^{1}and Harel Weinstein

^{2,a)}

### Abstract

One of the major factors distinguishing molecular processes *in vivo* from biochemical experiments *in vitro* is the effect of the environment produced by macromolecular crowding in the cell. To achieve a realistic modeling of processes in the living cell based on biochemical data, it becomes necessary, therefore, to consider such effects. We describe a protocol based on Brownian dynamics simulation to characterize and quantify the effect of various forms of crowding on diffusion and bimolecular association in a simple model of interacting hard spheres. We show that by combining the elastic collision method for hard spheres and the mean field approach for hydrodynamic interaction (HI), our simulations capture the correct dynamics of a monodisperse system. The contributions from excluded volume effect and HI to the crowding effect are thus quantified. The dependence of the results on size distribution of each component in the system is illustrated, and the approach is applied as well to the crowding effect on electrostatic-driven association in both neutral and charged environments; values for effective diffusion constants and association rates are obtained for the specific conditions. The results from our simulation approach can be used to improve the modeling of cell signaling processes without additional computational burdens.

The authors are grateful to Dr. Mihaly Mezei and Ernest L. Mehler for useful discussions. The work was supported by research grants from the National Institutes of Health.

I. INTRODUCTION

II. MODELS AND METHODS

A. The potential

B. Brownian dynamics simulation and diffusion constant calculation

C. Hydrodynamic interaction

D. Reaction rate calculation and calibration

III. RESULTS

A. Time-step-independent dynamics

B. Crowding reduces diffusion

C. Crowding slows down association

D. Larger crowding molecules produce lesser crowding effects

E. Importance of hydrodynamic interactions

F. Electrostatic interaction dominate in a charged environment

IV. DISCUSSION

### Key Topics

- Diffusion
- 63.0
- Proteins
- 17.0
- Biochemical reactions
- 9.0
- Colloidal systems
- 9.0
- Hydrodynamics
- 9.0

## Figures

Effect of time-step length on the calculation of the diffusion constant. Time is expressed as , where , and is plotted vs (in unit of ) for three different values of the volume fractions of the crowding molecules (0.10, 0.20, and 0.30). The is varied by keeping the number of hard sphere with radius of at 1001 and changing the edge length of the simulation box from 69 to 55 and , respectively. The dashed lines show results for a time step of 0.001, and the symbols are for 0.01.

Effect of time-step length on the calculation of the diffusion constant. Time is expressed as , where , and is plotted vs (in unit of ) for three different values of the volume fractions of the crowding molecules (0.10, 0.20, and 0.30). The is varied by keeping the number of hard sphere with radius of at 1001 and changing the edge length of the simulation box from 69 to 55 and , respectively. The dashed lines show results for a time step of 0.001, and the symbols are for 0.01.

Volume fraction dependence of the diffusion constant and association rate for a monodisperse hard sphere system. The number of hard spheres is 1001 and the simulation box size varies to achieve the required volume fraction. (a) was calculated according to Eq. (5) and from Eq. (8). The results for are shown by the symbols representing different methods: squares for our BD simulation, triangles for Cichocki and Hinsen (Ref. 34), and stars for Tokuyama (Ref. 49). The two lines represent corresponding theoretical predictions: continuous line from Tokuyama and Oppenheim (Ref. 42) and dashed line from Medina-Noyola (Ref. 48). (b) The mean value of at each volume fraction was obtained for all collision times, while the standard deviation was obtained from the distribution of the calculated rate based on 1000 reaction events. The circles are for , the squares for the corresponding , and the long dashed line is the fitted dependence of .

Volume fraction dependence of the diffusion constant and association rate for a monodisperse hard sphere system. The number of hard spheres is 1001 and the simulation box size varies to achieve the required volume fraction. (a) was calculated according to Eq. (5) and from Eq. (8). The results for are shown by the symbols representing different methods: squares for our BD simulation, triangles for Cichocki and Hinsen (Ref. 34), and stars for Tokuyama (Ref. 49). The two lines represent corresponding theoretical predictions: continuous line from Tokuyama and Oppenheim (Ref. 42) and dashed line from Medina-Noyola (Ref. 48). (b) The mean value of at each volume fraction was obtained for all collision times, while the standard deviation was obtained from the distribution of the calculated rate based on 1000 reaction events. The circles are for , the squares for the corresponding , and the long dashed line is the fitted dependence of .

