## Purpose

GMCT and GCEM form a pair of programs for studying the binding thermodynamics of macromolecular receptors. GCEM is a program for the automated preparation of the necessary input for GMCT from a continuum electrostatics / molecular mechanics model. The software allows a detailed modeling of complex ligand-receptor systems in structure based calculations. The primary targets of the software are biomolecular receptors like proteins that bind or transfer protons, electrons or other small-molecule ligands. The software might, however, also be useful for studying polyelectrolytes in a broader sense or other systems like Ising or Potts models. The properties of large systems can be studied using a variety of modern simulation methods. This makes the software ideally suited to study the thermodynamics of ligand binding and charge transfer processes in bioenergetic complexes and other complex biomolecular systems.

## The receptor model

The image on the right shows a schematic view of the receptor model of GMCT with all possible features at the example of a membrane-intrinsic receptor. The image gives explanations on mouse-over if your browser supports SVG.

• The receptor with the bound ligands is described in explicit detail while the receptor environment with the unbound ligands is modeled implicitly.
• Within the continuum electrostatics model of GCEM, the receptor and its environment can be further subdivided into regions of differing dielectric constants and accessibility to mobile ions. Details about the continuum electrostatics model are given → here.
• There can be multiple types of ligands, where the unbound ligands are represented implicitly by their electrochemical potentials.
• The receptor can possess multiple global conformations.
• The receptor can possess multiple sites.
• Each site can possess multiple instances, where an instance is a certain form of a site which is characterized by binding topology, charge distribution, conformation and the number of bound ligands of each type. With this description of the sites one can represent, e.g., multiple redox and/or protonation forms, tautomeric forms, sidechain rotamers and spin states.
• For a realistic modeling of trans-membrane proteins, it is possible to account for an electrochemical trans-membrane potential that consist of a chemical and an electrical component. The chemical component arises from concentration differences of compounds between the two phases, while the electrical component arises either from charge separation across the membrane or from an externally applied electrostatic potential, as for example in some electrophysiological experiments.

## Theoretical basis

Our software rests on a general formulation of binding theory in terms of electrochemical potentials, instead of not directly comparable quantities like pH value and reduction potential. This formulation increases the transparency of calculation results by making them particularly easy to interpret. A detailed derivation of the formalism can be found in → this thesis. The essentials of the formalism can be displayed below.

An analytical calculation of thermodynamic properties and observables is in practice impracticable already for systems of moderate size, because the number of possible microstates grows exponentially with the number of sites. Simulation methods make it possible to accomplish such calculations even for large systems, like the protein complexes occurring in bioenergetics, in acceptable time. Monte Carlo (MC) simulation methods are often particularly efficient.

## Simulation methods

Currently, GMCT offers two basic MC methods: Metropolis MC and Wang-Landau MC. Unique features of GMCT are accurate and efficient free energy calculation methods that can be used to calculate free energy differences for freely definable transformations and for the calculation of free energy measures of cooperativity. Namely, the free energy perturbation method, thermodynamic integration, the non-equilibrium work method and the Bennett acceptance ratio method have been implemented. The coupling between events in molecular systems can be quantified with covariances or free energy measures of cooperativity.

## Approximate semi-analytic methods

Since version 1.1, GMCT supports also calculations with the hybrid method that combines exact statistical mechanics within clusters of strongly interacting residues with a mean-field approach for the description of inter-cluster interactions. Details can be found in the user manual.

## Applications

A variety of modern Monte Carlo simulation methods can be used to study overall properties of the receptor as well as properties of individual sites. The description of the system in terms of discrete microstates of the receptor and chemical potentials of the ligands renders the simulations computationally very inexpensive relative to all-atom simulations. This computational efficiency enables very accurate calculations of receptor properties with low statistical uncertainty.

Properties of binding processes that can be calculated are for example binding probabilities (titration curves), binding free energies and binding constants. These properties can be computed from a microscopic viewpoint for studying the behavior of separate sites or groups of sites in the receptor or from a macroscopic viewpoint for studying the overall behavior of the receptor. Midpoint reduction potentials $$\mathcal{E}_{1/2}$$ and $$\mathrm{p}K_{1/2}$$ values can be derived from computed titration curves. Binding free energies can be expressed in terms of thermodynamically defined reduction potentials and $$\mathrm{p}K_{\mathrm{a}}$$ values. The free energy calculation methods can also be used to study charge transfer reactions, conformational transitions and any other process that can be described within the receptor model of GMCT .

A particularly interesting feature of GMCT is the possibility to calculate free energy measures of cooperativity that can be used to study the coupling of different processes in the receptor. An example of special interest in our lab is the coupling between binding and transfer processes of charged ligands in bioenergetic protein complexes.

