like Chain (WLC)(Kratky and Porod, 1943Fixman and Kovac, 1973Bustamante, et. al 1994)ExtensibleWorm-Like Chain (Odijk, 1995)FFr≈FFr≈FFrf≈(a, n)(a, n, ksegment)(p, n)(p, n, ksegment)MODEL SCHEMATIC FORMULAS()BB2contourBB-13k T 3k T=na aL1aExact Formula : ) = na coth x - where: x = (2)xkTkTLangevin Expansion : = (3)a= Inverse Langevin


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