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Second q-bio Summer School: Stochasticity in Biochemistry and Systems Biology

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In this theme, we will explore stoasticity in biochemical and systems biology modeling. As the subject is immense in its scope, we will be limited necessarily to exploring just a small section of the related topics. Specifically, we will review experimental manifestations of stochastic effects in biology, the methods used to treat them analytically and numerically, and effects of the stochasticity on behavior of certain systems.

This section of the summer school is organized by Ilya Nemenman. Please address all questions about this section of the summer school to its organizer.


Lecture 1

Scope
Stochastic effects in systems biology: Theoretical Foundations and Experimental Results, Part I
Lecturer
Brian Munsky
Synopsis
  • The importance of stochasticity in gene regulatory networks. Discussion of a couple key examples.
  • The physics behind stochastic chemical kinetics.
  • Connection between deterministic reaction rates and propensity functions.
  • Derivation of the Master Equation for discrete stochastic processes.
  • Analysis of the master equation for a simple transcription process.
  • Discussion of the importance of stochasticity in small populations.
Homework

Lecture 2

Scope
Stochastic effects in systems biology: Theoretical Foundations and Experimental Results, Part II
Lecturer
Brian Munsky
Synopsis
  • Solution of the master equation for systems with affine linear propensity functions.
  • Discussion of the effect of feedback.
  • Discussion of Kinetic Monte Carlo algorithms. Tau Leaping. Chemical Langevin equation. Time separation schemes. Hybrid methods.
  • Finite State projections techniques. Switch rate analysis.
  • Moment Closure techniques.
Homework

Lecture 3

Scope
Probability generating functional in stochastic kinetics
Lecturer
Nikolai Sinitsyn
Synopsis
  • Definitions of mesoscopic scale, mesoscopic systems and mesoscopic fluctuations. E. coli as a mesoscopic system.
  • Complexity of Markov chain equations. Definition and simplifying role of the probability generating function. Demonstration on a simple reaction-diffusion model.
  • Michaelis-Menten kinetics: definition, full derivation of the chemical flux generating function, including the treatment of boundary terms.
  • Brief review of recent single molecule experiments. Fano factor.
  • (If time permits). Brief review of analogy of generating function with quantum mechanical wave function. Applications to stochastic biochemical networks.
References
A comprehensive introduction to the method of generating functional can be found in
  • C. W. Gardiner. Handbook of Stochastic methods (Springer-Verlag Berlin Heidelberg, 2004).
Basics of the Michaelis-Menten kinetics can be found in many introductory biochemistry textbooks. A good exposition is in
  • David L. Nelson and Michael M. Cox, Lehninger Principles of Biochemistr (4th edition, W. H. Freeman, 2004).
Stochastic analog of the Michaelis-Menten kinetics was discussed in several recent publications. For example,
  • N. A. Sinitsyn and I. Nemenman, "The Berry phase and the pump flux in stochastic chemical kinetics", Euro. Phys. Lett. 77, 58001 (2007).
  • I. V. Gopich and A. Szabo, "Theory of statistics of kinetic transitions with application to single molecule enzyme catalysis", J. Chem. Phys. 124, 154712 (2006).
For a good example of a single molecule experiment on stochastic enzyme kinetics see
  • B. P. English et al. “Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited”, Nature Chemical Biology 2, 87-94 (2005).
Analogies between quantum mechanical and stochastic evolutions can be learned from
  • V. Elgart and A. Kamenev, “Rare event statistics in reaction-diffusion systems”, Phys. Rev. E 70, 041106 (2004).
Homework
Generating functional techniques

Lecture 4

Scope
Signal processing in biochemical networks: Fourier transforms, central limit heorem, and all that, Part I
Lecturer
Ilya Nemenman
Synopsis
  • Introduction of the phototransduction cascade
  • Derivation of the Langevin equation from the master equation
  • Linearization of the Langevin equation
  • Correlation functions and spectra
  • LInear noise approximation
  • Universal results and their significance:
    • Low-pass filtering
    • Fluctuation-dissipation theorem
  • Temporal integration as a way of reducing noise
  • Filtering at early stages of phototransduction
References

Lecture 5

Scope
Signal processing in biochemical networks: Fourier transforms, central limit theorem, and all that. Part II
Lectures
Ilya Nemenman
Synopsis
  • Noise suppression for early phototransduction
    • Are all of the ideas conceptually the same?
  • Constraints on the distinguishability of signals
  • Frequency-dependent gain of the push-pull amplifier (futile cycle)
    • Energy constraints, amplification, and bandwidth
    • Cascades vs. simple amplifiers
  • Filtering: intrinsic noise and extrinsic noise
  • Optimal filtering: Wiener's matched filter
  • Long-term feedback, adaptation, and band-pass filtering
  • Optimal adaptation: mean, and (maybe) variance.
References
Same as for lecture 4 above.

Lecture 6

Scope
Selective transport through biological channels: does theory work?
Lecturer
Anton Zilman
Brief Plan
  • Selective biological transport: active vs. passive transport
  • Several examples of passive but selective biological channels
  • Major issues: selectivity, efficiency, speed
  • Historical survey of methods and concepts
  • General formulation of the transport selectivity problem in terms of stochastic processes
  • Theoretical models of selectivity
  • Comparison of the theoretical predictions with experiments
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