We present a theoretical method to study how changes in pH shape the heterogeneous conformational ensemble explored by intrinsically disordered proteins (IDPs). The theory is developed in the context of coarse-grained models, which enable a fast, accurate, and extensive exploration of conformational space at a given protonation state. In order to account for pH effects, we generalize the Molecular Transfer Model (MTM), in which conformations are re-weighted using the transfer free energy, which is the free energy necessary for bringing to equilibrium in a new environment a ``frozen'' conformation of the system. Using the semi-grand ensemble, we derive an exact expression of the transfer free energy, which amounts to the appropriate summation over all the protonation states. Because the exact result is computationally too demanding to be useful for large polyelectrolytes or IDPs, we introduce a mean-field (MF) approximation of the transfer free energy. Using a lattice model, we compare the exact and MF results for the transfer free energy and a variety of observables associated with the model IDP. We find that the precise location of the charged groups (the sequence), and not merely the net charge, determines the structural properties. We demonstrate that some of the limitations previously noted for MF theory in the context of globular proteins are mitigated when disordered polymers are studied. The excellent agreement between the exact and MF results poises us to use the method presented here as a computational tool to study the properties of IDPs and other biological systems as a function of pH.

中文翻译：

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pH对固有紊乱蛋白的影响的分子转移模型：理论与应用

我们提供了一种理论方法来研究pH的变化如何塑造由内在无序蛋白（IDP）探索的异质构象集合。该理论是在粗粒度模型的背景下开发的，它可以在给定的质子化状态下快速，准确和广泛地探索构象空间。为了解决pH值的影响，我们推广了分子转移模型（MTM），其中使用转移自由能对构象进行了加权，转移自由能是使新环境中的``冻结''平衡所需的自由能。的系统构型。使用半盛大的合奏，我们得出了转移自由能的精确表达式，相当于所有质子化状态的适当总和。由于精确的计算结果对大型聚电解质或IDP的计算要求太高，因此我们引入了转移自由能的均值（MF）近似值。使用晶格模型，我们比较了传递自由能和与模型IDP相关的各种可观测值的精确和MF结果。我们发现，带电基团（序列）的精确位置决定了结构性质，而不仅仅是净电荷。我们证明，当研究无序聚合物时，可以减轻先前在球形蛋白上下文中针对MF理论所指出的局限性。精确结果和MF结果之间的出色一致性使我们能够使用此处介绍的方法作为计算工具来研究IDP和其他生物系统的性质与pH的关系。