Brief research statement
       Flyer, Emerging Modelling Methodologies in Medicine and Biology, Edinburgh, UK., July 2009

Modeling the generation of somatic mosaicism in the brain

Collaborators: M. McConnell, Fred Gage, Jerold Chun, and Jan Medlock

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Recent evidence suggests that somatic mosaicism &minus the presence of multiple distinct genomes within an individual &minus occurs in the mammalian brain, due to widespread genome rearrangement during embryonic development [e.g., see Muotri, et al., and Coufal et al. re retroelement activity (i.e., retrotransposition) and Rehen, et al., and Yurov, et al., re the somatic loss and/or gain of chromosomes (i.e., aneuploidy) in the brain]. By combining computational and analytical techniques for modeling branching processes, we are exploring how somatic genome rearrangement maps onto the branching patterns of neural stem and progenitor cell (NS/PC) lineage trees.

Briefly, according to a selection of basic branching process model assumptions, we randomly generate an ensemble of NS/PC lineage trees, such as the tree pictured. Overall, we impose that the computed ensembles of trees are consistent with population-averaged data on neuron production in the mouse. This provides a quantitative framework for addressing outstanding questions concerning variability in NS/PC daughter cell fate decision making. Also, these models provide a means for quantifying how this variability relates to the generation and composition of the neural genomic mosaic. Preliminary exploration of qualitative model behaviors within our computational framework has provided testable hypotheses and identified key experiments moving forward.

The biology of neurogenesis provides rich in vivo context for moving beyond the average cell perspective to develop a middle-out approach to modeling cell biological systems. With population-level data as constraints, we ultimately seek to incorporate simplified single-cell models and build explanatory models for the molecular basis of single-cell variability. Doing so will be necessary to understand how genetic alterations causally linked to a specific neurological disease actually cause the pathological cell loss or cell damage characteristic of the disease.


  • H. R. MacMillan and M. J. McConnell, 2010. "Seeing beyond the average cell: Branching process models of cell proliferation, differentiation, and death during mouse brain development. Theory in Biosciences, Submitted, Oct. 2009.
  • M. J. McConnell, H. R. MacMillan, and J. Chun, 2009. "Mathematical modeling supports substantial neural progenitor cell death." Neural Development. 4(28): 1-12. doi:10.1186/1749-8104-4-28. Link
  • M. A. Case, H. R. MacMillan, 2009. "On simulating the generation of mosaicism during mammalian cerebral cortical development." Journal of Biological Systems. 17(1): 27-62. doi:10.1142/S0218339009002740. Link. PDF.

Biogeochemistry of constructed wetlands for water remediation

Collaborators: Brandon Pelfrey, Scott Brame, John Rodgers, and James Castle.

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Selenium is a common contaminant of municipal, agricultural, and industrial waters. Oxidated states of selenium (i.e., selenate [SeVI], selenite [SeIV]) are soluble and toxic in high concentrations. Upon accepting electrons, these states are transformed (i.e, reduced) to immobile and less bioavailable elemental selenium [Se] and selenide [Se-II;]. This reduction occurs either due to abiotic electron donors, like iron [Fe-], or in concert with microbial oxidation of organic matter. In wetlands treatment systems, the latter mechanism results in selenate deposits within the microbes that reside in detrital biofilms, thereby keeping solubale oxidated selenium from entering the food chain.

As a consequence, wetlands designed to facilitate the microbial-mediated reduction of SeVI and SeIV hold promise as a sustainable means of treating agricultural, municipal, and industrial waters high in selenium. Toward the optimal design of such constructed wetlands, we are developing multiscale, multiphase, and multispecies mathematical models of selenium transport and fate to better understand the processes of advection, diffusion, reaction, volatilization, and biotransformation of selenium in free-surface wetlands densely buffered with typha latifolia (i.e., ``cat tails''). In particular, our models incorporate the effects of wetland architecture, plant buffering and litter, biofilm structure and composition, redox potential (eH), hydrogen ion activity (pH), soluble oxygen, temperature, and environmental perturbations.


  • B. Pelfrey, S. Brame, H.R. MacMillan, J. Castle, and J. Rodgers, 2010. "Modeling microbial-mediated reduction of selenium in pilot-scale constructed wetlands." In preparation.

Reduced-order models of the cellular response to DNA damage

Collaborators: Mike McConnell.