Survival probability for reacting molecules obtained from the BD simulations. was calculated according to Eq. (7) and plotted vs simulation time (in milliseconds). For both and , the edge length of the simulation box was . For free association there are no crowding molecules, while for there are 999 spheres of the same size as the reacting particles.

Survival probability for reacting molecules obtained from the BD simulations. was calculated according to Eq. (7) and plotted vs simulation time (in milliseconds). For both and , the edge length of the simulation box was . For free association there are no crowding molecules, while for there are 999 spheres of the same size as the reacting particles.

Effect of simulation box size on the calculated diffusion constant and association rate in the monodisperse hard sphere suspension. All spheres have a radius of . The open symbols indicate values for and closed symbols for . Squares, ; triangles, ; diamonds, .

Effect of simulation box size on the calculated diffusion constant and association rate in the monodisperse hard sphere suspension. All spheres have a radius of . The open symbols indicate values for and closed symbols for . Squares, ; triangles, ; diamonds, .

Effect of crowding molecule size on the diffusion constant and association rate of the interacting spheres. The open symbols indicate values for and closed symbols for . Squares, crowding molecule (including data from , 501, and 251); diamonds, (, 500, and 250); triangles, (, 500, and 250). All lines are fitted to the data to show the dependence for (solid) and (dashed).

Effect of crowding molecule size on the diffusion constant and association rate of the interacting spheres. The open symbols indicate values for and closed symbols for . Squares, crowding molecule (including data from , 501, and 251); diamonds, (, 500, and 250); triangles, (, 500, and 250). All lines are fitted to the data to show the dependence for (solid) and (dashed).

Effect of hydrodynamic interactions. (a) The diffusion constant calculated from BD simulation with HI correction for monodisperse system with sphere radius of is shown in comparison with results from other methods: Solid line, Tokuyama and Oppenheim (Ref. 42); dashed line, Medina-Noyola (Ref. 48); squares, our BD simulation with HI correction. (b) Dependence on volume fraction of the effective diffusion constant and association rate calculated with the HI contribution. Diamonds, crowding sphere ; squares, ; triangles, . The open symbols represent the diffusion constant values and closed symbols show the association rate values. Lines are fitted to represent the dependence of (dashed) and (solid).

Effect of hydrodynamic interactions. (a) The diffusion constant calculated from BD simulation with HI correction for monodisperse system with sphere radius of is shown in comparison with results from other methods: Solid line, Tokuyama and Oppenheim (Ref. 42); dashed line, Medina-Noyola (Ref. 48); squares, our BD simulation with HI correction. (b) Dependence on volume fraction of the effective diffusion constant and association rate calculated with the HI contribution. Diamonds, crowding sphere ; squares, ; triangles, . The open symbols represent the diffusion constant values and closed symbols show the association rate values. Lines are fitted to represent the dependence of (dashed) and (solid).

(Color online) Effect of electrostatic attraction between reacting molecules on the association rate with neutral and charged crowding molecules. Squares, ; diamonds, ; triangles, , all for the neutral crowding environment. Circles are for in the charged crowding environment (half with positive charge and the other half with negative charge); time step is . Open symbols are for calculations without the HI component and closed ones are for values calculated with HI.

(Color online) Effect of electrostatic attraction between reacting molecules on the association rate with neutral and charged crowding molecules. Squares, ; diamonds, ; triangles, , all for the neutral crowding environment. Circles are for in the charged crowding environment (half with positive charge and the other half with negative charge); time step is . Open symbols are for calculations without the HI component and closed ones are for values calculated with HI.

## Tables

Relative contribution to the crowding effect on diffusion and association in a monodisperse hard sphere system from HS (the excluded volume) and HI (the hydrodynamic interactions).

Relative contribution to the crowding effect on diffusion and association in a monodisperse hard sphere system from HS (the excluded volume) and HI (the hydrodynamic interactions).

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