Abstract
The Lifshitz equation for the confinement of a linear polymer in a spherical cavity of radius R has the form of the Schrödinger equation for a quantum particle trapped in a potential well with flat bottom and infinite walls at radius R. We show that the Lifshitz equation of a confined annealed branched polymer has the form of the Schrödinger equation for a quantum harmonic oscillator. The resulting confinement energy has a 1/R4 dependence on the confinement radius R, in contrast to the case of confined linear polymers, which have a 1/R2 dependence. We discuss the application of this result to the problem of the confinement of single-stranded RNA molecules inside spherical capsids.
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Notes
To help the reader, we provide here details of this “translation”. Suppose that the external potential φ2(r) is zero inside a sphere of radius R ≫ a and very large (infinite) outside. Replacing \(\hat {g} \simeq 1 + (a^{2}/6) \nabla ^{2}\), (2) becomes
$$\left( 1 - \frac{1}{p^{\ast}} \right) \tilde{\psi} = - \frac{a^{2}}{6} \nabla^{2} \tilde{\psi} . $$The Schrödinger equation inside such a potential well reads
$$\epsilon \psi = - \frac{\hbar^{2}}{2 m} \nabla^{2} \psi . $$From the known ground state energy for the quantum case, we can find p∗ by replacing \(\hbar ^{2}/2m \to a^{2}/6\) and 𝜖 → 1 − (1/p∗). This gives 1/p∗ = 1 − Ca2/3R2. Using formula (1), one finds a confinement free energy Ω = N lnp∗≃ Ca2/3R2.
In the original equations derived in the work [16] and also reproduced in [15], (3a) include also factors Λ1 and Λ3—the fugacities of the end-points and branch-points, respectively. For the system at hand, this is not necessary, because for any tree the number of ends is always equal to the number of branched points plus two: they are the same in large N limit. In the earlier paper [15], we got rid of these factors by a simple re-naming of variables. In a more general situation, where one may have one, two- and three-valent units in the polymer, i.e., our building set includes both dimers of Fig. 1 and trimers of Fig. 3, then we need Λ1, Λ2, and Λ3 to control relative abundance of these building blocks.
In order to obtain this result, start from (1) and insert it into the left hand side of (8). Recall that \(p^{\ast }_{\text {free}} = 1/2\) and Ωfree = −N ln 2 and insert this into the first term in the right hand side. One finds that \(1/p^{\ast } = 2 e^{-\varphi _{0} - \omega } \simeq 2 e^{-\varphi _{0}} \left (1 - \omega \right )\). Insert this into (3a) and (3b) with external potentials (6) and replace \(\hat {g} \to 1 + (a^{2}/2D) \nabla ^{2}\).
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Acknowledgments
RB would like to acknowledge support from the National Science Foundation under DMR Grant 1309423. The work of AYG was supported partially by the MRSEC Program of the National Science Foundation under Award Number DMR-1420073. RB and AYG thank the Aspen Center for Physics where part of this work was done with the support of the National Science Foundation under Grant No. PHY-1066293.
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Grosberg, A.Y., Bruinsma, R. Confining annealed branched polymers inside spherical capsids. J Biol Phys 44, 133–145 (2018). https://doi.org/10.1007/s10867-018-9483-x
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DOI: https://doi.org/10.1007/s10867-018-9483-x