ABSTRACT
Quantitative Biology: From Molecular to Cellular Systems edited by Michael E. Wall © 2012 CRC Press / Taylor & Francis Group, LLC. ISBN: 978-1-4398-2722-2
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 10.1.1 Measurement of Single-Cell Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 235 10.1.2 Using Measurements of Cellular Variability to Infer System
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 10.1.3 Chapter Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
10.2 Mesoscopic Modeling of Biomolecular Reactions . . . . . . . . . . . . . . . . . . . . . . . 237 10.2.1 The Chemical Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 10.2.2 Kinetic Monte Carlo Methods (Stochastic Simulation Algorithm) . . . . . 240
10.3 Analyzing Population Statistics with FSP Approaches . . . . . . . . . . . . . . . . . . . 242 10.3.1 Notation for the FSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 10.3.2 FSP Theorems and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.3.3 The FSP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.3.3.1 Choosing the Initial Projection Space . . . . . . . . . . . . . . . . . . . . . 244 10.3.3.2 Updating the Projection Space . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.3.4 Advancements to the FSP Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.4 Description of the FSP Two-Species Software . . . . . . . . . . . . . . . . . . . . . . . . . . 246 10.4.1 System Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.4.2 Generating Stochastic Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.4.3 Solving the Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10.4.3.1 Dening the Full Master Equation . . . . . . . . . . . . . . . . . . . . . . . 248 10.4.3.2 Dening the Projected Master Equation . . . . . . . . . . . . . . . . . . 249 10.4.3.3 Solving the Projected Master Equation . . . . . . . . . . . . . . . . . . . 249 10.4.3.4 Updating the Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 10.4.3.5 Analyzing FSP Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.5 Examples of Stochastic Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251 10.5.1 Example 1: Using Autoregulation to Reduce Variability in Gene
Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251 10.5.2 Example 2: Using Nonlinearities and Stochasticity to Amplify or
Damp External Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10.5.3 Example 3: Stochastic Toggle Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 10.5.4 Example 4: Stochastic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.1 Introduction Phenotypical diversity may arise despite clonal genetics for a number of reasons, many of which are due to uctuations in the environment: for example, cells nearer to nutrient sources grow faster, and those subjected to heat, light, or other inputs will respond accordingly. But even cells in carefully controlled, homogenous environments can exhibit diversity, and a strong component of this diversity arises from the rare and discrete nature of genes and molecules involved in gene regulation. In particular, many genes have only one or two copies per cell and may be inactive for large portions of the cell cycle. e times at which these genes turn on or o depend upon many random or chaotic events, such as thermal motion, molecular competitions, and upstream uctuations.
