Combinatorial State Equations and Gene Regulation

Jay Raol and Steven J. Cox

Computational and Applied MathematicsRice University

Outline• Phage Lambda

• Combinatorial State Equations

• Applications to Phage Lambda

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

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 genome1. Lytic Phase2. Lysogenic Phase

from Ptashne, “A Genetic Switch”

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

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

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DNA-protein Interactions

• Let us make several assumptions regarding the nature of proteins and the production of mRNA

1. mRNA production is determined by the (active) state of the gene

2. mRNA production reaches a maximal rate

3. Transcription Factors (TF) act independently of each other

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

Gene States

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

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?

• 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

Defining the state of gene A

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

Cooperativity

• There are 3! possible states

from Ptashne, “A Genetic Switch” 1992

Complex States

• Calculating P(sn) can become tedious for large number of proteins.

• P(sn) can be generalized to incorporate a simple statistical formulation

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

Boolean Logic

• Similarly, we find that

• For example, if for proteins A or B were required for activation using only one binding spot, then

ODE Model

• Keep track of everyway mRNA is produced and destroyed1. Transcription

2. Degradation

• Similar assumptions for proteins (Goodwin) apply

Full system

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

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

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

Phage Lambda Model

Phage λ: entire system

from McAdams and Shapiro, Science 1995

Filling in the Kinetics

• Consider the regulation at the PL promotor

• Using the CSE, we can describe the regulation of N

Kinetics of CI

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

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.

Anti-sense mRNA

• The production of anti-sense mRNA is

• Anti-sense mRNA q by CII inhibit the production Q

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

Results for Cro

• During lysis, Cro out competes CI

• During lysogeny, CI is able to out compete Cro

Results for CI

• CI behaves similarily as Cro

• The rapid decline of CI during lysis is a factor of CII, CIII, recA, …

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

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

Future Work

• Study the behavior of these systems– Relationship with master equation– Adding noise to the system– Describing network behavior

Acknowledgements

• This work is supported by the NSF VIGRE grant