^{1,2}, Denitsa Alamanova

^{3}, Volkhard Helms

^{3}and Ulrich S. Schwarz

^{1,2,a)}

### Abstract

We study the formation of protein-protein encounter complexes with a Langevin equation approach that considers direct, steric, and thermal forces. As three model systems with distinctly different properties we consider the pairs barnase:barstar, cytochrome -cytochrome c peroxidase, and p53:MDM2. In each case, proteins are modeled either as spherical particles, as dipolar spheres, or as collection of several small beads with one dipole. Spherical reaction patches are placed on the model proteins according to the known experimental structures of the protein complexes. In the computer simulations, concentration is varied by changing box size. Encounter is defined as overlap of the reaction patches and the corresponding first passage times are recorded together with the number of unsuccessful contacts before encounter. We find that encounter frequency scales linearly with protein concentration, thus proving that our microscopic model results in a well-defined macroscopic encounter rate. The number of unsuccessful contacts before encounter decreases with increasing encounter rate and ranges from 20 to 9000. For all three models, encounter rates are obtained within one order of magnitude of the experimentally measured association rates. Electrostatic steering enhances association up to 50-fold. If diffusional encounter is dominant (p53:MDM2) or similarly important as electrostatic steering (barnase:barstar), then encounter rate decreases with decreasing patch radius. More detailed modeling of protein shapes decreases encounter rates by 5%–95%. Our study shows how generic principles of protein-protein association are modulated by molecular features of the systems under consideration. Moreover it allows us to assess different coarse-graining strategies for the future modeling of the dynamics of large protein complexes.

We thank Christian Korn for many helpful discussions. This work was supported by the Volkswagen Foundation through Grant Nos. I/80469 and I/80470 to V.H. and U.S.S., respectively. J.S. and U.S.S. are supported by the Center for Modeling and Simulation in the Biosciences (BIOMS) at Heidelberg and by the Karlsruhe Institute of Technology (KIT) through its Concept for the Future.

I. INTRODUCTION

II. MODELS AND METHODS

A. Modeling proteins at different levels of detail

B. Diffusion properties

C. Langevin equation and simulation method

D. Anisotropic versus isotropic diffusion

E. System size and time step adaption

F. Electrostaticinteractions

G. Parametrization

III. RESULTS

A. Encounter frequency and encounter rate

B. Finite size effects

C. Alignment during encounter

D. Three bimolecular systems with different physicochemical interface properties

E. Size of the reaction patches

IV. DISCUSSION

### Key Topics

- Proteins
- 74.0
- Diffusion
- 38.0
- Electrostatics
- 27.0
- Langevin equation
- 13.0
- Biochemical reactions
- 12.0

## Figures

Scheme to visualize the different variants of the model for the three considered model systems. The color code is yellow for barnase, cytochrome c, and p53; green for barstar, cytochrome c peroxidase, and MDM2. The respective reaction patches are shown in white. only includes a simple steric interaction. has an additional effective electrostatic interaction, here denoted with red arrows showing the direction of the dipole of the model particles. In , the excluded volume is modeled in more detail as a collection of smaller beads. The transparent blue spherical surface marks the volume used in and for the sake of comparison. Finally, the bottom row shows surface representations of the atomistic structures taken from the protein database atomistic structures taken from the protein data bank (PDB).

Scheme to visualize the different variants of the model for the three considered model systems. The color code is yellow for barnase, cytochrome c, and p53; green for barstar, cytochrome c peroxidase, and MDM2. The respective reaction patches are shown in white. only includes a simple steric interaction. has an additional effective electrostatic interaction, here denoted with red arrows showing the direction of the dipole of the model particles. In , the excluded volume is modeled in more detail as a collection of smaller beads. The transparent blue spherical surface marks the volume used in and for the sake of comparison. Finally, the bottom row shows surface representations of the atomistic structures taken from the protein database atomistic structures taken from the protein data bank (PDB).

