# Energy Minimization of Discrete Protein Titration State Models Using Graph Theory

## Abstract

There are several applications in computational biophysics which require the optimization of discrete interacting states; e.g., amino acid titration states, ligand oxidation states, or discrete rotamer angles. Such optimization can be very time-consuming as it scales exponentially in the number of sites to be optimized. In this paper, we describe a new polynomial-time algorithm for optimization of discrete states in macromolecular systems. This algorithm was adapted from image processing and uses techniques from discrete mathematics and graph theory to restate the optimization problem in terms of maximum flow-minimum cut graph analysis. The interaction energy graph, a graph in which vertices (amino acids) and edges (interactions) are weighted with their respective energies, is transformed into a flow network in which the value of the minimum cut in the network equals the minimum free energy of the protein, and the cut itself encodes the state that achieves the minimum free energy. Because of its deterministic nature and polynomial-time performance, this algorithm has the potential to allow for the ionization state of larger proteins to be discovered.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1331738

- Report Number(s):
- PNNL-SA-111332

Journal ID: ISSN 1520-6106; 453040220

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Physical Chemistry. B, Condensed Matter, Materials, Surfaces, Interfaces and Biophysical Chemistry; Journal Volume: 120; Journal Issue: 33

- Country of Publication:
- United States

- Language:
- English

- Subject:
- protein; graph; energy minimization

### Citation Formats

```
Purvine, Emilie AH, Monson, Kyle E., Jurrus, Elizabeth R., Star, Keith T., and Baker, Nathan A.
```*Energy Minimization of Discrete Protein Titration State Models Using Graph Theory*. United States: N. p., 2016.
Web. doi:10.1021/acs.jpcb.6b02059.

```
Purvine, Emilie AH, Monson, Kyle E., Jurrus, Elizabeth R., Star, Keith T., & Baker, Nathan A.
```*Energy Minimization of Discrete Protein Titration State Models Using Graph Theory*. United States. doi:10.1021/acs.jpcb.6b02059.

```
Purvine, Emilie AH, Monson, Kyle E., Jurrus, Elizabeth R., Star, Keith T., and Baker, Nathan A. Thu .
"Energy Minimization of Discrete Protein Titration State Models Using Graph Theory". United States.
doi:10.1021/acs.jpcb.6b02059.
```

```
@article{osti_1331738,
```

title = {Energy Minimization of Discrete Protein Titration State Models Using Graph Theory},

author = {Purvine, Emilie AH and Monson, Kyle E. and Jurrus, Elizabeth R. and Star, Keith T. and Baker, Nathan A.},

abstractNote = {There are several applications in computational biophysics which require the optimization of discrete interacting states; e.g., amino acid titration states, ligand oxidation states, or discrete rotamer angles. Such optimization can be very time-consuming as it scales exponentially in the number of sites to be optimized. In this paper, we describe a new polynomial-time algorithm for optimization of discrete states in macromolecular systems. This algorithm was adapted from image processing and uses techniques from discrete mathematics and graph theory to restate the optimization problem in terms of maximum flow-minimum cut graph analysis. The interaction energy graph, a graph in which vertices (amino acids) and edges (interactions) are weighted with their respective energies, is transformed into a flow network in which the value of the minimum cut in the network equals the minimum free energy of the protein, and the cut itself encodes the state that achieves the minimum free energy. Because of its deterministic nature and polynomial-time performance, this algorithm has the potential to allow for the ionization state of larger proteins to be discovered.},

doi = {10.1021/acs.jpcb.6b02059},

journal = {Journal of Physical Chemistry. B, Condensed Matter, Materials, Surfaces, Interfaces and Biophysical Chemistry},

number = 33,

volume = 120,

place = {United States},

year = {Thu Sep 01 00:00:00 EDT 2016},

month = {Thu Sep 01 00:00:00 EDT 2016}

}