Two-gene network

two-gene feedback network

Principal Investigator

PI: João P. Hespanha
Email: hespanha @
Tel: +1 (805) 893-7042
Fax: +1 (805) 893-3262

Electrical & Computer Engineering Dept. (ECE)
University of California, Santa Barbara (UCSB)

Postal address:
Room 5157, Harold Frank Hall
Dept. of Electrical & Computer Eng.
University of California
Santa Barbara, CA 93106-9560 USA

quick links

Project Summary

This project's goal is to develop formal models for the analysis of a wide class of stochastic systems that arises in biology ranging from the dynamics of biological processes that occur at the molecular/cellular level to the interacting populations of organisms within an ecosystem.

This project has a high potential for strong societal impact, as a formal understanding of the principles behind gene regulation can lower the high costs involved into the experimental effort that is currently needed in drug discovery by the pharmaceutical industry. In addition, in an ecology context, dynamic population models provide formal tools for the efficient management of environmental resources.

The research proposed will make significant contributions to the modeling and analysis of stochastic systems. In particular, the following fundamental issues will be addressed:

  • Development of a modeling framework for stochastic population dynamics based on approximate moment closure techniques.
  • Development of a theory to simplify complex gene regulatory networks based on time-scales separation.
  • Development of formal tools to detect oscillations in stochastic biological systems and to understand their dependence on system parameters.

The proposed activities will have a strong educational component aimed at motivating undergraduate students to pursue advanced degrees in the engineering sciences. This will be achieved through the following initiatives:

  • Increasing the research content of current UCSB undergraduate courses.
  • Expanding our undergraduate Summer Internships program that brings to our laboratory current UCSB undergraduate students and minority students from local community colleges.


All the results, including papers, reports, and software are available freely to the research community through the world-wide-web. The course materials (including lecture notes, homeworks, laboratory materials, etc.) are also freely available to the academic community.

The publications based upon research funded by this project can be found at the following URL:

This material is based upon work supported by the National Science Foundation under Grant No. CNS-0720842. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Relevant Research Topics


As computers, digital networks, and embedded systems become ubiquitous and increasingly complex, one needs to understand the coupling between logic-based components and continuous physical systems. This prompted a shift in the standard control paradigm — in which dynamical systems were typically described by differential or difference equations — to allow the modeling, analysis, and design of systems that combine continuous dynamics with discrete logic. This new paradigm is often called hybrid, impulsive, or switched control.

hybrid systems example

The top diagram is a hybrid automaton model of a queuing system that receives jobs at a rate r and dispatches them at a rate B.

Below the hybrid automaton, we can see the corresponding modelica code.

This and other examples of hybrid systems can be found at the web site of the UCSB course ECE229 - Hybrid and Switched Systems .

Good starting points to learn about hybrid control systems include

  • the web site for the UCSB course ECE229 - Hybrid and Switched Systems
  • the following tutorial paper on hybrid control (mostly devoted to stability issues):
    J. Hespanha. Stabilization Through Hybrid Control. In Heinz Unbehauen, Encyclopedia of Life Support Systems (EOLSS), volume Control Systems, Robotics, and Automation, 2004. [bibtex] [pdf]

Our research covers several aspects of hybrid/switched systems:

  • construction of mathematical models to capture switching between discrete modes, delays, stochasticity, etc.
  • development of formal tools to analyze hybrid systems
  • development of methodologies to design control algorithms for hybrid systems.

Publications on this work can be found at the following URL:

While some of our work on hybrid systems is of a theoretical nature, it is motivated by several high-impact application areas, including networked control systems, cooperative control of autonomous systems, communication networks, and systems biology. Details on some of these application areas are included below.


Systems biology seeks to understand living organisms by modeling and analyzing the complex interactions of genes, proteins, and other cell elements. These interactions occur through biochemical reactions that take place inside the cell or close to the cell membrane. Particularly crucial are the chemical reactions that participate in the complex regulatory mechanism that control cell functions such as the heat shock response, which protects a cell against environmental stresses (heat, cold, oxygen deprivation, etc.); apoptosis, which leads to a programmed cell death with minimal harm to nearby cells; chemotaxis, which permits a cell to move in search of food or to flee from poisons; or cell division, which results in two daughter cells from a single parent cell.

Ultimately, the goal of systems biology is to transform the methodology used for drug discovery, which is currently dominated by mass experimentation. By enlarge, when faced with a new disease or condition, drug developers expose compromised cell cultures to a large number of chemical compounds in the hope of finding a substance that "treats" the disease. Finding such a substance, triggers a second phase of experiments aimed at making sure that this substance does not harm individual cells or organs. In addition, a mechanism must be found to deliver the treatment to the right cells. The goal of systems biology is to guide this effort so that most effort is spent searching among the most promising types of substances and making sure that all cell functions that could be affected by the potential treatment are not negatively affected.