Logarithmic plot of the distribution of the FPT to encounter between a single pair of barnase and barstar model particles in a cubic boundary box of edge length , representing a concentration of for each protein. The dashed line represents a single exponential fit to the data points, which shows the expected behavior with respect to the encounter frequency .

Logarithmic plot of the distribution of the FPT to encounter between a single pair of barnase and barstar model particles in a cubic boundary box of edge length , representing a concentration of for each protein. The dashed line represents a single exponential fit to the data points, which shows the expected behavior with respect to the encounter frequency .

Simulated encounter frequencies for a single pair of barnase and barstar model particles in cubic boundary boxes of different sizes representing different concentrations. The dashed line is a linear fit to the data.

Simulated encounter frequencies for a single pair of barnase and barstar model particles in cubic boundary boxes of different sizes representing different concentrations. The dashed line is a linear fit to the data.

Encounter frequency for different numbers of barnase:barstar pairs leaving the size of the boundary box constant. The data points show the encounter frequencies as obtained from simulations, while the dashed line represents the function , where is the number of particle pairs and is a fitted prefactor.

Encounter frequency for different numbers of barnase:barstar pairs leaving the size of the boundary box constant. The data points show the encounter frequencies as obtained from simulations, while the dashed line represents the function , where is the number of particle pairs and is a fitted prefactor.

Different alignment states during the encounter process. proteins are completely unaligned. In state , referred to as *contact* in this paper, the proteins are translationally aligned, i.e., they are close enough to actually encounter (denoted by the overlap of the lightened area around the model particles), but lack the correct orientation. proteins reached the encounter meaning that the reactive patches are in translational and rotational alignments.

Different alignment states during the encounter process. proteins are completely unaligned. In state , referred to as *contact* in this paper, the proteins are translationally aligned, i.e., they are close enough to actually encounter (denoted by the overlap of the lightened area around the model particles), but lack the correct orientation. proteins reached the encounter meaning that the reactive patches are in translational and rotational alignments.

Logarithmically plotted distribution of the number of approaches between a barnase and a barstar particle with incorrect rotational alignment. The dashed line is an exponential fit to the data.

Logarithmically plotted distribution of the number of approaches between a barnase and a barstar particle with incorrect rotational alignment. The dashed line is an exponential fit to the data.

Double-logarithmic plot of the distribution of resting and return times of the translationally aligned state ( in Fig. 5).

Double-logarithmic plot of the distribution of resting and return times of the translationally aligned state ( in Fig. 5).

Correlation plot of encounter rate and mean number of contacts with all the data from Table II.

Correlation plot of encounter rate and mean number of contacts with all the data from Table II.

Encounter rates in dependency of the patch size for the barnase:barstar model system in the variant.

Encounter rates in dependency of the patch size for the barnase:barstar model system in the variant.

Comparison of and similar to Fig. 9 for small patch sizes. For larger patch sizes there is no substantial difference.

Comparison of and similar to Fig. 9 for small patch sizes. For larger patch sizes there is no substantial difference.

## Tables

Protein structures and parameters used in the study. The coordinates of the patches and the dipole moment are given relative to the center of mass.

Protein structures and parameters used in the study. The coordinates of the patches and the dipole moment are given relative to the center of mass.

Encounter rates which have been averaged over several simulations at different concentrations as given in the text. The values are given in for the three different versions of our model. are average values for the number of unsuccessful contacts before encounter. is basically independent of the concentration. Therefore it is again averaged over the different simulations for each of the chosen systems. The errors were determined by one standard deviation from the eight values obtained at different concentrations. Some of the choices for the patch radius were not applicable to , as for these cases an encounter was completely prevented by the detailed excluded volume model.

Encounter rates which have been averaged over several simulations at different concentrations as given in the text. The values are given in for the three different versions of our model. are average values for the number of unsuccessful contacts before encounter. is basically independent of the concentration. Therefore it is again averaged over the different simulations for each of the chosen systems. The errors were determined by one standard deviation from the eight values obtained at different concentrations. Some of the choices for the patch radius were not applicable to , as for these cases an encounter was completely prevented by the detailed excluded volume model.

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