Spatial Modeling in Systems Biology

Phil Colella

Computational Research Division

Lawrence Berkeley National Laboratory

Joint work with: Joe Grcar, Peter Schwartz (LBNL / CRD); Adam Arkin, Matt Onsum, Eric Alm (LBNL / PBD); David Adalsteinsson (UNC)

Spatial Modeling - Goals

• Modeling of spatial effects: chemotaxis, metabolism, locomotion, cellular communities.

• Multiple physical processes – Chemical and transport processes– Mechanical processes

• Predictive models of high fidelity, or at least known fidelity• Validation

A Deterministic Two-Compartment Model

• Reaction-diffusion equations in the cytoplasm

• Flux boundary conditions coupling the cytoplasm and membrane

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• Reaction-diffusion equations on the membrane

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Typical of a number of proposed spatial models for cells, differing in solution approaches, e.g. stochastic vs. deterministic.

Algorithmic Innovations

• Fast grid generation for realistic geometries in 3D. We convert image data to cut-cell description on a computational grid using level-set feature detection.

Algorithmic Innovations

• PDE on surfaces: solve on annular region using cut-cell discretization:

Gradient Sensing in Neutrophils

• Goal: identify a chemical / transport mechanism for gradient sensing of chemoattractants by neutrophils– Adaptation: function of gradient, not of the average level of

chemoattractant.– Nonlocking: can respond to changes in the environment.– Model related to known reaction mechanisms in cells.

Gradient Sensing in Neutrophils

• Levchenko and Iglesias (2002): 1D reaction-diffusion model.

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• Issues: – Dependence on idealized geometry.– Surface variables and volume variables are indistiguishable in model. In reality, PTEN is transported in cytosol, while other species are bound to the membrane.

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Gradient Sensing in Neutrophils

3D simulation results (P3 on membrane)

• Qualitatively, our results agree with the L&I results with respect to adaptivity, and nonlocking properties.• Quantitatively different: 1D predicts a nonlinear amplification of the signal, while the 3D model shows linear dependence of the P3 gradient on the chemoattractant gradient (latter is observed in experiments).

Septatation and Sporulation

• Simulate the chemical signal that drives the onset of sporulation, starting from 1D model.• Stiff coupling between diffusion and singular chemical reaction at poles causes numerical problems for operator splitting (need Newton-Krylov).

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• More recent experimental data suggests an entirely different model (E. Alm)

Spatial Modeling Algorithmic Requirements

• Some quantities are at such low concentrations that macroscopic models based on averaging over many particles are not valid; leading to stochastic models (Gillespie, SDE; MCell).

• The preceding statement is not true for all quantities - hybrid models in physical space (AMAR) or state space.

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• Mechanical, transport coupling to discrete structures (DNA, flagella) in prokaryotes.

Spatial Modeling Algorithmic Requirements

• All of the above apply to eukaryotes, and more:

– Enormous spatial heterogeneity, geometric diversity: multiple 3D, 2D, 1D and 0D geometric structures, all coupled together.– Multiple mechanical, transport processes: fluid dynamics, discrete mechanics, diffusion, discrete channels.

University of Nebraska “virtual cell” website

– Appropriate representation depend on the scales of interest, level of detail.

Actin cortex image, UCL

Institutional Requirements

• A substantial amount of algorithmic and software infrastructure.

• Need models that can be validated by experimental data.

• Support of a large amount of retail science using modeling. Emphasis on agile development, fast turnaround, high throughput.