What makes finding cures for diseases especially challenging is the fact that cells are exquisitely regulated mechanisms with multiple feedback loops. Suppose for example that it is discovered that a particular disease develops because a set of cells is lacking protein X. A naive cure would be to inject X into the blood stream in an attempt to increase its concentration. However, this can actually have a completely opposite effect if the body interprets the high concentration of X in the blood as a signal that this protein is being overproduced and shuts down the natural production of X. This is not unlike the apparent paradox that results from placing a heater next to the temperature feedback sensor of a central heating systems and suddenly realizing that the whole building got much colder.

Gene feedback

Simple gene regulatory network, where a gene G produces mRNA (transcription), which in turn produces a protein X (translation). The regulation of the production of X is achieved by a negative feedback loop that results from the fact that the protein X dimerizes to produce X2, which acts as a repressor transcription factor that inhibits transcription by preventing the RNA polymerase from binding to the gene G, thus effectively inactivating it by preventing the production of mRNA. To the right-hand side of the network, we see a set of chemical reactions that can embody this mechanism.

The goal of our research has been to develop tools to analyze complex networks of biochemical reactions. Motivated by the above observations, we are especially interested in constructing dynamical models that highlight the feedback mechanisms in cell regulation and that provide a qualitative and quantitative understanding of how the different genes, proteins, and other cell elements contribute to the observed behavior (phenotype).

Gene regulatory mechanisms typically involve a large number of distinct chemical species, but it is common for some of these species to be represented by just a few molecules, which can invalidate models based on the deterministic chemical rate equation. Our work has been using tools developed for Stochastic Hybrid Systems to construct differential equations that accurately model the stochastic effects present in biochemical networks.

Moment dynamics

The left plot shows the evolution of the mean and variance of the number of molecules of a particular chemical species involved in a bio-chemical network obtained from an ODE that approximates the moment dynamics. In the middle plot we see a typical sample path obtained through a Monte Carlo simulation and in the right plot an histogram obtained from a large number of such simulations. The moment dynamics ODE model was obtained using the package StochDynTools developed by our team and the Monte Carlo simulations were obtained with Petzold's StochKit.

Publications on this work can be found at the following URL:

Software to compute moment dynamics can be found at the following URL:

Relevant Courses

  • ECE147C — Control Systems Design Project & ME106A — Advanced Mechanical Engineering Laboratory (Spring’04, Spring'05, Spring’06, Spring'07)
  • ECE229 — Hybrid and Switched Systems (Winter’04, Fall’05), see
  • ECE594D — Modeling and Control of Large-Scale Distributed Systems (Winter'02)
  • ECE594D — Hybrid Control and Switched Systems (Spring'02)

Recent talks & events


“Stochastic Hybrid Systems: Modeling, analysis, and applications to networks and biology” Electrical Engineering and Computer Science Seminar, UC Berkeley, May 1, 2006. [slides]

“Stochastic Modeling of Chemical Reactions (and more…),” UC Santa Barbara Theoretical Ecology Seminar, Mar. 17, 2006. [slides]


Course on Modeling Analysis and Design of Hybrid Control Systems at the HYCON Graduate School on Control from the European Embedded Control Institute, February 12-16, 2007.

Hybrid Systems Biology. Workshop for the 45th IEEE Conference on Decision and Control, San Diego, CA, December 12, 2006.

9th International Workshop on Hybrid Systems: Computation and Control (HSCC 2006), Santa Barbara, California, from March 29--31, 2006.

Stochastic Hybrid Systems: Theory and Applications. Workshop for the 43rd IEEE Conference on Decision and Control, December 13, 2004.

Students, Postdocs, and Visitors


Shaunak Bopardikar, BT/MT in Mechanical Engineering 2004 (Indian Institute of Technology, Bombay), started PhD in Fall 2005 (co-advised with Prof. Francesco Bullo).

Alexandre Mesquita, Undergraduate Degree in Electrical Engineering 2006 (Divisão de Engenharia Eletrônica, Instituto Tecnológico de Aeronáutica - ITA), started PhD in Fall 2006.


Abhyudai Singh, PhD 2008, BT in Mechanical Engineering (Indian Institute of Technology, Kaput), currently Postdoctoral Scholar, University of California, San Diego (as of Oct. 2008).

Prabir Barooah, PhD 2007, BT 1996 (Indian Institute of Technology, Kanpur), currently Assistant Professor at the Department of Mechanical Engineering, University of Florida, Gainsville (as of Sep 2007).


Daniel Klein, PhD 2007 (University of Washington, Seattle, WA).


This list only contains visitors that stayed at UCSB for 2 weeks or longer (list sorted by date of last departure)

Duarte Antunes, PhD student, Inst. Superior Técnico, Lisbon, Portugal, 8/20/07-12/20/07, 4/4/08-6/28/08, 10/1/08-12/12/08.

Pietro Tessi, PhD student, University of Florence, Italy, 9/3/08-2/25/09

Prof. Kenji Hirata, Dept. of Mechanical Engineering, Nagaoka University of Technology, Japan, 9/1/08-8/31/09.