Multimedia : Podcast : Sacrificial Bonds in Biological Materials; Fantner, et al. Biophys. J , 1411

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1 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE I LECTURE 2: THEORETICAL ASPECTS O SINGLE MOLECULE ORCE SPECTROSCOPY 2 : EXTENSIBILITY AND THE WORM LIKE CHAIN (WLC) Outline : REVIEW LECTURE #19 : THEORETICAL ORMULATIONS OR THE ELASTICITY O SINGLE POLYMER CHAINS (JC)...2 VARIOUS MATHEMATICAL ORMS OR THE INEXTENSIBLE JC- GRAPHICAL COMPARISON... 3 EECT O a AND n on INEXTENSIBLE JC... 4 EXTENSIBLE JC... 5 WORM LIKE CHAIN (WLC) MODEL... 6 ITS TO EXPERIMENTAL SINGLE MOLECULE ORCE SPECTROSCOPY DATA... 7 Objectives: To understand how extensibility o chain segments aects the JC elasticity model and to understand the dierences between the JC and WLC models Readings: Course Reader Document 31, CR Documents are the original theoretical papers or reerence, English translations o CR 33 and 36 are available as handouts. Multimedia : Podcast : Sacriicial Bonds in Biological Materials; antner, et al. Biophys. J. 26 9,

2 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE REVIEW LECTURE 19 : MODELS OR SINGLE POLYMER CHAIN ELASTICITY MODEL SCHEMATIC ORMULAS reely- Jointed Chain (JC) (Kuhn, 1934 Guth and Mark, 1934) Extensible reely-jointed Chain (Smith, et. al, 1996) Worm-Like Chain (WLC) (Kratky and Porod, 1943 ixman and Kovac, 1973 Bouchiat, et al. 1999) r (a, n) r (a, n, k segment ) r (p, n) 3kBT 3kBT Gaussian : (r) = 2 r = r (1) na alcontour Non - Gaussian : 1 a Exact ormula : r( ) = na coth ( x) - where : x = (2) x kbt kt B Langevin Expansion : (r) = β (3) a -1 r β = L = Inverse Langevin unction na r 9 r 297 r 1539 r = na 5 na 175 na 875 na -1 kt r B High Stretch Approximation : (r) = 1- (4) a Lcontour Non - Gaussian : kt B r (r) = L a ; L = L + n -1 total contour L total ksegment Exact : Numerical Solution kt B r 1 1 Interpolation ormula : (r) = 2 p + Lcontour 4 r 4 1 L contour Extensible Worm-Like Chain (Odijk, 1995 Wang, et al. 1997) r (p, n, k segment ) kt Interpolation ormula : = L total = L contour + n ksegment r ; 4 r 4 1 L total B (r) p 2 Ltotal 2

3 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE COMPARISON O VARIOUS MATHEMATICAL ORMS OR THE INEXTENSIBLE REELY JOINTED CHAIN (JC) MODEL orce (nn) high stretch (4) dashed elastic r elastic coth (3) Langevin (2) Gaussian (1) orce (nn) /3L contour high stretch (4) dashed 3/4L contour coth (3) Gaussian Langevin (2) Distance (nm) Distance (nm) Surace separation distance, D= r, chain end-to-end distance; sign convention (-) or attractive back orce, however some scientists plot as (+); e.g. Zauscher (podcast) (1) Gaussian physically unrealistic; orce continues to increase orever beyond L contour, valid or r,d<1/3 L contour (3) Langevin Series Expansion; inite orce beyond L contour (physically unrealistic); valid or r,d<3/4 L contour (4) High stretch approximation underestimates orce or r,d<3/4l contour, valid or r,d>3/4l contour -.5 3

4 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE EECT O a and n on INEXTENSIBLE JC 3kBT Gaussian : (r) = 2 r na elastic (nn) a =.1 nm a =.2 nm a =.3 nm a =.6 nm a = 1.2 nm a = 3. nm elastic (nn) n=1 n=2 n=3 n=4 n= (let) Elastic orce versus displacement as a unction o the statistical segment length, a, or the non-gaussian JC model (L contour = 2 nm) and (right) elastic orce versus displacement as a unction o the number o chain segments, n, or the non-gaussian JC model (a =.6 nm) 4

5 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE EXTENSIBLE REELY JOINTED CHAIN (JC) MODEL elastic r elastic - Take into account a small amount o longitudinal (along chain axis) enthalpic deormability (monomer/bond stretching) o each statistical segment, approximate each statistical segment as a linear elastic entropic spring (valid or small deormations) with stiness, k segment springs is series, orces are equal, strain additive; k segment ; segment = k segment δ segment solve or: δ segment = segment /k segment Add displacement term to L contour : L total = L contour + n 123 =na ksegment extension beyond Lcontour due to enthalpic stretching o chain segments n= number o statistical segments kt r (r) = L B -1 a Ltotal Now we have three physical (itting) parameters; a, n, k segment elastic (nn) non- Gaussian JC extensible non- Gaussian JC Schematic o the stretching o an extensible reely jointed chain and (b) the elastic orce versus displacement or the extensible compared to non-extensible non- Gaussian JC (a =.6 nm, n = 1, k segment = 1 N/m) - note units 5

6 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE WORM LIKE CHAIN (WLC) MODEL (*Kratky-Porod Model) "Directed random walk"- segments are correlated, polymer chains intermediate between a rigid rod and a lexible coil (e.g. DNA) - takes into account both local stiness and long range lexibility -chain is treated as an isotropic, homogeneous elastic rod whose trajectory varies continuously and smoothly through space as opposed to the jagged contours o the JC γ (l w ) 1 p= persistence length, length over which statistical segments remain directionally correlated in space Exact : Numerical Solution kt B r 1 1 Interpolation ormula : (r) = 2 p + Lcontour 4 r 4 1 L contour -WLC stier at higher extensions, orce rises sooner than JC since statistical segments are constrained, can also make an extensible orm o WLC replace L contour by L total as beore or JC -In reality the JC and WLC are very similar and just produce slightly dierent values o the local chain stiness elastic (nn) r (l w ) n non-gaussian JC WLC 6

7 3.52 Nanomechanics o Materials and Biomaterials Tuesday 5/1/7 Pro. C. Ortiz, MIT-DMSE ITS TO EXPERIMENTAL SINGLE MOLECULE ORCE SPECTROSCOPY DATA AM retract data on single polymer chain o polystyrene in toluene; comparison to reely-jointed Chain Model (a =.68 nm) - right plot aata normalized by L contour to create a master plot chain (nn) n chain (nn) L chain /L contour = Distance (nm) D / L contour 7