h1. The Importance of Activator-Repressor Balance in Various Models of Sequestration for Circadian Clocks

h5. Mary Fletcher, Roxana Gheorghe, Olivia Lang, Stephanie Taylor

To understand protein networks, scientists use mathematical models to simulate the various molecules and their interactions. These mathematical models are differential equations for the concentrations of molecules over time.  When constructing a model, it is important to choose a level of abstraction that makes models as simple as possible, without losing their key robustness properties. For oscillating systems, the ability to maintain oscillations is key. In a recent publication Kim and Forger (2012) discuss the relationship between the ratio of repressor and activator proteins and the robustness of a system’s oscillations and conclude that a 1:1 ratio is best. Here, we examine three additional models of circadian processes (Leloup and Goldbeter, 2003; Mirsky et al, 2009; and Ueda et al 2000) and seek to determine which model structures lead to the situation in which a 1:1 repressor to activator ratio gives rise to a more robust system. For each of the models, we modify the mode of repression using three different levels of abstraction: explicit sequestration, implicit sequestration, and direct Hill repression.

h3. Background

h4. What is a circadian clock?

Plants and animals repeat certain biological process with a 24 hour period. The organism can use concentrations of certain proteins to determine the time of day.

h4. How do the concentrations change throughout the day?


\* change image to the image created in the presentation.

Caption: A model of simple repression where BMAL activates the transcription of the _Per_ gene and the PER protein represses the transcription of the _Per_ gene.

+Convention+: When reading mechanistic models or published papers concerning the expression of proteins, the convention is for the protein to be denoted by all caps and the gene that is being transcribed be denoted by italics. Lines with arrows indicate the element from the origin of the line is activating the element it is pointing to. Lines with “T”s at the end indicate the element from the origin of the line is repressing the element it is pointing to.

Existing proteins may be repressors and activators by regulating the rate at which other proteins are produced. This forms a gene regulatory network. When there are cycles in the network, periodic behavior can arise.


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\* insert sequestration figure

Caption: A simple model of sequestration

+Convention:+ The notation for a dimer (inactive complex) is often the two protein names with a slash between them. (e.g. PER/BMAL or PER/CRY)

*Sequestration* is a type of model that describes a mechanistic model in which the repressor indirectly represses the gene by repressing the activator. The activator controls the transcription of the gene and the repressor is not included in the transcription rate expression because its presence is accounted for in its manipulation of the activator concentration. The repressor controls activator concentration by binding with the activator to form an inactive complex, which cannot promote the transcription of the gene as shown in the figure above.

h4. How do you study the biological system?


Caption: The different colors show the concentrations of different states (proteins and mRNA) over time. Note the regular oscillations. Circadian clocks oscillate with period of 24 hour lengths.

Scientists develop mathematical models of the system and use computers to simulate the protein concentrations over time. The models say how the system is changing at any given state. In our case, we used differential expressions that describe the concentration of a given “state” or protein concentration based on the other state concentrations.


h4. What are we doing?

A recent paper by Kim and Forger discusses how an average 1:1 ratio between repressor and activator concentrations leads to the most robust systems. Kim and Forger termed this ratio property as the *stoichiometry* of a system.

We are interested in finding out whether this observation can be applied to all models. If it cannot, what conditions allow this observation to be true (i.e. “what is sequestration’s role in the relationship between stoichiometry and the robustness of a system?,” “is sequestration a condition for the stoichiometry observation to be true?”)?

h4. Circadian Models

We consider three models from other papers designed with different structures to compare the results of modifying the model to each sequestration model.

|| || Modeled after || Repression Model || Number of States ||
| Leloup and Goldbeter 2003 | Mammalian | Explicit sequestration | 16 |
| Mirsky 2009 | Mouse | Direct Hill | 21 |
| Ueda 2000 | _Drosophila_ | Direct Hill | 10 |

The “number of states” is the number of actors in our system and include mRNA and proteins. It indicates the complexity of a system, where more states results in a more complex system with a longer calculation time and fewer states will take less time to run.

h3. Repression Models

To simulate the circadian systems, we model the concentrations of PER protein and its additional repressors (CRY for mice, TIM for files) and activators (BMAL1 for mice, CLK/CYC for flies). Sequestration describes an interaction between the proteins where the activator and repressor bind together to form an inactive complex, thereby preventing each other from repressing or activating the transcription of DNA to form proteins.

h4. Explicit Sequestration

| This model gives the inactive complex (the bound repressor and activator) its own equation and returns the value of the concentration of the inactive complex in addition to the other states in the model. | !exlicitSeqEqn.png|border=1,width=150! | !explicit.png|border=1,width=200! |

h4. Implicit Sequestration

| By using the equation Kim and Forger used to modify the Goodwin model, we were able to simulate sequestration without giving the inactive complex its own state. !implicitSeqEqn.png|border=1,width=400! | !implicit.png|border=1,width=200! |

h4. Direct Hill Repression

| This model uses the repression equation found in many models and does not involve any sequestration. | \\  \\  !UedaHillEqn.png|border=1,height=150! \\  \\  \\  !MirskyHillEqn.png|border=1,height=150! | !ueda.png|border=1,width=200! \\  !mirsky.png|border=1,width=200! |

h3. Results

We ran the models with different parameter sets and recorded both the average ratio between repressors and activators and the amplitude of Per mRNA oscillation. High amplitude is an indicator that the model with those parameters is robust. We expect an ultrasensitive response for a stoichiometry of 1, which should create high-amplitude oscillations.


These are plots showing the relationship between average repressor/activator ratio and the amplitude of Per mRNA.  For each of 9 models, we ran 500 trials with different parameter sets.  Each x or . on the plots corresponds to one parameter set.  Notice that the Leloup with explicit sequestration has a very clear spike in amplitude for 1:1 stoichiometry.  Leloup with hill kinetics demonstrates this relationship to a lesser extent.  On its own, this would suggest that the explicit sequestration model of repression is the best at producing the desired relationship and the implicit model is worst.  The other basic circadian frameworks do not support this broad statement, however, so there must be more factors involved.