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In time, models of extracellular signaling and genetic regulation, embedded within simulated branching process models, will enable study of increasingly refined questions about the biology of neuron production in the brain. However, quantifying the associated uncertainty of resolving genetic mechanisms within ensembles of trees will require sparse and effective sampling of high-dimensional parameter spaces; this remains a significant computational challenge, coined the curse of dimensionality. Minimizing dimensionality and reducing the complexity of gene-regulatory and protein-interaction networks is therefore imperative, and we are pursuing a "functional" approach to model reduction of the cellular response to DNA damage. We restrict ourselves to the developmental cellular context of NS/PCs in the mouse.



  • H.R. MacMillan and M. J. McConnell, 2010. "A nest of qualitative DNA damage response networks." In preparation.

Meshfree least squares methods

Collaborators: Max Gunzburger and John Burkardt

Convergence is established for a meshfree first-order system least squares (FOSLS) partition of unity (PU) finite element method. By virtue of the partition of unity, local approximation gives rise to global approximation in H(div)&cap H(curl) . FOSLS provides local a posteriori error estimation to guide the judicious allotment of new degrees of freedom when adaptively enriching an initially-coarse point set used to construct a meshfree discretization. Hence, this synthesis can be used to leverage the advantages of meshfree approach. Beyond mechanics and classical PDE systems, application of PU methods to dynamic cell-centered biological simulations may hold special promise. This is due to natural interest in either moving (cell migration), eliminating (cell death), or adding (cell division) subsets of points used to build multilevel---with respect to biological organization---descriptions of various molecular factors and their impact on cell fate decisions.


  • H.R. MacMillan, M. D. Gunzburger, and J. D. Burkardt, 2007. "Meshfree First-order System Least Squares." Numerical Mathematics: Theory, Methods, and Applications. 1:29-43. Link.

Acetylcholine Reaction and Diffusion within a Neuromuscular Junction

Collaborators: Kaihsu Tai, Steve Bond, Nathan Baker, Andy McCammon, and Michael Holst.

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A robust infrastructure for solving time-dependent diffusion using the finite element package FEtk has been developed to simulate synaptic transmission in a neuromuscular junction with realistic postsynaptic folds. Simplified rectilinear synapse models serve as benchmarks in initial numerical studies of how variations in geometry and kinetics relate to endplate currents associated with fast-twitch, slow-twitch, and dystrophic muscles. The flexibility and scalability of FEtk affords increasingly realistic and complex models that can be formed in concert with expanding experimental understanding from electron microscopy. Ultimately, such models may provide useful insight on the functional implications of controlled changes in processes, suggesting therapies for neuromuscular diseases.


  • K. Tai, S. D. Bond, H. R. MacMillan, N. A. Baker, M. J. Holst and J. A. McCammon, 2003. "Finite Element Simulations of Acetylcholine Diffusion in Neuromuscular Junctions." Biophysical Journal. 84:2234-2241. Link.

First-order system least squares for and electrical impedance tomography

Collaborators: T. A. Manteuffel and Stephen F. McCormick.

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Electrical Impedance Tomography (EIT) belongs to a family of imaging methods that employ boundary measurements to distinguish interior spatial variation of an electromagnetic parameter. The associated inverse problem is notoriously ill-posed, due to diffusive effects in the quasi-static regime, when electrical impedance reduces to its real part, resistivity. The standard approach to EIT is output least squares (OLS). For a set of applied normal boundary currents, one minimizes the defect between the measured and computed boundary voltages associated, respectively, with the exact impedance and its approximation. In minimizing a boundary functional, OLS implicitly imposes the governing Poisson equation as an optimization constraint. To reconstruct resistivity or, equivalently, conductivity, we introduce a new first-order system least-squares (FOSLS) formulation that incorporates the elliptic PDE as an interior functional in a global unconstrained minimization scheme. We then establish equivalence of our functional to OLS and to an existing interior least-squares functional due to Kohn and Vogelius. That the latter may be viewed as a special dual approach (FOSLL*) is an interesting attribute of this equivalence. Because the FOSLS functional implicitly reflects the inherent loss of resolution away from the boundary, relative to the set of applied boundary tests, the need for artificial regularization may be avoided.


  • H. R. MacMillan, T. A. Manteuffel, and S. F. McCormick, 2004. "First-order system least squares and electrical impedance tomography," SIAM Journal of Numerical Analysis, 42(2): 461-483. Link.
  • H. R. MacMillan, 2001. "First-order system least squares and electrical impedance tomography." Dept. of Applied Mathematics, Univ. of Colorado at Boulder, Ph.D. Thesis. Link.

Stability of a pair of vortex filaments

Collaborators: A. Majda and R. McLaughlin.

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Parallel asymmetric vortices, such as those forming in the wake of aircraft, can persist depending on atmospheric conditions. This can present a disruptive down draft for smaller aircraft flying through such a wake. Here we conduct a stability analysis to examine the conditions of a persistent idealized wake.