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Free-energy analysis of the molecular binding into lipid membrane with the method of energy representation
1.The Structure of Biological Membranes, 2nd ed., edited by P. L. Yeagle (CRC, Boca Raton, FL, 2005).
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50.The drawback of the approach is that the solvation free energy is evaluated from an approximate functional.
51.A. Ben-Naim, Solvation Thermodynamics (Plenum, New York, 1987).
52.The one-to-one association is implied in Eq. (1). This is because both the membrane and solute are assumed to be dilute in the present formulation. When more than one solute molecule is involved in the membrane solution, it suffices to extend the development straightforwardly by viewing one of the solute molecules as the “solute” species and the others as the “solvent” species constituting the “mixed solvent” system.
53.The extension of the formulation is straightforward when the system contains cosolvents.
54.The condition of a fixed (total) chemical potential of the solute is realized, for example, when the counterpart of the equilibrium of the solute insertion is common to the membrane solution and neat solvent.
55.When is much larger than , is simply close to and is determined essentially by the solubility in the neat solvent. Indeed, when the affinity of the solute to the membrane is strong enough, the mole fraction in the membrane is at the order of 1 and the order of is given by the solubility in neat water for a sparingly soluble solute. In this case, (or ) is not very informative for the solute-membrane correlation.
58.In Eq. (7), one membrane aggregate is treated as a single particle. The configuration fluctuates within the aggregate, and the coordinates for the intra-aggregate motion are incorporated in the definition of .
59.Actually, an arbitrary position in the solute molecule can be taken as the molecular center. The value calculated from Eq. (8) or (9) is independent of the choice of the molecular center of the solute.
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65.W. Shinoda and M. L. Klein, manuscript in preparation. The equilibration scheme adopted is summarized as follows. First, 128 DMPC molecules were placed in lattice form with the conformation at the crystal phase. Then 3405 water molecules were added, and the system was equilibrated through an 80 ns molecular dynamics simulation in the ensemble with the membrane area corresponding to the experimental value (Ref.64). Finally, the number of water molecules was increased to 3886 and the equilibration was further performed for 6 ns.
68.W. Rawicz, K. C. Olbrich, T. McIntosh, D. Needham, and E. Evans, Biophys. J. 79, 328 (2000).
69.The area compressibility modulus is given by , where is the average of the total area of the membrane and is its mean square deviation (Ref. 44). When the experimental value of 234 dyne/cm is used for the modulus (Ref. 68) and the membrane area is considered to be per lipid, is with 128 DMPC molecules and is larger than the area expected to be occupied by the solute treated in the present work. According to the membrane simulations involving alcoholic or anesthetic solute (Refs. 21, 43, 47, and 48), the membrane area changes by a few percent when the mole ratio of the solute to the phospholipid is . In our simulation, the mole ratio is 1/128 (a single solute molecule in 128 DMPC molecules). The area expansion or contraction is thus negligible upon introduction of CO, , benzene, or ethylbenzene as the solute; the area can be safely taken to be identical between the solution and reference solvent systems.
74.For all the solutes and the regions of solute position, and can be constructed from a single trajectory of the membrane system without the solute since the solute insertion as a test particle does not affect the trajectory. This is useful to reduce the computational demand when a multiple set of solvations is to be treated against a common (mixed) solvent system. A similar remark applies to the calculation in neat water.
76.The free energies presented in Ref. 20 are different from those in this work. According to the notation in Eq. (2) of Ref. 20, Jedlovszky and Mezei showed in their paper while we show .
77.The cavity-insertion Widom method is highly efficient when it works. When the solute size is large, however, an appropriate cavity is not found in the molecular simulation system and the method does not work. For the DMPC-water system, benzene and ethylbenzene are too large and are beyond the applicability of the cavity-insertion Widom method. In the approximate procedure described in Secs. II and III, the overlapping configuration produced upon the solute insertion provides nonzero and at large energy coordinates and contributes to the solvation free energy as the excluded-volume effect.
78.When the solvation free energy is given by as a continuous function of and the region of interest is identified as , the value [as defined by Eq. (12)] in the region is expressed as . Note that is not a simple average (weighted sum) of . According to this equation, the -portion with smaller is more weighted in the integral to determine . For example, when increases linearly by in an interval of , is the minimum in the interval and the difference between and is ; the differences are and , respectively, with and 10. In previous works (Refs. 24 and 26), it is seen that is more favorable (more negative) in the inner portion (smaller ) within the interfacial region. Thus, the value in region IV is closer to at than at .
79.The contributions to from regions V–VI are , , , and in weight for CO, , benzene, and ethylbenzene, respectively.
80.The discussion concerning is based on the approximation expressed as Eqs. (19)–(28). The arguments concerning are exact within the statistical error, on the other hand, for the potential functions employed.
82.As shown in Refs. 30 and 81, the density dependence is steep for the contribution from the repulsive component of the solute-solvent interaction to the solvation free energy, while it is milder for the contribution from the attractive component. Correspondingly, when the water density decreases in the interfacial region of the membrane, the reduction of the repulsive effect is stronger and the solvation free energy can be negative.
83.The “excluded-volume” region can be identified over the energy coordinate as , where is rather arbitrary and is typically . When the integral over in Eqs. (24) and (25) is restricted to the domain corresponding to the excluded-volume region, the integral value is larger in the order of bulk water, region IV, and region II.
85.The change in the solvent-solvent interaction energy contributes to the thermodynamic enthalpy and energy. The excess partial molar enthalpy is the sum of the change in the intramolecular energy, , the solvent reorganization term, and the pressure-volume term (Ref. 84).
86.Handbook of Surface and Colloid Chemistry, 2nd ed., edited by K. S. Birdi (CRC, Boca Raton, FL, 2003).
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