### Abstract

In modeling many biological systems, it is important to take into account flexible structures which interact with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work, it is shown how the immersed boundary method of [C.S. Peskin, The immersed boundary method, Acta Num. 11 (2002) 1-39.] for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic (τ^{-3/2}) decay for long times [B.J. Alder, T.E. Wainwright, Decay of the Velocity Autocorrelation Function, Phys. Rev. A 1(1) (1970) 18-21]. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method. Specifically, we present simulations of osmotic effects of molecular dimers, worm-like chain polymer knots, and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force. The theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales.

Original language | English (US) |
---|---|

Pages (from-to) | 1255-1292 |

Number of pages | 38 |

Journal | Journal of Computational Physics |

Volume | 224 |

Issue number | 2 |

DOIs | |

State | Published - Jun 10 2007 |

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### Keywords

- Brownian dynamics
- Brownian ratchet
- Fluid dynamics
- Immersed boundary method
- Osmotic pressure
- Polymer knot
- Statistical mechanics
- Stochastic processes

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

*Journal of Computational Physics*,

*224*(2), 1255-1292. https://doi.org/10.1016/j.jcp.2006.11.015

**A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales.** / Atzberger, Paul J.; Kramer, Peter R.; Peskin, Charles.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 224, no. 2, pp. 1255-1292. https://doi.org/10.1016/j.jcp.2006.11.015

}

TY - JOUR

T1 - A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales

AU - Atzberger, Paul J.

AU - Kramer, Peter R.

AU - Peskin, Charles

PY - 2007/6/10

Y1 - 2007/6/10

N2 - In modeling many biological systems, it is important to take into account flexible structures which interact with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work, it is shown how the immersed boundary method of [C.S. Peskin, The immersed boundary method, Acta Num. 11 (2002) 1-39.] for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic (τ-3/2) decay for long times [B.J. Alder, T.E. Wainwright, Decay of the Velocity Autocorrelation Function, Phys. Rev. A 1(1) (1970) 18-21]. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method. Specifically, we present simulations of osmotic effects of molecular dimers, worm-like chain polymer knots, and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force. The theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales.

AB - In modeling many biological systems, it is important to take into account flexible structures which interact with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work, it is shown how the immersed boundary method of [C.S. Peskin, The immersed boundary method, Acta Num. 11 (2002) 1-39.] for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic (τ-3/2) decay for long times [B.J. Alder, T.E. Wainwright, Decay of the Velocity Autocorrelation Function, Phys. Rev. A 1(1) (1970) 18-21]. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method. Specifically, we present simulations of osmotic effects of molecular dimers, worm-like chain polymer knots, and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force. The theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales.

KW - Brownian dynamics

KW - Brownian ratchet

KW - Fluid dynamics

KW - Immersed boundary method

KW - Osmotic pressure

KW - Polymer knot

KW - Statistical mechanics

KW - Stochastic processes

UR - http://www.scopus.com/inward/record.url?scp=34248594739&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34248594739&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2006.11.015

DO - 10.1016/j.jcp.2006.11.015

M3 - Article

VL - 224

SP - 1255

EP - 1292

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -