Thermodynamic integration (TI), a powerful formalism for computing Gibbs free energy, has been implemented for many biophysical processes with alchemical schemes that require delicate human efforts to choose/design biasing potentials for sampling the desired biophysical events and to remove their artifactitious consequences afterwards. Theoretically, an alchemical scheme is exact but practically, an unsophisticated implementation of this exact formula can cause error amplifications. Small relative errors in the input parameters can be amplified many times in their propagation into the computed free energy [due to subtraction of similar numbers such as (105±5)−(100±5) = 5±7]. In this paper, we present an unsophisticated implementation of TI in 3n dimensions (3nD) (n = 1, 2, 3…) for the potential of mean force along a 3nD path connecting one state in the bound state ensemble to one state in the unbound state ensemble. Fluctuations in these 3nD are integrated in the bound and unbound state ensembles but not along the 3nD path. Using TI3nD, we computed the standard binding free energies of three protein complexes: trometamol in Salmonella effector SpvD (n = 1), biotin-avidin (n = 2), and Colicin E9 endonuclease with cognate immunity protein Im9 (n = 3). We employed three different protocols in three independent computations of E9-Im9 to show TI3nD's robustness. We also computed the hydration energies of 10 biologically relevant compounds (n = 1 for water, acetamide, urea, glycerol, trometamol, ammonium and n = 2 for erythritol, 1,3-propanediol, xylitol, biotin). Each of the 15 computations is accomplishable within 1 (for hydration) to 10 (for E9-Im9) days on an inexpensive GPU workstation. The computed results all agree with the available experimental data.

热力学集成（TI）是一种用于计算吉布斯自由能的强大形式主义，已通过炼金术方案应用于许多生物物理过程，这些方案需要人类做出精巧的努力来选择/设计用于对所需生物物理事件进行采样并消除其后人为后果的偏置电位。从理论上讲，炼金术方案是精确的，但实际上，对该精确公式的简单实施会导致误差放大。输入参数中的较小相对误差在传播到计算出的自由能时会被放大很多倍[由于相减了相似的数字，例如（105±5）-（100±5）= 5±7]。在本文中，我们介绍了TI在3n维（3nD）中的简单实现（ñ= 1、2、3…）表示沿着3nD路径的平均力的潜力，该路径将绑定状态集合中的一个状态连接到未绑定状态集合中的一个状态。这些3nD中的波动被集成在绑定和未绑定状态集合中，但不沿着3nD路径。使用TI3nD，我们计算了三种蛋白质复合物的标准结合自由能：沙门氏菌效应物SpvD中的曲美他莫（ñ = 1），生物素-亲和素（ñ = 2），以及带有关联免疫蛋白Im9的Colicin E9核酸内切酶（ñ= 3）。我们在E9-Im9的三个独立计算中采用了三种不同的协议，以显示TI3nD的鲁棒性。我们还计算了10种生物学上相关的化合物的水合能（ñ 对于水，乙酰胺，尿素，甘油，丁三醇，铵和 ñ对于赤藓糖醇，1,3-丙二醇，木糖醇，生物素= 2。在便宜的GPU工作站上，这15个计算中的每一个都可以在1（对于水合）到10（对于E9-Im9）天之内完成。计算结果均与可用的实验数据一致。