^{1}, Holger Merlitz

^{2}, Chen-Xu Wu

^{2}and Jörg Langowski

^{3}

### Abstract

In this paper a lattice model for the diffusional transport of particles in the interphase cell nucleus is proposed. Dense networks of chromatin fibers are created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model, and a self-avoiding random walk chain model with persistence length. By comparing a discrete and a continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter particle diffusion. The influence of the three dimensional geometry of the fiber network on the particle diffusion is investigated in detail while varying the occupation volume, chain length, persistence length, and walker size. It is shown that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and, to a lesser extent, the persistence length affect particle diffusion. It is demonstrated how the introduction of the effective chain occupancy, which is a convolution of the geometric chain volume with the walker size, eliminates the conformational effects of the network on the diffusion, i.e., when plotting the diffusion coefficient as a function of the effective chain volume, the data fall onto a master curve.

One of the authors (A.W.) thanks two of the authors (H.M. and C.W.) for their hospitality during a research stay at the Department of Physics at the Xiamen University. One of the authors (A.W.) was supported by a scholarship from the International PhD program of the German Cancer Research Center.

I. INTRODUCTION

II. MODELING VOLUME

A. Particles

B. Diffusional transport simulation

C. Test system with disconnected obstacles

III. RANDOM WALK CHAIN SYSTEMS

A. Model and chain simulation

B. Results

C. Conclusions: RW chains

IV. SELF-AVOIDING RANDOM WALK CHAIN SYSTEM

A. Model

B. Chain simulation

1. Bond fluctuation method: Single site model

2. Metropolis Monte Carlo procedure

C. Results

1. Chain relaxation

2. Translational diffusion coefficient

3. Particle diffusion in various environments

V. SUMMARY

### Key Topics

- Diffusion
- 77.0
- Random walks
- 24.0
- Polymers
- 22.0
- Cell nucleus
- 16.0
- Surface acoustic waves
- 15.0

## Figures

Comparison of the continuous and discrete random walk chain models and the test system (disconnected obstacles). Upper panel: Diffusion coefficient dependent on the geometric volume fraction. Lower panel: Diffusion coefficient dependent on the effective volume fraction. Circles: smallest particle, squares: medium sized particle, triangles: largest particle, and diamonds: medium sized particle in the test system. Blank: Discrete random walk chain model. Solid: Continuous random walk chain model.

Comparison of the continuous and discrete random walk chain models and the test system (disconnected obstacles). Upper panel: Diffusion coefficient dependent on the geometric volume fraction. Lower panel: Diffusion coefficient dependent on the effective volume fraction. Circles: smallest particle, squares: medium sized particle, triangles: largest particle, and diamonds: medium sized particle in the test system. Blank: Discrete random walk chain model. Solid: Continuous random walk chain model.

Mean square displacement of the largest particle vs time. The continuous lines are linear fits through the first five points of the curve (blank squares and blank triangles) to illustrate the deviation from linearity for the long-time diffusion in the denser systems. The dashed lines are power law fits yielding the anomalous diffusion exponent . Blank circles: Random walk chain model (effective volume with 68% and geometric volume with 20%). Solid circles: Continuous random walk chain model (effective volume with 68% and geometric volume with 20%). Blank squares: Random walk chain model (effective volume with 73%, geometric volume with 23%, and ). Solid squares: Continuous random walk chain model (effective volume with 78%, geometric volume with 26%, and ). Triangles: Self-avoiding random walk chain model (effective volume with 81%, geometric volume with 12.5%, chain length , , and ).

Mean square displacement of the largest particle vs time. The continuous lines are linear fits through the first five points of the curve (blank squares and blank triangles) to illustrate the deviation from linearity for the long-time diffusion in the denser systems. The dashed lines are power law fits yielding the anomalous diffusion exponent . Blank circles: Random walk chain model (effective volume with 68% and geometric volume with 20%). Solid circles: Continuous random walk chain model (effective volume with 68% and geometric volume with 20%). Blank squares: Random walk chain model (effective volume with 73%, geometric volume with 23%, and ). Solid squares: Continuous random walk chain model (effective volume with 78%, geometric volume with 26%, and ). Triangles: Self-avoiding random walk chain model (effective volume with 81%, geometric volume with 12.5%, chain length , , and ).

