Worm-like chain

File:Worm-like Chain.png The WLC model envisions a continuously flexible isotropic rod.{{cite book | author=Doi and Edwards | title=The Theory of Polymer Dynamics | year=1988}}{{cite book | author=Rubinstein and Colby | title=Polymer Physics | year=2003}}{{cite book | author=Kirby, B.J. | title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices | url=http://www.kirbyresearch.com/textbook}} This is in contrast to the freely-jointed chain model, which is only flexible between discrete freely hinged segments. The model is particularly suited for describing stiffer polymers, with successive segments displaying a sort of cooperativity: nearby segments are roughly aligned. At room temperature, the polymer adopts a smoothly curved conformation; at $T\; =\; 0$ K, the polymer adopts a rigid rod conformation.

For a polymer of maximum length $L\_0$, parametrize the path of the polymer as $s\; \backslash in(0,L\_0)$. Allow $\backslash hat\; t(s)$ to be the unit tangent vector to the chain at point $s$, and $\backslash vec\; r(s)$ to be the position vector along the chain, as shown to the right. Then:

:$\backslash hat\; t(s)\; \backslash equiv\; \backslash frac\; \{\backslash partial\; \backslash vec\; r(s)\; \}\{\backslash partial\; s\}$ and the end-to-end distance $\backslash vec\; R\; =\; \backslash int\_\{0\}^\{L\_0\}\backslash hat\; t(s)\; ds$ .

The energy associated with the bending of the polymer can be written as:

$E\; =\; \backslash frac\; \{1\}\{2\}k\_B\; T\; \backslash int\_\{0\}^\{L\_0\}\; P\; \backslash cdot\; \backslash left\; (\backslash frac\; \{\backslash partial^2\; \backslash vec\; r(s)\; \}\{\backslash partial\; s^2\}\backslash right\; )^\{2\}\; ds$

where $P$ is the polymer's characteristic persistence length, $k\_B$ is the Boltzmann constant, and $T$ is the absolute temperature. At finite temperatures, the end-to end distance of the polymer will be significantly shorter than the maximum length $L\_0$. This is caused by thermal fluctuations, which result in a coiled, random configuration of the undisturbed polymer.

The polymer's orientation correlation function can then be solved for, and it follows an exponential decay with decay constant 1/P:

$\backslash langle\backslash hat\; t(s)\; \backslash cdot\; \backslash hat\; t(0)\backslash rangle=\backslash langle\; \backslash cos\; \backslash ;\; \backslash theta\; (s)\backslash rangle\; =\; e^\{-s/P\}\backslash ,$

File:Crossover from flexible to stiff behavior in the worm-like chain polymer model.svg

A useful value is the mean square end-to-end distance of the polymer:

\langle R^{2} \rangle = \langle \vec R \cdot \vec R \rangle

= \left\langle \int_{0}^{L_0} \hat t(s) ds \cdot \int_{0}^{L_0} \hat t(s') ds' \right\rangle = \int_{0}^{L_0} ds \int_{0}^{L_0} \langle \hat t(s) \cdot \hat t(s') \rangle ds'= \int_{0}^{L_0} ds \int_{0}^{L_0} e^{-\left | s - s' \right | / P} ds'

\langle R^{2} \rangle

= 2 PL_0 \left [ 1 - \frac {P}{L_0} \left ( 1 - e^{-L_0/P} \right ) \right ]

Note that in the limit of $L\_0\; \backslash gg\; P$, then $\backslash langle\; R^\{2\}\; \backslash rangle\; =\; 2PL\_0$. This can be used to show that a Kuhn segment is equal to twice the persistence length of a worm-like chain. In the limit of $L\_0\; \backslash ll\; P$, then $\backslash langle\; R^\{2\}\; \backslash rangle\; =\; L\_0^2$, and the polymer displays rigid rod behavior. The figure to the right shows the crossover from flexible to stiff behavior as the persistence length increases.

File:Worm-like Chain Model vs. DNA.svg

Experimental data from the stretching of Lambda phage DNA is shown to the right, with force measurements determined by analysis of Brownian fluctuations of a bead attached to the DNA. A persistence length of 51.35 nm and a contour length of 1560.9 nm were used for the model, which is depicted by the solid line.{{cite journal|last1=Bouchiat|first1=C|title=Estimating the Persistence Length of a Worm-Like Chain Molecule from Force-Extension Measurements|journal=Biophysical Journal|volume=76|issue=1|pages=409–413|date=1999|doi=10.1016/S0006-3495(99)77207-3|pmid=9876152|pmc=1302529|bibcode=1999BpJ....76..409B|doi-access=free}}

Other biologically important polymers that can be effectively modeled as worm-like chains include:

