Preface to the First English Edition

Landau, L.D.; Lifshitz, E.M.

doi:10.1016/b978-0-08-057069-3.50004-8

Cited by 997 publications

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“…19 We start with a Cartesian coordinate system with its origin coinciding with the center of the capsid. A point r inside, the capsid shell is given by where R 0 -h/2 e r e R 0 + h/2, 0 e φ < 2π, and where we are only interested in a small section 0 e θ , 1.…”

Section: Discussionmentioning

confidence: 99%

“…Equating σ rr to zero leads to the following equation ε can be determined from this equation by demanding that it is satisfied to leading order in θ. We first compute the relevant components of the strain tensor 19 and we then find and The remaining components of the strain tensor are u rθ ) u θr ) O(θ) and u rφ ) u φr ) O(θ). These last two equations are consistent with the fact that σ θr ≈ u θr and σ φr ≈ u φr should be small.…”

Section: Discussionmentioning

confidence: 99%

“…The free-energy density in the small section is equal to 19 from which the free energy per unit area of neutral surface can be calculated by integration…”

Section: Discussionmentioning

confidence: 99%

“…Note that the elastic constant is proportional to (1 -ν) -1 and hence is different from the flexural rigidity or cylindrical rigidity, 19 which is proportional to (1 -ν 2 ) -1 . However, we are dealing with the deformation of a surface where the deformed surface does not contain the same amount of material as the (smaller) undeformed surface.…”

Section: Entropic Elasticity Of Armsmentioning

confidence: 99%

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Multishell Structures of Virus Coat Proteins

et al. 2010

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Under conditions of low ionic strength and a pH ranging between about 3.7 and 5.0, solutions of purified coat proteins of cowpea chlorotic mottle virus (CCMV) form spherical multishell structures in the absence of viral RNA. The outer surfaces of the shells in these structures are negatively charged, whereas the inner surfaces are positively charged due to a disordered cationic N-terminal domain of the capsid protein, the arginine-rich RNA-binding motif that protrudes into the interior. We show that the main forces stabilizing these multishells are counterion release combined with a lower charge density in the RNA-binding motif region of the outer shells due to their larger radii of curvature, arguing that these compensate for the outer shells not being able to adopt the smaller, optimal, radius of curvature of the inner shell. This explains why the structures are only stable at low ionic strengths at pHs for which the outer surface is negatively charged and why the larger outer shells are not observed separately in solution. We show how to calculate the free energy of shells of nonoptimal radius of curvature from the elastic properties of the native shell. The spacing between shells is determined mainly by the entropic elasticity of the RNA-binding motifs. Although we focus on CCMV multishells, we also predict the solution conditions under which multishells formed by CCMV coat protein mutants with a lower RNA-binding motif charge are stable, and we examine other viruses as well. We conclude that at a given surface charge density, the boundaries separating regions of stable multishells with different numbers of shells shift to lower ionic strengths upon either increasing the length of the RNAbinding motif, increasing the stiffness of the shells, or decreasing the charge per RNA-binding motif.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…The free-energy density in the small section is equal to 19 from which the free energy per unit area of neutral surface can be calculated by integration…”

Section: Discussionmentioning

confidence: 99%

Section: Entropic Elasticity Of Armsmentioning

confidence: 99%

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Multishell Structures of Virus Coat Proteins

et al. 2010

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“…coli tethered by antifilament antibody, the time constant for relaxation is -0.15 s, giving a torsional spring constant of -4 x dyn-cm rad-l. The spring constant of a short hollow tube is related to the torsional rigidity, p, through the formula k = pdR4-r4)/2L, where r and R are the inner and outer tube radii, res ectively, and L is the tube length (20). Taking R = 70 f;, r = 30 A, and L = 0.2 pm, along with the experimental value for k, we obtained p = 2 x 10' dyn cm-2, which is 3-4 orders of magnitude lower than estimates derived from isolated filaments of Salmonella typhimurium (14).…”

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confidence: 99%

Compliance of bacterial polyhooks measured with optical tweezers

1991

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In earlier work, a single‐beam gradient force optical trap (“optical tweezers”) was used to measure the torsional compliance of flagella in wild‐type cells of Escherichia coli that had been tethered to glass by a single flagellum. This compliance was nonlinear, exhibiting a torsionally soft phase up to 180°, followed by a torsionally rigid phase for larger angles. Values for the torsional spring constant in the soft phase were substantially less than estimates based on the rigidity determined for isolated flagellar filaments. It was suggested that the soft phase might correspond to wind‐up of the flagellar hook, and the rigid phase to wind‐up of the stiffer filament. Here, we have measured the torsional compliance of flagella on cells of an E. coli strain that produces abnormally long hooks but no filaments. The small‐angle compliance of these cells, as determined from the elastic rebound of the cell body after wind‐up and release, was found to be the same as for wild‐type cells. This confirms that the small‐angle compliance of wild‐type cells is dominated by the response of the hook. Hook flexibility is likely to play a useful role in stabilizing the flagellar bundle.

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