Upper panel: Start conformation of the Monte Carlo algorithm. Forty six cubes, chains with , are homogeneously distributed on the lattice. Lower panel: Relaxed conformation after time steps of the Monte Carlo algorithm combined with the bond fluctuation method. The chains constitute a dense network, comparable with chromosome territories inside the cell nucleus. To highlight the topological structure of the chain network, the 46 chromosomes are alternately colored here with red and blue.

Upper panel: Start conformation of the Monte Carlo algorithm. Forty six cubes, chains with , are homogeneously distributed on the lattice. Lower panel: Relaxed conformation after time steps of the Monte Carlo algorithm combined with the bond fluctuation method. The chains constitute a dense network, comparable with chromosome territories inside the cell nucleus. To highlight the topological structure of the chain network, the 46 chromosomes are alternately colored here with red and blue.

Energy distribution as a function of time . Geometric occupation volume: 6.4%. Circles, ; squares, . Blank, ; solid, .

Energy distribution as a function of time . Geometric occupation volume: 6.4%. Circles, ; squares, . Blank, ; solid, .

Relaxation time vs chain length on a logarithmic scale. Geometric occupation volume: 6.4%. The solid line is a power law fit with an exponent of 2.5. Squares, ; circles, .

Relaxation time vs chain length on a logarithmic scale. Geometric occupation volume: 6.4%. The solid line is a power law fit with an exponent of 2.5. Squares, ; circles, .

Upper panel: Anomalous translational diffusion exponent as a function of chain length . Geometric occupation volume: 6.4%. Circles, ; squares, ; triangles, . Lower panel: Center of mass vs time . Squares, and ; circles, and . Solid, cubic lattice; blank, cubic lattice. Straight line, linear fit; dashed line, power law fit.

Upper panel: Anomalous translational diffusion exponent as a function of chain length . Geometric occupation volume: 6.4%. Circles, ; squares, ; triangles, . Lower panel: Center of mass vs time . Squares, and ; circles, and . Solid, cubic lattice; blank, cubic lattice. Straight line, linear fit; dashed line, power law fit.

Comparison of the random walk chain model and the self-avoiding random walk chain model. Upper panel: Diffusion coefficient dependent on the geometric volume fraction. Lower panel: Diffusion coefficient dependent on the effective volume fraction. Blank: Random walk chain model. Solid: Self-avoiding random walk chain model. Diamonds: middle sized particle in the test system. Circles: smallest particle, squares: middle sized particle, and triangles: largest particle. Stars: largest particle in the SAW system with different persistence lengths of 0.5, 1, 2, and 3 with a constant geometric occupation volume of 6.12%, respectively. Crosses: largest particle in the SAW system with different persistence lengths of 0.5, 1, 2, and 3 with a constant geometric occupation volume of 7.95%, respectively.

Comparison of the random walk chain model and the self-avoiding random walk chain model. Upper panel: Diffusion coefficient dependent on the geometric volume fraction. Lower panel: Diffusion coefficient dependent on the effective volume fraction. Blank: Random walk chain model. Solid: Self-avoiding random walk chain model. Diamonds: middle sized particle in the test system. Circles: smallest particle, squares: middle sized particle, and triangles: largest particle. Stars: largest particle in the SAW system with different persistence lengths of 0.5, 1, 2, and 3 with a constant geometric occupation volume of 6.12%, respectively. Crosses: largest particle in the SAW system with different persistence lengths of 0.5, 1, 2, and 3 with a constant geometric occupation volume of 7.95%, respectively.

Upper panel: Diffusion coefficient as a function of chain length for different occupation volumes. Solid symbols: smallest particle. Blank symbols: largest particle. Circles denote 6.4% occupation volume, squares with 8% occupation volume, and triangles with 10% occupation volume. Lower panel: Diffusion coefficient as a function of persistence length for different occupation volumes. Circles, 46 chains with (4.6% occupation volume); squares, 46 chains with (6.12% occupation volume); triangles, 46 chains with (7.95% occupation volume).

Upper panel: Diffusion coefficient as a function of chain length for different occupation volumes. Solid symbols: smallest particle. Blank symbols: largest particle. Circles denote 6.4% occupation volume, squares with 8% occupation volume, and triangles with 10% occupation volume. Lower panel: Diffusion coefficient as a function of persistence length for different occupation volumes. Circles, 46 chains with (4.6% occupation volume); squares, 46 chains with (6.12% occupation volume); triangles, 46 chains with (7.95% occupation volume).

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