epi-scenario-6

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Scenario 6: Regnets for gene expression and regulation

A fundamental theory in molecular biology, known as the ‘the central dogma’, describes gene expression as the flow of information from DNA to RNA (through a step called transcription), and from RNA to proteins (through a step called translation). The processes of transcription, translation, and protein degradation, depend on the amount of RNA (specifically, mRNA), and proteins, and mathematically can be described as the rate of change of the concentrations of these materials. This scenario will explore several simple models of gene expression and regulation.

Chen Model: An early linear transcription model of gene expression was proposed by Chen et. al. in 1999, which considered only a feedback loop from proteins, to transcription, but ignores any feedback to translation. For the purpose of this scenario, we make the following modifications to the original model:

• We consider three types of mRNA and proteins
• Type 1 mRNA produce Type 1 proteins, which are involved in the transcription of all types of mRNA.
• Type 2 mRNA produce Type 2 proteins, and Type 3 mRNA produce Type 3 proteins. Type 2 and 3 proteins are not involved in transcription.

Figure 33. Modified version of the model by Chen et. al., which describes a dynamic system of gene regulation with feedback on transcription, but not translation.

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The modified Chen model is described in Figure 33 and can be expressed by the following set of differential equations:

$$\begin{align} \frac{dr_1}{dt} &= C_1 p_1 - V_1 r_1 \\ \frac{dr_2}{dt} &= C_2 p_1 - V_2 r_2 \\ \frac{dr_3}{dt} &= C_3 p_1 - V_3 r_3 \\ \frac{dp_1}{dt} &= L r_1 - U_1 p_1 \\ \frac{dp_2}{dt} &= L_2 r_2 - U_2 p_2 \\ \frac{dp_3}{dt} &= L_3 r_3 - U_3 p_3 \end{align}$$

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Where:

• $r_i$ represent the concentration of Type $i$ mRNA, in units $nMolar$
• $p_i$ represent the concentration of Type $i$ protein, in units $nMolar$
• $C_i$ represents the transcription rates for Type $i$ mRNA, in units $\frac{\text{nMolar of mRNA}}{\text{nMolar of protein} \cdot \text{minute}}$
• $L_i$ represents the translation rates for Type $i$ protein, in units $\frac{\text{nMolar of protein}}{\text{nMolar of mRNA} \cdot \text{minute}}$
• $V_i$ is the natural degradation rate of Type $i$ mRNA, in units $\frac{1}{\text{minute}}$
• $U_i$ is the natural degradation rate of Type $i$ protein, in units $\frac{1}{\text{minute}}$

Hunt Model: In this scenario, we also consider an extended model of gene expression, proposed by Hunt et. al., which incorporates multiple additional feedback loops into the protein synthesis process:

• Type 2 proteins help stabilize all mRNA against degradation (in effect lowers degradation rate)
• Each type of protein regulates the production of its own protein type, during both transcription and translation processes

Figure 44. Representation of the Hunt Model.

The Hunt Model is described in Figure 44, and defined by the following set of differential equations:

$$\begin{align} \frac{dr_1}{dt} &= \frac{1}{1 + \left(\frac{p_1^2}{a_1^2}\right)} C_1 p_1 - \frac{1}{1 + \frac{p_2}{b_1}} V_1 r_1 \\ \frac{dr_2}{dt} &= \frac{1}{1 + \frac{p_2}{a_2}} C_2 p_1 - \frac{1}{1 + \frac{p_2}{b_2}} V_2 r_2 \\ \frac{dr_3}{dt} &= \frac{1}{1 + \frac{p_3}{a_3}} C_3 p_1 - \frac{1}{1 + \frac{p_2}{b_3}} V_3 r_3 \\ \frac{dp_1}{dt} &= \frac{1}{1 + \frac{p_1}{d_1}} L_1 r_1 - U_1 p_1 \\ \frac{dp_2}{dt} &= \frac{1}{1 + \frac{p_2}{d_2}} L_2 r_2 - U_2 p_2 \\ \frac{dp_3}{dt} &= \frac{1}{1 + \frac{p_3}{d_3}} L_3 r_3 - U_3 p_3 \end{align}$$

Where:

• $r_i$ represent the concentration of Type $i$ mRNA, in units $nMolar$
• $p_i$ represent the concentration of Type $i$ protein, in units $nMolar$
• $C_i$ represents the transcription rates for Type $i$ mRNA, in units $\frac{\text{nMolar of mRNA}}{\text{nMolar of protein} \cdot \text{minute}}$
• $L_i$ represents the translation rates for Type $i$ protein, in units $\frac{\text{nMolar of protein}}{\text{nMolar of mRNA} \cdot \text{minute}}$
• $V_i$ is the natural degradation rate of Type $i$ mRNA, in units $\frac{1}{\text{minute}}$
• $U_i$ is the natural degradation rate of Type $i$ protein, in units $\frac{1}{\text{minute}}$
• $a_i$ is an effectiveness factor for feedback on transcription (i.e., suppresses the rate of transcription), for Type $i$ mRNA. The bigger the value of $a_i$, the smaller the impact of feedback and the less the transcription rate is decreased. The units are $nMolar$.
• $d_i$ is an effectiveness factor for feedback on translation (i.e., suppresses the rate of translation), for Type $i$ proteins. The bigger the value of $d_i$, the smaller the impact of feedback and the less the translation rate is decreased. The units are $nMolar$.
• $b_i$ is an effectiveness factor for feedback on mRNA degradation (i.e., Type 2 proteins act as stabilizers that lower the rate of degradation, for Type $i$ mRNA). The bigger the value of $b_i$, the smaller the impact of feedback and the less the degradation rate is decreased. The units are $nMolar$.

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Table 1. Initial and parameter values for Questions 1-2

i	$r_i (0)$ (nM)	$p_i (0)$ (nM)	$C_i$ ($\frac{\text{nM of mRNA}}{\text{nM of protein} \cdot \text{min}}$)	$L_i$ ($\frac{\text{nM of protein}}{\text{nM of mRNA} \cdot \text{min}}$)	$V_i$ (1/min)	$U_i$ (1/min)	$a_i$ (nM)	$b_i$ (nM)	$c_i$ (nM)
1	3	100	0.03	2	0.03	0.15	60	120	120
2	6	500	0.03	2	0.03	0.15	140	140	150
3	5	1	0.024	2	0.03	0.015	170	180	260

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1. Chen Model: Using the relevant values in Table 1 (not all will be applicable), simulate the Chen Model for a time period of 10 hours, and plot the trajectories of all mRNA and protein types. Comment on the nature of the trajectories and biological realism of the outcomes. Do the trajectories make sense given the model?

2. Simulations with the Hunt Model: For all parts, use the relevant values in Table 1, simulate for a time period of 10 hours, and plot the trajectories of all mRNA and protein types.
   a. Simulate the full Hunt Model. How do the outcomes compare with Q1?
   b. Now consider a Special Case 1, where we assume b_i,≫p_2 and d_i≫p_i (i.e., the stabilization of mRNA, and feedback from proteins to translation, are negligible). What does the set of differential equations reduce to? Repeat the simulation for this Case 1 Hunt Model. How do the outcomes compare with Q2a (the Full Hunt Model)? Do the results make sense given the assumptions made?
   c. Consider Special Case 2, where we assume a_i,d_i ≫pi (i.e., assume the feedback from proteins to transcription and translation, is negligible). What does the set of differential equations reduce to? Simulate this Case 2 Hunt Model. How do the outcomes compare with Q2a,b? Do the results make sense given the assumptions made?
   d. Consider Special Case 3, where we assume a_i,≫p_i and b_i≫p_2 (i.e., assume the feedback from proteins to transcription and stabilization of mRNA, are negligible). What does the set of differential equations reduce to? Simulate this Case 3 Hunt Model. How do the outcomes compare with Q2a-c? Do the results make sense given the assumptions made?
   e. Consider Special Case 4, where we assume d_i,≫p_i (i.e., assume the feedback from proteins to translation is negligible). What does the set of differential equations reduce to? Simulate this Case 4 Hunt Model, and if needed, extend the simulation length to ensure you can determine whether the system reaches an equilibrium state or not. How do the outcomes compare with Q2a-d? Do the results make sense given the assumptions made?

3. For Questions 1-2, some of the simulation outcomes should include exponential growth trajectories. For those cases, explore changes in Table 1 parameters that will result in changing the outputs from exponential growth trajectories to more bounded behavior. What is the biological interpretation of these parameter changes? (Note that there are multiple sets of parameters that will fulfill this criteria, and there is no single correct answer).

4. (Optional Challenge). For one or more of the systems considered in Q1-2, do a formal stability analysis (i.e., determine equilibrium points, their stability, and their biological interpretation).