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.

Mathematical models of pattern formation have begun to play a valuable role in understanding morphogenetic processes in both normal and disease conditions. have been successfully applied to a number of phenomena. We will review several applications of Partial Differential Equation models, particularly to the formation of focal lesions in vascular calcification, which are driven by Bone Morphogenetic Proteins and their inhibitors.

However, most applications (including ours), have focused on Turing patterns, which arise as primary bifurcations of periodic patterns from a uniform equilibrium state. These linear instabilities are only the first level of the 'pattern zoo'. We will discuss further bifurcations 'far from Turing' and their associated patterns (holes, isolated spots, etc.), including some applications to physiology.

We will also discuss PDE models of branching morphogenesis in vasculature, including applications to defective branching and/or defective connections, as seen in a number of disease conditions, such as arteriovenous malformations, uneven caliber arteries, and other disease states.

Stochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this talk I will focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species.

I will describe an algorithm for inferring kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). A general formula will be derived for calculating the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. I will illustrate the utility of the method with two examples: a birth-death model and an auto-regulatory gene network.