1. Reaction network theory: One of the most challenging issues systems biologists face is the extraordinarily complicated structure of a system. Thus, characterizing the system structure is a major open question in the field as the structure induces emergent phenotypes and behaviors of the system's dynamics.

In chemical reaction network theory, a biochemical system is described with a graphical configuration called a reaction network that consists of constituent species, complexes and reaction.

These graphical configurations are used to model broad classes of interaction systems including signaling systems, viral infections, metabolism, neuronal networks, population models, etc. For an example of such a network, see Figure 1A below. As it has been noticed that intrinsic noise significantly contributes to the dynamical behavior associated with a reaction network, stochastic models associated with a reaction network in biology have become increasingly relevant. The effects of noise are especially large if the abundance of a species in the system is low. %Many important biochemical models consist of species with low copy numbers inside an individual cell.

2. Stochastic epigenetic models: Immune responses involve dramatic activation of signal-dependent transcription factors (SDTFs), and improper regulation can play roles in disease processes such as autoimmunity, diabetes, and cancer. Multiple bio-molecular signaling networks, such as NFkB, ERK, or p53, show stimulus-specific dynamic control that affects cell behavior. The transcription factor NFkB is an SDTF that acts as a central mediator in inflammatory gene programs. In macrophages, NFkB activation can demonstrate oscillatory or non-oscillatory dynamics \cite{hoffmann2002ikappab}, which correlate with the propensity to form open enhancers and promoters (Figure 2A). It remains unclear how the information contained in signaling dynamics is decoded at the level of chromatin. Both the temporal dynamics and molecular stochasticity of NFkB signaling may serve to modulate the epigenome to influence cell states.

We describe stochastic models (Figure 2B) to explore mechanisms that may explain how oscillatory or non-oscillatory signaling dynamics lead to changes in chromatin accessibility. The fundamental unit of chromatin is the nucleosome that involves 14 main DNA-histone interactions (Figure 2A). Tracking the first hitting time to state 14 in Figure 2B, we verified experimental results about the effects of two different temporal dynamics of NFkB activity as shown in Figure 2C. We also used the stochastic model to explore how the location of the NFkB binding site affects nucleosome eviction upon non-oscillatory NFkB activities. As shown in Figure 2D, we found under reasonable assumptions that the mean first eviction time as a function of NFkB binding location reflects the experimental observations in Figure 2D(right) from chromatin accessibility data. These models of epigenome dynamics, combined with genomic sequencing analysis, allow predictive understanding of how nucleosomes at specific genomic locations respond to different patterns of SDTF signaling to produce alterations to chromatin states in normal and disease conditions.

3. Designing a control circuit: We propose synthetic controllers that can be added to a given chemical reaction network in order to control a given species of interest under both deterministic and stochastic regimes. The controllers designed in this project are inspired by a property called absolute concentration robustness(ACR). We provide both a theoretical framework and computational simulations in several specific biochemical systems to show that an ACR controller can shift all positive steady state values of a target species towards a desired value.

For the deterministic model, the concentrations of species in the controlled system can be driven to the desired steady state. We also show that in stochastic networks, the ACR controller can account for the intrinsic noise in the chemical reaction. We approximate the behavior of the target species using a reduced chemical reaction model derived through multiscaling analysis. Our stochastic analysis assumes certain topological conditions of the controlled network that are described using the so-called deficiency of the system. These two theoretical tools will be combined to calculate the behavior of the reduced system, as well as to show that the behavior of the target species in the reduced system approximates that of the original network. Using computational simulations we also explore the robust perfect adaptation of the target species in the controlled system, a highly desirable goal in control theory.

As an application of our controllers, we consider a receptor-ligand binding model which generates a downstream response with protein P* (Figure 3 A). We show that the ACR system in Figure 3A controls the inactive receptor $R_0$ in order to obtain the desired amount of active protein $P^*$. The main idea of this control is that the controlled system is approximated with the reduced system shown in Figure 3B. Species $R_0$ in the controlled system shown in Figure 3D (left) at a sufficiently large finite time $t$ is well approximated by the Poisson distribution centered at mu/theta for any input R0. Consequently the protein $P$ distribution is also robustly stabilized as shown in Figure 3D (right). On the other hand both mean and variance of $R_0$ in the original system vary with respect to different inputs (Figure 3C, left), and this causes the distribution of $P^*$ to change accordingly (Figure 3C, right).