# APh 161: Dynamics of Transcriptional Regulation

This page contains my lecture outline and notes for a set of lectures that I will be giving in APh 161, Rob Phillips's course on Physical Biology of the Cell. This page is mainly intended as a place for me to keep my notes, but might be useful as a reference for the lecture (the final lecture notes will be posted on the APh 161 web page).

## Goals

• Motivate the role of dynamics and feedback in the physical biology of the cell
• Provide mathematical tools for analyzing the dynamics of transcriptional regulation in the cell
• Work through some case studies that serve as model systems for feedback control in cells

## Lecture 1: Feedback and Dynamics in Cells

This lecture will be a powerpoint-style lecture that tries to motivate the role of dynamics and feedback in cells and describes how we can analyze models to obtain insights into the dynamics. We focus on the lac operon as the main object of study.

#### Lecture Outline

1. How cells regulate function
2. Motivating example #1: lac operon
3. Going from cartoons → numbers: rates and transfer curves
4. The role of dynamics: stability, performance, robustness
5. Methods of modeling: chemical kinetics, statistical mechanics, Hill functions
6. Worked example: deriving a dynamical model for lac

#### Lecture Materials

• Ch 19 of the course text
• N. Yildirim, M. Santillan, D. Horike and M. C. Mackey, "Dynamics and bistability in a reduced model of the lac operon". Chaos, 14(2):279-292, 2004.
• J. M. G. Vilar, C. C. Guet and S. Leibler, "Modeling network dynamics: the lac operon, a case study. Journal of Cell Biology, 161:471-476, 2003.

#### Notes

• I'll mainly be using materials from others for the overview parts of this lecture. I hope to get some of Christina Smolke's very nice slides from the CSHL "Engineering Principles in Biological Systems" meeting in Dec 2006, plus use things from Rob's slides from previous years. - ended up cutting back on the "feedback circuits" aspects of this lecture.
• I need to figure out a good way of going from the equilibrium stat mech analysis to the dynamics using Hill functions, etc. The main issue is figure out how to rigorously transition to the rate and to find some data that support the approach. Done. Did this through the master equation and talking about energy based approaches + vibrational frequency.
• The mini-review paper by Vilar, Guet and Leibler provides a good summary of some of the approaches and limitations of modeling, using lac. It doesn't go into much detail, so mainly useful as a broad overview paper. Eric: we should put this on the APh 161 web page. It is good background reading.
• The paper by Yildirim et al takes a fairly detailed model of the lac operon that was developed by Yildirim and Mackey (Biophysical Journal, 2003) and simplifies it to get a three dimensional model that they claim captures the main features of interest. They make use dynamical systems concepts (bistability, parameteric stability diagrams, bifurcations) to study the system. They compare their results to data and explore its predictive abilities. I spend the second half of the lecture looking at this model.
• Rob suggested looking at the work of Mahaffy at SDSU, who has done some nice dynamical systems analysis of lac, including taking into account time delays. I looked through his 1999 paper with Simeonov and this is a nice analysis. One thing that is missing compared with Yilidirim et al is the tie to experimental data for determining the various constants in the model. One advantage is that the paper uses a much more streamlined version of the dynamics. I'll try to touch on this in the lecture, but I think the model may be missing some important features if you want to ask about bistability, rates of turn-on, ,etc.
• I'm having a bit of trouble tracking down a couple of numbers that I need for my analysis:
• Rate of transcription: how fast is RNA transcribed by RNAP in bp/sec. The paper by Vogel and Jensen (J. Bacteriology, 1994) looks like a good source. They show ~30-80 bp/sec depending on growth rate of the cell (which depends on medium).
• Binding rates for repressor onto the DNA. Presumably this is very fast, which allows us to replace the dynamics with a simple equilibrium calculation. Rob pointed me at a paper (cited in lecture notes; don't have it at hand right now) that gave a diffusion rate of up to 1000 bp/sec, which is what I used in the lecture.

## Tutorial Interlude: Dynamical Systems 101

On Wednesday night there was an optional session for anyone interested in reviewing basic concepts and techniques in dynamical systems.

## Lecture 2: Case Studies and Calculations

#### Lecture Outline

1. Genetic switch (KPT, 19.3.2)
• Motivation$\lambda$ → synthetic biology
• Questions that we want to ask
• Modeling of the switch
• Analysis and answers to questions/predictions
2. Represillator
• Motivation: circadian rythm → synthetic biology
• Questions that we want to ask
• Modeling of the represillator
• Analysis and answers to questions/predictions
3. Recap: the big picture

#### Lecture Materials

• Chapter 19 of course text (especially 19.3)
• J. L. Cherry and F. R. Adler, "How to make a Biological Switch". J. Theoretical Biology, 203(117-133), 2000.
• M. B. Elowitz and S. Leibler, "A synthetic oscillatory network of transcriptional regulators". Nature 403, 335-338, 2000.
• T. S. Gardner, C. R. Cantor and J. J. Collins, "Construction of a genetic toggle switch in Escherichia coli". Nature, 2000.

#### Notes

• The first part of this lecture will be a "stick in the sand" style lecture - working out everything by hand, on the board. The follows the material in Chapter 19 of the text.
• Rough timing for the lecture:
• Switch motivation ($$\lambda$$ → synthetic): 15 min (PPT)
• Switch derivation: 45 min (board)
• Repressilator: 20 min (MATLAB)
• Wrap up: 5 min
##### Annotated references
• J. E. Ferrell, Jr, "Self-perptuating states in signal transduction: positive feedback, double-negative feedback and bistability". Current Opinion in Chemical Biology, 6:140-148, 2002. This is a nice paper showing some of the mechanisms of bistability, focusing on different implementation schemes (graphs and pictures, no equations).