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Hill’s small systems nanothermodynamics: a simple macromolecular partition problem with a statistical perspective

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Abstract

Using a simple example of biological macromolecules which are partitioned between bulk solution and membrane, we investigate T.L. Hill’s phenomenological nanothermodynamics for small systems. By introducing a system size-dependent equilibrium constant for the bulk-membrane partition, we obtain Hill’s results on differential and integral chemical potentials μ and \(\hat{\mu}\) from computations based on standard Gibbsian equilibrium statistical mechanics. It is shown that their difference can be understood from an equilibrium re-partitioning between bulk and membrane fractions upon a change in the system’s size; it is closely related to the system’s fluctuations and inhomogeneity. These results provide a better understanding of nanothermodynamics and clarify its logical relation with the theory of statistical mechanics.

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Notes

  1. A difference between differential and integral forms of macromolecular interaction also appears in the theory of binding: Thermodynamic binding can go to zero while molecular interaction is maximized [9].

  2. To be mathematically more rigorous, one needs to solve the problem with self-consistency among (3), (4) and: \(V(N,p,T)=\left(\partial G/\partial p\right)_{N,T}\).

  3. Both steps in this gedankenexperiment are carried out under isothermal conditions. They cannot be realized in a laboratory. They are different from the adiabatic (isoentropic) and isothermal processes in the derivation of the fundamental equation of thermodynamics.

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Acknowledgements

I thank Prof. Dick Bedeaux for a stimulating week of discussions in June 2008 at XXI Sitges Conference (Spain), Profs. Ralph Chamberlin and Signe Kjelstrup for discussions that stimulated and renewed my interest in nanothermodynamics. I would like to acknowledge my friend and teacher (via his writing), Terrell Hill, whose work has had a major influence on mine, not only in nanothermodynamics, but also in nonequilibrium steady-state cycle kinetics and theory of muscle contraction [26–28].

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  1. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195-2420, USA

    Hong Qian

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  1. Hong Qian
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Correspondence to Hong Qian.

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Qian, H. Hill’s small systems nanothermodynamics: a simple macromolecular partition problem with a statistical perspective. J Biol Phys 38, 201–207 (2012). https://doi.org/10.1007/s10867-011-9254-4

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