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Research Papers

Large Amplitude Thermal Fluctuations of Confined Semiflexible Biopolymer Filaments

[+] Author and Article Information
F. Jonsdottir

School of Engineering and Natural Sciences,
University of Iceland,
Reykjavik 107, Iceland
e-mail: fj@hi.is

L. B. Freund

Department of Materials Science and Engineering,
University of Illinois at Urbana-Champaign,
1304 West Green Street,
Urbana, IL 61801
e-mail: lbf@illinois.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 16, 2014; final manuscript received September 8, 2014; accepted manuscript posted September 11, 2014; published online September 24, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(11), 111006 (Sep 24, 2014) (6 pages) Paper No: JAM-14-1374; doi: 10.1115/1.4028535 History: Received August 16, 2014; Revised September 08, 2014; Accepted September 11, 2014

The phenomenon of thermal fluctuations of biopolymers has been of active interest for some time with a view toward understanding the effect of filament confinement, migration, and bonding. In this study, we focus our attention on planar fluctuations of a single filament between parallel confining surfaces. Filament slopes, with respect to the centerline of the channel, commonly exceed 0.1 in magnitude and therefore fall outside the range of small amplitude fluctuations. Consequently, large amplitudes are anticipated from the outset. Determination of the partition function leads to the quantitative dependence of free energy and other thermodynamic parameters on the degree of confinement.

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Figures

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Fig. 2

(a) Parallel rigid surfaces separated by a distance 2c define the spatial limits on fluctuation of a filament confined to the plane between the surfaces. (b) The lower sketch illustrates a representative shape for a filament of contour length L. The lateral deflection h(x) at distance x along the centerline of the gap is a convenient means for describing shape.

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Fig. 1

The upper portion shows the configuration of an initially straight filament of length L bent into a semicircular shape. If the elastic bending stiffness of the filament is B then the end moment required to maintain that shape is πB/L. The lower portion shows an approximate shape for the same filament consisting of six links, each of length b=L/6, and the bending moment transmitted across each joint is κ=B/b per radian of angle change.

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Fig. 5

A histogram showing the distribution of energies calculated for a total of 105 trial filament shapes for c=30b and λp=15b. The unit of energy along the abscissa is kT and the bin width in those units is 2. The ordinate n(E) represents the number of trial configurations resulting in an elastic energy value falling within a certain bin as the height of that bin. In the present case, the mean value of energy is found to be approximately μ=335.8 and the standard deviation from the mean is found to be approximately σ=32.7.

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Fig. 3

Graphical representation of the transformation rule for mapping a random distribution of numbers α in the range of 0 < α < 1 into a set of numbers β that are normally distributed. The latter are required to construct realistic random conformations of a filament.

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Fig. 4

Sample of a part of one filament shape generated by means of the scheme outlined in the text for the case of c=20b and κ=20 kT. The portion shown consists of approximately 500 segments, each of length b. Note that slopes in the large deflection range are common.

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Fig. 6

Plot of the filament free energy A normalized by kT versus one-half of the width of the confining channel c normalized by the parameter b. Results are shown for four values of the normalized persistence length λp/b for a value of n = 1000.

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