How to Correct Systematic Errors in Constant-pH Ensemble Simulations

The constant-pH ensemble method is a popular algorithm to study acid-base equilibria in charge regulating soft matter systems including weak polyelectrolytes and proteins. However, the method suffers from systematic errors in simulations with explicit ions, which lead to a symmetry-breaking between chemically equivalent implementations of the acid-base equilibrium. Here, we show that this artifact of the algorithm can be corrected a-posteriori by simply shifting the pH-scale. We present two analytical methods as well as a numerical method using Widom insertion to obtain the correction. By numerically investigating various sample systems, we assess the range of validity of the analytical approaches and show that the Widom approach always leads to consistent results, even when the analytical approaches fail. Overall, we provide practical guidelines on how to use constant-pH simulations to avoid systematic errors, including cases where special care is required, such as polyampholytes and proteins.