• double-stranded DNA (persistence length 40-50 nm) and RNA (persistence length 64 nm){{cite journal | author=J. A. Abels and F. Moreno-Herrero and T. van der Heijden and C. Dekker and N. H. Dekker | title=Single-Molecule Measurements of the Persistence Length of Double-Stranded RNA | journal=Biophysical Journal| year=2005| volume=88| issue=4 | pages=2737–2744| doi=10.1529/biophysj.104.052811 | pmid=15653727 | pmc=1305369 | bibcode=2005BpJ....88.2737A}}
• single-stranded DNA (persistence length 4 nm) {{cite journal|last1=Bernard|first1=Tinland|title=Persistence Length of Single-Stranded DNA|journal=Macromolecules|date=1997|volume=30|issue=19|page=5763|doi=10.1021/ma970381+|bibcode=1997MaMol..30.5763T}}
• unstructured RNA (persistence length 2 nm) {{cite journal|last1=Chen|first1=Huimin|last2=Meisburger|first2=Steve P.|title=Ionic strength-dependent persistence lengths of single-stranded RNA and DNA|journal=PNAS|volume=109|issue=3|pages=799–804|date=2011|doi=10.1073/pnas.1119057109|pmid=22203973|pmc=3271905|bibcode=2012PNAS..109..799C|doi-access=free}}
• unstructured proteins (persistence length 0.6-0.7 nm){{cite journal | author=L. J. Lapidus and P. J. Steinbach and W. A. Eaton and A. Szabo and J. Hofrichter | title=Single-Molecule Effects of Chain Stiffness on the Dynamics of Loop Formation in Polypeptides. Appendix: Testing a 1-Dimensional Diffusion Model for Peptide Dynamics | journal=Journal of Physical Chemistry B| year=2002| volume=106| pages=11628–11640| doi=10.1021/jp020829v}}
• microtubules (persistence length 0.52 cm) {{cite journal|last1=Gittes|first1=F|title=Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape|journal=Journal of Cell Biology|volume=120|issue=4|pages=923–934|date=1993|doi=10.1083/jcb.120.4.923|pmid=8432732|pmc=2200075}}
• filamentous bacteriophage{{Cite journal|last1=Khalil|first1=A. S.|last2=Ferrer|first2=J. M.|last3=Brau|first3=R. R.|last4=Kottmann|first4=S. T.|last5=Noren|first5=C. J.|last6=Lang|first6=M. J.|last7=Belcher|first7=A. M.|date=2007|title=Single M13 bacteriophage tethering and stretching|journal=Proceedings of the National Academy of Sciences|language=en|volume=104|issue=12|pages=4892–4897|doi=10.1073/pnas.0605727104|issn=0027-8424|pmc=1829235|pmid=17360403|doi-access=free}}

Upon stretching, the accessible spectrum of thermal fluctuations reduces, which causes an entropic force acting against the external elongation.

This entropic force can be estimated from considering the total energy of the polymer:

$H\; =\; H\_\{\backslash rm\; elastic\}\; +\; H\_\{\backslash rm\; external\}=\; \backslash frac\; \{1\}\{2\}k\_B\; T\; \backslash int\_\{0\}^\{L\_0\}\; P\; \backslash cdot\; \backslash left\; (\backslash frac\; \{\backslash partial^2\; \backslash vec\; r(s)\; \}\{\backslash partial\; s^2\}\backslash right\; )^\{2\}\; ds\; -\; xF$.

Here, the contour length is represented by $L\_0$, the persistence length by $P$, the extension is represented by $x$, and external force is represented by $F$.

Laboratory tools such as atomic force microscopy (AFM) and optical tweezers have been used to characterize the force-dependent stretching behavior of biological polymers. An interpolation formula that approximates the force-extension behavior with about 15% relative error is:{{cite journal|last1=Marko|first1=J.F.|last2=Siggia|first2=E.D.|title=Statistical mechanics of supercoiled DNA|journal=Physical Review E |volume=52|issue=3|pages=2912–2938|date=1995|doi=10.1103/PhysRevE.52.2912|pmid=9963738|bibcode=1995PhRvE..52.2912M}}

:$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; \backslash frac\; \{x\}\; \{L\_0\}\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; \backslash frac\; \{x\}\{L\_0\}$

A more accurate approximation for the force-extension behavior with about 0.01% relative error is:

:$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; \backslash frac\; \{x\}\; \{L\_0\}\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; \backslash frac\; \{x\}\{L\_0\}\; +\; \backslash sum\_\{i=2\}^\{i=7\}\; \backslash alpha\_i\; \backslash left(\; \backslash frac\{x\}\{L\_0\}\; \backslash right)\; ^i$,

with $\backslash alpha\_2\; =\; -0.5164228$, $\backslash alpha\_3\; =\; -2.737418$, $\backslash alpha\_4\; =\; 16.07497$, $\backslash alpha\_5\; =\; -38.87607$, $\backslash alpha\_6\; =\; 39.49944$, $\backslash alpha\_7\; =\; -14.17718$.

