Future discovery and control in biology and medicine will come from the mathematical modeling of large-scale molecular biological data, such as DNA microarray data, just as Kepler discovered the laws of planetary motion by using mathematics to describe trends in astronomical data [1].

In this talk, I will first describe novel generalizations of the matrix and tensor computations that underlie theoretical physics (e.g., [2,3]). In my Genomic Signal Processing Lab we are developing these computations for comparison and integration of multiple high-dimensional datasets recording different aspects of, e.g., the cell division cycle and cancer.

Second, I will describe the prediction of a previously unknown mechanism of regulation by using these computations to uncover a genome-wide pattern of correlation between DNA replication initiation and mRNA expression during the cell cycle [4,5]. This computational prediction was recently experimentally verified by analyzing global mRNA expression levels in synchronized cultures under conditions that prevent DNA replication initiation without delaying cell cycle progression [6].

Last, I will describe the computational prognosis of brain cancers by using these computations to compare global DNA copy numbers in patient-matched normal and tumor samples from the Cancer Genome Atlas [7].

1. Alter, PNAS 103, 16063 (2006); http://dx.doi.org/10.1073/pnas.0607650103
2. Alter, Brown & Botstein, PNAS 100, 3351 (2003); http://dx.doi.org/10.1073/pnas.0530258100
3. Ponnapalli, Saunders, Van Loan and Alter, under review.
4. Alter & Golub, PNAS 101, 16577 (2004); http://dx.doi.org/10.1073/pnas.0406767101
5. Omberg, Golub & Alter, PNAS 104, 18371 (2007); http://dx.doi.org/10.1073/pnas.0709146104
6. Omberg, Meyerson, Kobayashi, Drury, Diffley & Alter, MSB 5, 312 (2009); http://dx.doi.org/10.1038/msb.2009.70
7. Lee & Alter, 60th Annual Meeting of the American Society of Human Genetics (ASHG), Washington, DC, November 2-6, 2010.

Sensory systems in individual living cells, as well as in multi-cellular organisms, employ a variety of adaptation mechanisms in order to produce behaviors that are invariant to certain characteristics of environmental inputs, such as symmetries or background signal levels, while at the same time allowing the extraction of relevant features of these inputs. These mechanisms and behaviors are responsible for phenomena ranging from chemotaxis in bacteria to the logarithmic sensitivities to forces, sounds, and vision in humans revealed through psychophysical measurements.

Much of our recent research has been devoted to the understanding of feedforward and feedback circuits that produce adaptation behavior. While closely related to standard concepts in control theory such as disturbance rejection, completely new questions arise. For example, while the internal model principle (IMP) would predict that feedback systems must be present in order to guarantee robust adaptation, the lack of separation between plant and controller components makes the significance of the IMP questionable. More so, the need to perform coordinate changes to exhibit the internal model (transformations which, if at all possible, require strong nonsingularity and global properties on vector fields) typically leads to uninterpretable variables. Moreover, questions such as the invariance of transient behaviors to symmetries appear not to have been systematically studied in this context. We will discuss one such behavior (fold-invariance) and mention new experimental results that confirm theoretical predictions.

Heteroepitaxial growth is a process where crystals are grown
one layer at a time using a molecular beam in a vacuum.
When strain is present these systems can form three dimensional
islands often called quantum dots. This is a nanoscale process
and continuum models struggle to capture many of the phenomena
that occur. Kinetic Monte Carlo (KMC) is an alternative approach which
is quite promising but has had limited use in strained systems
due to a variety computational bottle necks. In this talk, I will
outline KMC models for strained epitaxial growth and
how one can go about performing simulations in an efficient manner.