Paul J. Atzberger

Paul J. Atzberger

Assistant Professor
Department of Mathematics
Department of Mechanical Engineering
PhD Mathematics · Courant Institute, New York University · 2003

Research Interests

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, micro-rheology, molecular motors, and microfluidic / nanofluidic devices.

For a more detailed discussion of our work see the research summaries.

[Research Summaries]

Talks (Video)

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations (October 2009) [IMA video / slides].
- Multiscale Modeling and Simulation of Soft Materials (November 2008) [IMA video / slides].

Publications:

1. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]
2. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF]
3. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (submitted). [PDF]
4. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., Physica D: Nonlinear Phenomena, Vol. 238, Iss. 14, July, pp. 1168-1179, (2009),[PDF] [DOI]
5. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009), [PDF] [DOI]
6. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]
7. Error Analysis of a Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations , Atzberger, P.J. and Kramer, P.R., Mathematics and Computers in Simulation, Vol. 79, Iss. 3, pg. 379 -- 408, (2008). [PDF] [DOI]
8. A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF] [DOI]
9. Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF] [DOI]
10. Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). PDF] [DOI]
11. A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF] [DOI]
12. Velocity Correlations of a Thermally Fluctuating Brownian Particle: A Novel Model of the Hydrodynamic Coupling , Atzberger, P.J., Phys. Lett. A, Vol. 351, Iss. 4-5, 6, March, pp. 225-230, (2006). [PDF] [DOI]
13. A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF] [DOI]

We gratefully acknowledge the following sources of support:

*This material is based upon work supported by the National Science Foundation under Grant No. DMS-0635535. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Numerical Codes:

Stochastic Immersed Boundary Method (SIB) : Release 1.0 (alpha)

Class Websites and Course Notes:

  Mathematical Finance Course Notes and Resources:
  course notes and supplemental materials

  Numerical Analysis I (Summer 2009)  
  MTWR 12:30pm - 1:35pm.
  course website

  Finite Difference Methods for Partial Differential Equations (Spring 2009)
  TR 11:00am - 12:15pm.
  course website

  Numerical Analysis II (Winter 2009)
  TR 9:30am - 10:45am.
  course website

  Introduction to Stochastic Analysis and Applications (Fall 2008)
  TR 11:00am - 12:15pm.
  course website

  Numerical Analysis I (Fall 2008)
  TR 9:30am - 10:45am.
  course website

  Introduction to Mathematical Biology (Winter 2008)
  TR 8:00am - 9:15pm, Old Little Theatre, Rm. 160B.
  course website

  Finite Element Methods (Fall 2007)
  TR 11:00am - 12:15pm, South Hall Rm. 6635.
  course website

  Introduction to Complex Variables (Fall 2007)
  TR 8:00am - 9:15pm, GIRV Rm. 1116.
  course website

  Additional Class Websites:
  classes taught 2003 - 2007

Advising:

  Postdoctoral Researchers:
    - Chetan Pahlajani

  Graduate Students:
    - David Valdman
    - Pat Plunkett
    - Jon Karl Sigurdsson
    - Per Danzl (co-advisor)
    - David Boy (co-advisor)

  Undergraduate Students:
    - Joe Rudzinski
    - Justin Shrake

  Administration:
    - Mathematical Finance Majors (Undergraduate Faculty Adviser)

Research Summaries:

Paul J. Atzberger

Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:

A recent talk on the Stochastic Immersed Boundary Method can be found here [IMA video / slides].

The Stochastic Immersed Boundary Method (SIB) is a numerical approach for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions [AVI Movie Unknot] [AVI Movie Trefoil Knot] [AVI Movie Figure Eight Knot], (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling [AVI Movie Membrane Sheet], [AVI Movie Membrane Vesicle], [AVI Movie Membrane Vesicle Formation], (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load [AVI Movie Motor with No Load], [AVI Movie Motor with Intermediate Load], [AVI Movie Motor with Large Load]. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations.

Publications

• Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF]
• A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF] [DOI]
• Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF] [DOI]
• A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF] [DOI]
• On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]
• Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]

Osmotic Phenomena / Microfluidics:

Osmosis refers to a general phenomenon in mechanical systems where pressures and forces are generated in a fluid or gas solvent by solute particles which are confined by semi-permeable membrane boundaries or electric fields. Such forces and pressures play a fundamental role in many biological systems and technological devices. Some examples in biology include mechanisms responsible for exchange of nutrients and wastes in microscopic capillaries in tissues and means of propulsion in cell motility. In technological applications, microscopic devices have been designed which utilize osmotic effects to pump fluids or actuate forces via swelling. The classical theories for osmotic pressure, such as van't Hoff's relation, become inaccurate for microscopic systems where solute particles interact on length scales comparable to the size of the confining chamber. Classical theories of osmosis also typically assume systems are in thermodynamic equilibrium.

To study systems in regimes applicable to biological systems and microscopic devices we are developing theory to describe microscopic osmotic phenomena both in equilibrium and non-equlibrium settings. Some recent work includes studying how osmotic pressures generated by polymer solutes depend on polymer topology, stiffness, and excluded volume. In related work, a pumping mechanism for a microfluidic device has been proposed which uses reversible chemical reactions to drive fluid flows.

Publications

• A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., Physica D: Nonlinear Phenomena, Vol. 238, Iss. 14, July, pp. 1168-1179, (2009),[PDF] [DOI]
• Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). PDF] [DOI]

Applications in Molecular Biology and Biotechnology:

Technological advances such as optical traps and molecular tagging are yielding a wealth of quantitative information about the mechanics and localization of processes in biological systems at the single molecule level. In cell biology an interesting class of molecules are motor proteins which act essentially as microscopic machines within cells to form cellular structures, transport materials, or generate forces. We are using stochastic modeling, analysis, and simulation to study the basic mechanisms underlying how molecular motor proteins operate with an effort toward making connections with available experimental data. In previous work we have studied the kinesin motor protein and developed a coarse-grained mechanical model consistent with available optical trap experimental data, see papers.

Aptamers are short segments of DNA or RNA which bind to proteins. In applications, specific aptamers (nucleic acid sequences) are sought which bind to a desired target protein with high affinity and specificity. Applications include purification methods for proteins, analytics in biosensor devices, and target validation in pharmaceutical drug development. Obtaining aptamers with strong affinity and specificity binding a given target molecule poses a significant challenge in practice. In collaboration with the H.T. Soh Laboratory, Department of Mechanical Engineering along with the undergraduate Joe Rudzinski (CCS Program, UCSB) mathematical approaches are being used to study experimental methods for the selection of DNA-Aptamers from a library of random sequences.

Publications

• Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009), [DOI] [PDF]
• A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF] [DOI]

Spatially Adaptive Numerical Methods for Stochastic Partial Differential Equations:

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions or domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. As a demonstration we present results for a stochastic reaction-diffusion system with two chemical species undergoing Gray-Scott reactions subject to intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize a quad-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

• Growth of a noise-induced spatial pattern resolved with adaptive meshing [AVI Movie].

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are essentially spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are using these approaches to develop a variety of stochastic numerical methods for the study of spatially extended stochastic systems.

Publications

• Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF]

For a more in-depth discussion of this work and information about additional projects please see the publications section.

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