Mechanical transduction via a single soft polymer

Ruizheng Hou, Nan Wang, Weizhu Bao, and Zhisong Wang

Phys. Rev. E 97, 042504 – Published 25 April 2018

Abstract

Molecular machines from biology and nanotechnology often depend on soft structures to perform mechanical functions, but the underlying mechanisms and advantages or disadvantages over rigid structures are not fully understood. We report here a rigorous study of mechanical transduction along a single soft polymer based on exact solutions to the realistic three-dimensional wormlike-chain model and augmented with analytical relations derived from simpler polymer models. The results reveal surprisingly that a soft polymer with vanishingly small persistence length below a single chemical bond still transduces biased displacement and mechanical work up to practically significant amounts. This “soft” approach possesses unique advantages over the conventional wisdom of rigidity-based transduction, and potentially leads to a unified mechanism for effective allosterylike transduction and relay of mechanical actions, information, control, and molecules from one position to another in molecular devices and motors. This study also identifies an entropy limit unique to the soft transduction, and thereby suggests a possibility of detecting higher efficiency for kinesin motor and mutants in future experiments.

Received 29 September 2017
Revised 6 February 2018

DOI:https://doi.org/10.1103/PhysRevE.97.042504

Physics Subject Headings (PhySH)

Chain stiffness Entropy production Nanomechanical devices Persistence length Random walks

Physics of Living SystemsStatistical Physics & ThermodynamicsPolymers & Soft MatterInterdisciplinary Physics

Authors & Affiliations

Ruizheng Hou^1,*, Nan Wang², Weizhu Bao^2,3, and Zhisong Wang^3,4

¹School of Science and Institute of Quantum Optics and Quantum Information, Xi'an Jiaotong University, Shaan Xi 710049, China
²Department of Mathematics, National University of Singapore, Singapore 119076
³NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 119076
⁴Department of Physics, National University of Singapore, Singapore 117542

^*houruizheng@xjtu.edu.cn

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Issue

Vol. 97, Iss. 4 — April 2018

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Images

Figure 1
Illustration of the single-polymer model for orientation-based transduction (a) and the neck linker zippering in kinesin (b). (a) The red line denotes a polymer with one end (A end) fixed at the origin of the coordinate frame ( $x$ , $y$ , and $z$ axes) and the other end (B end) free. The two vectors $t_{A}$ and $t_{B}$ denote the orientations of the two ends. The force applied in the middle of the polymer represents a mechanical load. The gray spherical symbols on the $z$ axis denote the association units for the B end. (b) The dashed lines indicate the two feet of kinesin and the subunits ( $α / β$ tubulin) that compose the track (microtubule filaments). The broad red line is the neck linker which connects the two feet. The black arrow denotes the zippering effect that controls the orientation of the linker. “+” and “−” denote the polarity of the track.
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Figure 2
Biased displacement of the B end. The polymer is modeled as a WLC in 3D space (Fig. 1) with the A end oriented $(t_{A})$ along the $z$ axis. The contour length $(l_{c})$ of the polymer is 10 nm. (a) The position distribution of the B end in the $x − z$ plane or $y − z$ plane is shown for a persistence length $(l_{p})$ of 1 nm. The white cross in the center of the figure marks the origin point of the coordinate system. The overall distribution is symmetric around the $z$ axis. (b) The upper panel shows the free energy of the polymer versus the B-end position along the $z$ axis $[r_{B} = (0, 0, z_{B})]$ for different values of persistence length. The symbols are the measured binding barriers [42] for kinesin's mobile foot connected to the other track-bound foot via the neck linker. The lower panel shows the difference between the polymer's free energy when its B end is at a forward site $(z_{B} = l_{step})$ or an equally distanced backward site ( $z_{B} = - l_{step}$ , with $l_{step}$ as the step size).
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Figure 3
The bias generated by very soft polymers. The contour length $(l_{c})$ of the WLC and the FJC is 10 nm. (a),(b) The bias versus persistence length $(l_{p})$ toward the soft-polymer limit $(l_{p} \to 0)$ is shown. The WLC is in 3D space (a) or on a 2D plane (b). The results include biases at nine different step sizes $(l_{step})$ , from $0.8 l_{c}$ to 0 with the interval of $0.1 l_{c}$ , which are denoted by different colors and symbols. The symbols are numerical results and the lines are fittings by a fourth-order polynomial function. (c) The symbols are derived by the fittings in (a),(b) at the soft-polymer limit $(l_{p} \to 0)$ . The limit is marked by the black arrows in (a),(b). The solid lines are derived from the FJC model at the soft-polymer limit $(b \to 0)$ . The dashed lines are derived from Gaussian approximation [Eq. (1)] at the same limit.
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Figure 4
Transduction of work with load attached in the middle. The contour length of the polymer is 10 or 20 nm as indicated in the figure. (a) Typical load dependences of the bias. The symbols are numerical results from the 3D WLC model and the lines are fittings. (b),(c) Comparison between the decreasing slope ( $d$ ) of the bias and the load displacement $(l_{step})$ (b), as well as between the maximum work and the bias without load attached (c). The maximum work is derived by the load $(f_{stall})$ that cancels the bias (a). The symbols are numerical results and the lines indicate the two quantities in the comparison are equal.
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Figure 5
Transduction of work with load attached arbitrarily. Comparison between the load displacement and the decreasing slope of the bias (a), as well as between the bias without load attached and the maximum work (b). $s_{f}$ denotes the load-attached location. The colors denote results from different models and with different load-attached locations. The symbols are numerical results and the lines indicate the two quantities in the comparison are equal. The contour length of the polymer is 10 nm.
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Figure 6
Limits of work transduction. The contour length of the polymer is 10 nm. (a) Typical results obtained from the 3D WLC model for the displacement bias at the B end ( $Δ F_{out}$ ) generated from a limited input energy ( $Δ F_{in}$ ) for controlling the A-end orientation. The maximum displacement bias at the B end $(Δ F)$ generated from the fixed A-end orientation along the $z$ axis (black lines) are shown for comparison. (b) The comparison between the load displacement and the decreasing slope of the bias $Δ F_{out}$ against load (upper panel), as well as between the bias $Δ F_{out}$ with no load attached and the maximum work (lower panel). The load is attached in the middle of the polymer. The symbols are numerical results and the lines indicate equality between the two quantities. (c) Transduction efficiency $(η)$ calculated using Eq. (6) from the results in (a) and similar results for three more $Δ F_{in}$ values.
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Physical Review E

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