GMCT can also be helpful in setting up and complementing molecular dynamics (MD) simulations. The preparation of a protein structure for MD simulations does often require the specification of protonation states and tautomeric states occupied by titratable residues. This information can be obtained from Metropolis MC calculations with GMCT . In addition, protonation state calculations can be used to assess whether the modeling of a protein or other polyelectrolyte could require a constant-pH MD method.

## Documentation

GMCT and the continuum electrostatics / molecular mechanics model of GCEM are described in →this paper. See the subdirectory doc of the GMCT distribution for detailed documentation, including the theoretical basis of both programs (basically comprises the above mentioned paper with more detail at some points plus a detailed description of all programs). Information about the extended MEAD library utilized by GCEM and usage of the program can be found →here. Examples for the usage of GMCT can be found in the directory examples of the GMCT distribution. Examples for the usage of GCEM can be found in the directory examples/gcem of the MEAD distribution.

GMCT version 1.2.3 offers the user more influence on the convergence control in Wang-Landau Monte Carlo simulations.
The MEAD distribution was updated to version 2.3.1. The main changes in this version are a bugfix for the application GCEM and the additional application "get-cavities" for assigning solvent accessible volume within the transmembrane region of a molecule. The tool facilitates the setup of calculations on membrane inserted systems with GCEM. Details can be found on the web page of the distribution.

The software is free software in the sense of the definition given by the Free Software Foundation.

GMCT and GCEM are distributed under the terms of the GNU Affero General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. For more details, see the documentation of GMCT found in the directory doc of the distribution.

## Feedback and bug reports

Your opinion and hints are very welcome. Please provide detailed information about the program input and output when reporting a bug. Often, running a program with a higher verbosity level (set the parameter blab to 3) helps to clarify the source of an error.

## Related software

There is a variety of software that utilizes a continuum electrostatics approach for the description of binding equilibria. In principle, program input for GMCT can also be generated with alternative continuum electrostatics software some of which is included in the list below (some reformatting and unit conversion could be necessary). Likewise, the energy terms computed by GCEM should also be usable with other simulation software, as far as the features are supported. The following list contains the corresponding software known to us, grouped according to the authoring research groups. For details about the software, please refer to the respective linked websites.
• Donald Bashford wrote the original MEAD software package. Besides GCEM, MEAD contains an older program called Multiflex that can also be used for the calculation of continuum electrostatics energies, but does not provide all of GCEM's , features. The program Redti can be used for the analytic computation of titration curves using an approximate approach.
• XMCTI is a program from Paul Beroza that can be used for Metropolis MC simulations of protonation equilibria.
• HYBRID is a program from Michael Gilson that can be used for the computation of titration curves with an approximate method.
• The group of Ernst-Walther Knapp provides a pair of software for continuum electrostatics calculations on protonation and reduction equilibria. Karlsberg allows Metropolis MC titrations as function of pH value and reduction potential. Replica exchange parallel tempering and biased MC are also possible. TAPBS allows the calculation of the electrostatic energies used by Karlsberg by interfacing to the software APBS.
• Jens Erik Nielsen is the author of various related software. The WhatIf pKa package allows protonation state calculations within a continuum-electrostatics model, where the continuum electrostatics part utilizes the Delphi software. A newer variant of the package named pdb2pka utilizes APBS for the continuum electrostatics part.
• MCCE is a program for continuum electrostatics calculations on protonation equilibria from the group of Marilyn Gunner. The program allows Metropolis Monte Carlo titrations of protonation and reduction equilibria. MCCE uses the programs DELPHI or APBS for the continuum electrostatics calculations.
• The group of Antonio Baptista provides several programs for continuum electrostatics calculations on protonation and reduction equilibria. The programs Petite and MCRP perform Metropolis MC simulations of protonation and reduction equilibria. Correlations between pairs of sites can also be computed. The electrostatics part of their software utilizes the MEAD software package.
• The group of Alexei Stuchebrukhov is also developing a pair of software for continuum electrostatics calculations on protonation and reduction equilibria. Monte allows Metropolis MC titrations and replica exchange parallel tempering. pKip allows the calculation of the electrostatic energies used by Monte by interfacing to the software APBS. Currently, the programs do not seem to be publicly available.
• DOPS is a program for continuum electrostatics calculations on protonation equilibria from the Antosiewiczs group. The name appeared in an older publication, but the program seems not to be publicly available.
• Zap TK is a commercial programming library which can be used for continuum electrostatics calculations on protonation equilibria and also seems to contain functionality for Metropolis MC titrations. The library can be licensed from OpenEye Scientific Software.