Combinatorial State Equations and Gene Regulation

1. Combinatorial State Equations and Gene Regulation Jay Raol and Steven J. Cox Computational and Applied Mathematics Rice University

2. Outline • Phage Lambda • Combinatorial State Equations • Applications to Phage Lambda

3. Phage lambda • Why model phage lambda? • Genetic network is small and relatively well understood compared to more complex organisms • Still exhibits complex enough behavior to make modeling it non-trivial • Allows for the testing of modeling techniques • Goodwin (1963), McAdams and Shapiro (1995), … • Applications to engineering bacteria • Exhibits regulatory behavior found in all high level organisms

4. Phage lambda • Phage lambda is a virus that attacks E. coli • It inserts its DNA into the bacterial genome • A decision is made by the genome • Lytic Phase • Lysogenic Phase from Ptashne, “A Genetic Switch”

5. Lysogeny or Lytic? • This decision by phage lambda involves two primary players and a host of supporting proteins and gene • Depending on environmental conditions, phage lambda favors the production of cro or cII • Once the phage is committed, it proceeds down a path that either destroys the cell, or lays dormant waiting for activation

6. Methodology • Questions to consider • How to model protein-DNA interactions? • How to model protein-protein interactions? • How to model evolution of a given system? • protein-protein interactions: Law of Mass Action • [a] + [b] → [ab] • k is the reaction rate

7. DNA-protein Interactions • Let us make several assumptions regarding the nature of proteins and the production of mRNA • mRNA production is determined by the (active) state of the gene • mRNA production reaches a maximal rate • Transcription Factors (TF) act independently of each other

8. Gene States • The state of a gene is simply a possible configuration of the TFs which bind to the gene • The probability of a given state is determined by the concentration and affinity of the respective TF’s • If a state governs transcription of a gene, then over a period a time, the probability of that state will be proportional to the transcription rate

9. Gene States • If we assign weights for each state based on protein concentrations and their binding affinities, then the probability of the state is

10. Defining a state of the network • Consider the following small network: gene a promotes itself and inhibits gene b. Gene b promotes itself and inhibits gene a. • What is happening at Gene a?

11. Defining the state of gene A • represent the three possible states. For the state with no a or b present (s0), a basal rate normalized to 1 is assumed. • What is the probability of an activating state? • Where is the value of the activated state, and is the sum of the values of all states

12. Cooperativity • Break down the binding into different states with different values • At the Or operon, cII binds to three different sites with three different affinities. This corresponds to three different states from Ptashne, “A Genetic Switch” 1992

13. Cooperativity • There are 3! possible states from Ptashne, “A Genetic Switch” 1992

14. Complex States • Calculating P(sn) can become tedious for large number of proteins. • P(sn) can be generalized to incorporate a simple statistical formulation

15. Boolean Logic • Consider Gene a and Gene b. If both protein A and B are required at the operon for activation, then • Let and then

16. Boolean Logic • Similarly, we find that • For example, if for proteins A or B were required for activation using only one binding spot, then

17. ODE Model • Keep track of everyway mRNA is produced and destroyed • Transcription • Degradation • Similar assumptions for proteins (Goodwin) apply

18. Full system

19. Generalizing Goodwin • This formulation of gene state probabilities is a generalization of Goodwin “epigenetic systems” • Goodwin oscillator and other similar ODE examples of gene regulation can be reproduced this way

20. Phage Lambda: N protein • For an example, lets look at the equations that govern n protein and its gene • YN indicates the gene n recruits the transcription machinery alone. It also promotes itself. Cro and CII inhibit the producton of N

21. Phage Lambda: Full System • Many existing models in the literature • Many are capable of capturing at least the general dynamics of protein expression • Many consist of hybrid discrete and continuous parts • This approach makes understanding their dynamics more difficult • We shall consider one of the first phage lambda network models and compare it to the combinatorial state equation approach

22. Phage Lambda Model

23. Phage λ: entire system from McAdams and Shapiro, Science 1995

24. Filling in the Kinetics • Consider the regulation at the PL promotor • Using the CSE, we can describe the regulation of N

25. Kinetics of CI • In this case, we seek to describe the regulation of CI

26. Dimerization and CI • Cro and CI actually bind to the Or as dimers. • We model this as a separate reaction in the form dictated by the law of mass action.

27. Anti-sense mRNA • The production of anti-sense mRNA is • Anti-sense mRNA q by CII inhibit the production Q

28. Parameters • Parameters were found in existing literature • Patshne 1992, McAdams and Akers 1998, … • The remaining parameters were found using by fitting the model to data using least squares optimization

29. Results for Cro • During lysis, Cro out competes CI • During lysogeny, CI is able to out compete Cro

30. Results for CI • CI behaves similarily as Cro • The rapid decline of CI during lysis is a factor of CII, CIII, recA, …

31. Results for CII • During lysogeny, a rapid growth in CII ensures production of CI. Afterwards, CI once established, shuts down production of CII • During lysis, CII can not induce enough CI to establish CI over Cro

32. Conclusion • Combinatorial state equations simply incorporate several aspects of modeling • Boolean logic at genes • Continuity of mRNA transcription (i.e. saturation curves) • Within the framework of ODE’s • Generalize existing methodologies (i.e. Goodwin, …) • Allows for more general analysis of network behavior • Qualitatively “passes” the test of phage lambda

33. Future Work • Study the behavior of these systems • Relationship with master equation • Adding noise to the system • Describing network behavior

34. Acknowledgements • This work is supported by the NSF VIGRE grant