Application of Jarzynski’s equality in simple versus complex systems Turgut Bas ßtug ˘, Serdar Kuyucak * School of Physics, University of Sydney, NSW 2006, Australia Received 8 January 2007; in final form 24 January 2007 Available online 2 February 2007 Abstract Jarzynski’s equality provides a non-equilibrium method for calculation of free energies, which could have wide-ranging applications in complex molecules. Tests of Jarzynski’s equality have mostly focused on simple systems or near-equilibrium processes. Here we present a benchmark test that compares the results of free energy calculations for a potassium ion in carbon nanotube and gramicidin channel, which can be identified with simple and complex systems. A good agreement is found between the free energy results obtained using Jarzynski’s equality and established methods in the carbon nanotube, but discrepancies occur in gramicidin, indicating relaxation prob- lems in application of Jarzynski’s equality to complex biomolecular processes. Ó 2007 Published by Elsevier B.V. Free energy is a fundamental descriptor of physical sys- tems, and calculation of free energy differences is essential for understanding molecular processes in physics, chemis- try and biology. The traditional equilibrium methods used for this purpose, e.g. free energy perturbation and thermo- dynamic integration [1,2], have recently been extended to a non-equilibrium method based on Jarzynski’s equality [3]. This so-called fast growth method [4] has been tested in some simple systems [5–7,9,8,10,11], which indicated gener- ally positive results with regard to its accuracy but no gains in efficiency have been found compared to the traditional methods. Potentially a more rewarding application of Jarzynski’s equality is the calculation of free energy profile of a ligand along a reaction coordinate using steered molecular dynamics simulations [12,13]. Here advantages of the new method in calculation of the potential of mean force (PMF) compared to the alternative—umbrella sampling with weighted histogram analysis method [2]—are much more obvious. Instead of equilibrating each umbrella win- dow, of which there could be many, one needs to equili- brate only the initial state, and the computations are trivially parallelized. Thus even if there are no gains in computational efficiency, its simplicity alone would be suf- ficient grounds for using Jarzynski’s equality in PMF cal- culations. Indeed there have been numerous applications of Jarzynski’s equality to calculate the PMF of ligands in various ligand-protein systems [14–22]. Tests of Jarzynski’s equality have so far been carried out in simple systems or for near-equilibrium processes. Despite some doubts having been raised on formal aspects of Jar- zynski’s equality [23] and its applicability to biomolecular processes in practice [24], no comparable tests have been done in complex systems—although such applications have attracted the most attention. Problems with convergence of the PMF results in biological systems have been stressed in recent review articles [25,26]. Here we address this issue using ion permeation in channels as a testing ground. What makes this proposition particularly interesting is the avail- ability of two channels with similar gross structure and function, namely, a (6, 6) armchair carbon nanotube [27] and gramicidin A [28], which can be identified with simple and complex systems, respectively. As seen in Fig. 1, both systems form a narrow cylindrical hole across which an ion and water molecules can pass in a single-file configura- tion. The main difference between the two systems is in their interactions with the ion-water complex: the carbon atoms in the nanotube have only short range Lennard–Jones 0009-2614/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.cplett.2007.01.078 * Corresponding author. Fax: +61 2 9351 7726. E-mail address: serdar@physics.usyd.edu.au (S. Kuyucak). www.elsevier.com/locate/cplett Chemical Physics Letters 436 (2007) 383–387