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 S.
N1 - Funding Information:
The author P.J.A. was supported by NSF VIGRE Postdoctoral Research Fellowship Grant DMS – 9983646 and NSF Mathematical Biology Grant DMS – 0635535. The authors thank Eric Vanden-Eijnden and David Cai for helpful discussions on analytical and physical aspects of the work, and Tom Bringley for a careful reading of a preliminary draft. The authors also thank David McQueen for discussions concerning the immersed boundary method and Yuri Lov for lending computational resources used to obtain some of the numerical results. We are especially indebted to George Oster, whose vision of direct numerical simulation of osmotic phenomena via random forces inspired this work.
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
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U2 - 10.1016/j.jcp.2006.11.015
DO - 10.1016/j.jcp.2006.11.015
M3 - Article
AN - SCOPUS:34248594739
SN - 0021-9991
VL - 224
SP - 1255
EP - 1292
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 2
ER -