A simple and accurate approximation for the force-extension behavior with about 1% relative error is:{{cite journal|last=Petrosyan|first=R.|title=Improved approximations for some polymer extension models|journal=Rehol Acta|year=2017|volume=56|pages=21–26|doi = 10.1007/s00397-016-0977-9 |arxiv=1606.02519|s2cid=100350117}}

:$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; \backslash frac\; \{x\}\; \{L\_0\}\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; \backslash frac\; \{x\}\{L\_0\}-0.8\; \backslash left\; (\backslash frac\; \{x\}\; \{L\_0\}\; \backslash right\; )^\{2.15\}$

Approximation for the extension-force behavior with about 1% relative error was also reported:

:$\backslash frac\; \{x\}\; \{L\_0\}\; =\; \backslash frac\; \{4\}\{3\}\; -\; \backslash frac\; \{4\}\; \{3\backslash sqrt\{\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; +\; 1\}\}\; -\backslash frac\; \{10e^\{\backslash sqrt[4]\{900\backslash frac\; \{k\_\{B\}T\}\; \{FP\}\}\}\}\; \{\backslash sqrt\{\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\}\; \backslash left(e^\{\backslash sqrt[4]\{900\backslash frac\; \{k\_\{B\}T\}\; \{FP\}\}\}-1\; \backslash right)^\{2\}\; \}\; +\; \backslash frac\; \{\backslash left\; (\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\backslash right)^\{1.62\}\}\; \{3.55+3.8\backslash left\; (\; \backslash frac\; \{FP\}\; \{k\_\{B\}T\}\backslash right)^\{2.2\}\}$

The elastic response from extension cannot be neglected: polymers elongate due to external forces. This enthalpic compliance is accounted for the material parameter $K\_0$, and the system yields the following Hamiltonian for significantly extended polymers:

$H\; =\; H\_\{\backslash rm\; elastic\}+H\_\{\backslash rm\; enthalpic\}+H\_\{\backslash rm\; external\}\; =\; \backslash frac\; \{1\}\{2\}k\_B\; T\; \backslash int\_\{0\}^\{L\_0\}\; P\; \backslash cdot\; \backslash left\; (\backslash frac\; \{\backslash partial\; ^2\backslash vec\; r(s)\; \}\{\backslash partial\; s^2\}\backslash right\; )^\{2\}\; ds\; +\; \backslash frac\; \{1\}\{2\}\backslash frac\; \{K\_0\}\{L\_0\}\; x^\{2\}\; -\; xF$,

This expression contains both the entropic term, which describes changes in the polymer conformation, and the enthalpic term, which describes the stretching of the polymer due to the external force.

Several approximations for the force-extension behavior have been put forward, depending on the applied external force. These approximations are made for stretching DNA in physiological conditions (near neutral pH, ionic strength approximately 100 mM, room temperature), with stretch modulus around 1000 pN.{{cite journal|last=Wang|first=Michelle D.|author-link=Michelle Wang|author2=Hong Yin|author3=Robert Landick|author4=Jeff Gelles|author5=Steven M. Block|year=1997|title=Stretching DNA with Optical Tweezers|journal=Biophysical Journal|volume=72|issue=3|pages=1335–1346|bibcode=1997BpJ....72.1335W|doi=10.1016/S0006-3495(97)78780-0|pmc=1184516|pmid=9138579}}{{Cite journal | doi=10.1007/s12551-016-0236-4| pmid=28303166| pmc=5331113|title = Single-molecule studies of high-mobility group B architectural DNA bending proteins| journal=Biophysical Reviews| volume=9| issue=1| pages=17–40|year = 2017|last1 = Murugesapillai|first1 = Divakaran| last2=McCauley| first2=Micah J.| last3=Maher| first3=L. James| last4=Williams| first4=Mark C.}}

For the low-force regime (F < about 10 pN), the following interpolation formula was derived:{{cite journal|last=Marko|first=J.F.|author2=Eric D. Siggia |title=Stretching DNA|journal=Macromolecules|year=1995|volume=28|issue=26|pages=8759–8770|bibcode = 1995MaMol..28.8759M |doi = 10.1021/ma00130a008 }}

$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; \backslash frac\; \{x\}\; \{L\_0\}\; +\; \backslash frac\; \{F\}\{K\_0\}\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; \backslash frac\; \{x\}\{L\_0\}\; -\; \backslash frac\; \{F\}\{K\_0\}$.

For the higher-force regime, where the polymer is significantly extended, the following approximation is valid:{{cite journal|last=Odijk|first=Theo|title=Stiff Chains and Filaments under Tension|journal=Macromolecules|year=1995|volume=28|issue=20|pages=7016–7018|bibcode = 1995MaMol..28.7016O |doi = 10.1021/ma00124a044 |url=http://resolver.tudelft.nl/uuid:2766c66f-fec4-4e02-81cd-12aec69d56b6}}

$x\; =\; L\_0\; \backslash left\; (\; 1\; -\; \backslash frac\; \{1\}\; \{2\}\; \backslash left\; (\; \backslash frac\; \{k\_\{B\}T\}\{FP\}\; \backslash right\; )^\{1/2\}\; +\; \backslash frac\; \{F\}\{K\_0\}\backslash right\; )$.

As for the case without extension, a more accurate formula was derived:

$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; l\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; l\; +\; \backslash sum\_\{i=2\}^\{i=7\}\; \backslash alpha\_i\; \backslash left(\; l\; \backslash right)\; ^i$,

with $l\; =\; \backslash frac\{x\}\{L\_0\}\; -\; \backslash frac\{F\}\{K\_0\}$. The $\backslash alpha\_i$ coefficients are the same as the above described formula for the WLC model without elasticity.

Accurate and simple interpolation formulas for the force-extension and extension-force behaviors for the extensible worm-like chain model are:

:$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; \backslash frac\; \{x\}\; \{L\_0\}+\; \backslash frac\; \{F\}\{K\_0\}\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; \backslash frac\; \{x\}\{L\_0\}\; -\; \backslash frac\; \{F\}\{K\_0\}-0.8\; \backslash left\; (\backslash frac\; \{x\}\; \{L\_0\}-\; \backslash frac\; \{F\}\{K\_0\}\; \backslash right\; )^\{2.15\}$

:$\backslash frac\; \{x\}\; \{L\_0\}\; =\; \backslash frac\; \{4\}\{3\}\; -\; \backslash frac\; \{4\}\; \{3\backslash sqrt\{\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; +\; 1\}\}\; -\backslash frac\; \{10e^\{\backslash sqrt[4]\{900\backslash frac\; \{k\_\{B\}T\}\; \{FP\}\}\}\}\; \{\backslash sqrt\{\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\}\; \backslash left(e^\{\backslash sqrt[4]\{900\backslash frac\; \{k\_\{B\}T\}\; \{FP\}\}\}-1\; \backslash right)^\{2\}\; \}\; +\; \backslash frac\; \{\backslash left\; (\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\backslash right)^\{1.62\}\}\; \{3.55+3.8\backslash left\; (\; \backslash frac\; \{FP\}\; \{k\_\{B\}T\}\backslash right)^\{2.2\}\}\; +\; \backslash frac\; \{F\}\{K\_0\}$

• Ideal chain
• Polymer
• Polymer physics

{{Reflist|30em}}

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• {{cite journal | last1 = Marko | first1 = J. F. | last2 = Siggia | first2 = E. D. | year = 1995 | title = Stretching DNA | journal = Macromolecules | volume = 28 | issue = 26| pages = 8759–8770 | doi=10.1021/ma00130a008 | bibcode=1995MaMol..28.8759M}}
• {{cite journal | last1 = Bustamante | first1 = C. | last2 = Marko | first2 = J. F. | last3 = Siggia | first3 = E. D. | last4 = Smith | first4 = S. | year = 1994 | title = Entropic elasticity of lambda-phage DNA | journal = Science | volume = 265 | issue = 5178| pages = 1599–1600 | pmid = 8079175 | doi=10.1126/science.8079175|bibcode = 1994Sci...265.1599B | doi-access = free }}
• {{cite journal | last1 = Wang | first1 = M. D. | last2 = Yin | first2 = H. | last3 = Landick | first3 = R. | last4 = Gelles | first4 = J. | last5 = Block | first5 = S. M. | year = 1997 | title = Stretching DNA with optical tweezers | journal = Biophysical Journal | volume = 72 | issue = 3| pages = 1335–1346 | pmid = 9138579 | doi=10.1016/S0006-3495(97)78780-0 | pmc=1184516|bibcode = 1997BpJ....72.1335W }}
• C. Bouchiat et al., [http://www.biophysj.org/cgi/content/abstract/76/1/409 "Estimating the Persistence Length of a Worm-Like Chain Molecule from Force-Extension Measurements"], Biophysical Journal, January 1999, p. 409-413, Vol. 76, No. 1

Category:Polymer physics

Category:Polymers

Category:Biophysics