I am an Assistant Professor of Applied Mathematics at the University of New Mexico, broadly interested in the development and analysis of models for self-organization in living systems. The best way to get in touch with me is by email.
|Spring 2016||MATH 181: Elements of Calculus II|
|MATH 513/463: Introduction to Partial Differential Equations|
|Spring 2015||MATH 375: Introduction to Numerical Computing|
|MATH 412: Nonlinear Dynamics and Chaos|
|Fall 2014||MATH 375: Introduction to Numerical Computing|
|MATH 321: Linear Algebra|
|Spring 2014||MATH 375: Introduction to Numerical Computing|
|MATH 466: Mathematical Methods in Science and Engineering|
Phyllotaxis, the arrangement of organs (e.g. leaves, bracts, petals, seeds) on the surface of a plant, has been a target of scientific inquiry for many centuries.
Classical models explain phyllotaxis as the optimization of some quantity (e.g. packing efficiency, contact pressure, entropy) by the plant in order to be best suited for survival. While they can reproduce many of the arrangements observed in nature, these models fail to admit that the only tools the plant has to work with are the underlying biochemical and mechanical mechanisms of growth.
Fortunately, these mechanisms are now beginning to be understood. Recent work1,2 has shown that the PIN1 protein plays an role in the transport of auxin, an important growth hormone. Under certain conditions, PIN1 can move auxin along its concentration gradient, resulting in an instability of the uniform concentration. Phylla begin to form where the concentration of auxin is locally maximal.
The governing equation for the fluctuation of the auxin concentration3 turns out to be strikingly close to the gradient flow given by which has the form of a modified Swift-Hohenberg equation. A peculiarity of plant growth distinguishes this from the traditional case, though, in that the solution is laid down as an annular front between a patterned and an unpatterned region.
On the head of a sunflower, for instance, seeds are laid down on a disc-shaped surface with an unpatterned central region and an initial patterned state on the outer boundary. The images above are pseudocolor plots of obtained by numerically integrating its governing equation. Remarkably, the point configurations of maxima coincide almost exactly with those configurations generated by the classical models of phyllotaxis!
1. H. Jönsson, M.G. Heisler, B.E. Shapiro, E.M. Meyerowitz and E. Mjolsness. An Auxin-Driven Polarized Transport Model for Phyllotaxis. Proceedings of the National Academy of Sciences, 2006.
2. D. Reinhardt, E. Pesce, P. Stieger, T. Mandel, K. Baltensperger, M. Bennett, J. Traas, J. Friml and C. Kuhlemeier. Regulation of Phyllotaxis by Polar Auxin Transport. Nature, 2003.
3. A.C. Newell, P. Shipman and Zhiying Sun. Phyllotaxis: Cooperation and Competition Between Mechanical and Biochemical Processes. Journal of Theoretical Biology, 2008.
M. Pennybacker, P. D. Shipman, and A. C. Newell
Phyllotaxis: Some Progress, but a Story Far From Over
Physica D, 2015
A.C. Newell and M. Pennybacker
Fibonacci Patterns: Common or Rare? (Open Access)
Procedia IUTAM, 2013
M. Pennybacker and A.C. Newell
Phyllotaxis, Pushed Pattern-Forming Fronts, and Optimal Packing
Physical Review Letters, 2013
P.D. Shipman, Zhiying Sun, M. Pennybacker, and A.C. Newell
How Universal Are Fibonacci Patterns?
European Physical Journal D, 2011
E.A. Gibson, M. Pennybacker, A.I. Maimistov, I. Gabitov, and N.M. Litchinitser
Resonant Absorption in Transition Metamaterials: Parametric Study
Journal of Optics, 2011
T. Gibson, M. Pennybacker, I. Mozjerin, I. Gabitov, V.M. Shalaev, N.M. Litchinitser
Design Optimization of Transition Metamaterials
Photonic Metamaterials and Plasmonics, OSA Technical Digest, 2010