Stochastic Modeling and Control of Biological Systems: The Lactose Regulation System ofEscherichia Coli

Abstract

Abstract-In this paper, we present a comprehensive framework for stochastic modeling, model abstraction, and controller design for a biological system. The first half of the paper concerns modeling and model abstraction of the system. Most models in systems biology are deterministic models with ordinary differential equations in the concentration variables. We present a stochastic hybrid model of the lactose regulation system of E. coli bacteria that capture important phenomena which cannot be described by continuous deterministic models. We then show that the resulting stochastic hybrid model can be abstracted into a much simpler model, a two-state continuous-time Markov chain. The second half of the paper discusses controller design for a specific architecture. The architecture consists of measurement of a global quantity in a colony of bacteria as an output feedback and manipulation of global environmental variables as control actuation. We show that controller design can be performed on the abstracted (Markov chain) model and implementation on the real model yields the desired result.

From discrete to continuous and back: Abstractions and mesoscopic phenomena in cells

“…In the stochastic case, one would still like to be able to think of the lac system as bistable, but with appropriately modified transition rules. This is exactly what we did in the case of the lac operon, connecting the high level discrete abstraction with the underlying stochastic phenomenon [7]. By performing first-principles simulations we were able to derive estimates for the spontaneous transition rate between the induced and uninduced states.…”

“…For more details on the simulations discussed here we refer to [7]. Our mixed simulations allowed us to perform many individual runs, equivalent to simulations of small colonies of 100-1000 cells.…”

Stochastic hybrid systems for studying biochemical processes

“…However, fast reactions or reactions containing high-copy molecular species can be modelled using ODEs (or in some cases SDEs) resulting in a reduced approximate SHS where the dynamics of the continuous state is no longer trivial (figure 2). These reduced SHSs provide much faster simulation times than the original SHS, with only a marginal decrease in accuracy (Neogi 2004;Salis & Kaznessis 2005;Chen et al 2009) and have been used to model a wide array of biological processes including lactose regulation in Escherichia coli (Julius et al 2008), the human immunodeficiency virus transactivation network (Griffith et al 2006) and synthetic gene networks (Bortolussi & Policriti 2008).…”

“…This bifurcation at the population level occurs owing to an underlying bi-stability in the lactose metabolic network, with stochastic fluctuations in regulatory molecules causing single cells to converge to either one of the two stable steady states. Starting from a full SHS model of the lactose metabolic network, Julius et al (2008) built a reduced model that can quickly predict the fraction of induced and uninduced cells in response to a given concentration of lactose. This reduced model allowed the design of control feedback laws that can robustly steer a colony of cells to a desired fraction of induced cell, using external lactose as a control input.…”

Stochastic modeling and identification of emergent behaviors of an Endothelial Cell population in angiogenic pattern formation

“…Bacteria propulsion of micro objects Sitti, 2007, 2008), for example, is based on stochastic modeling and control of an array of bacteria. Several authors have addressed the dynamics of bacteria gene regulation systems (Raser and O'Shea, 2004;Losick and Desplan, 2008;Shahrezaei and Swain, 2008), and Escherichia coli has been modeled as a hybrid stochastic system (Julius et al, 2008). Inspired by skeletal muscles and motor control, stochastic recruitment and broadcast feedback have been developed for controlling a population of anonymous agents (Odhner and Asada, 2009) and have been applied to artificial muscle actuators with cellular structure (Ueda et al, 2007).…”