A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells

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.title[
# A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells
]
.author[
### <strong>Max Chumley</strong> <br> — <br> Mechanical Engineering <br> Michigan State University <br> —
]
.date[
### 5-16-2023
]

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# Acknowledgements 
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<div style="height: 25px;"></div>
<p style="font-size:9px">&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;</p>
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<p style="text-align: left; font-size:20px">This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550- 30 22-1-0007</p>

<p style="font-size:9px">&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;</p>
]

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# Protein Synthesis

<!-----------------
  Script: 
  An essential component in living cells is the ability to make proteins for the organism. This mainly occurs through protein synthesis in the cell where a pool of ribosomes utilize resources in the cell and traverse the messenger ribonucleic acid or mRNA strands. These strands contain the necessary code for synthesizing a protein and the ribosomes read this code to make the protein as the mRNA is traversed. 
  
  It has been observed in experimental settings that when the cell is starved of resources it will exhibit metabolic cycling where the protein production rates oscillate.

We aimed to model this process as a time delay system due to the metabolic cycling process taking approximately 40 minutes in experiments. 
  
  This animation demonstrates ribosomes traversing the mRNA for three proteins and when they reach the end the protein is produced.
----------------->

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# Outline
![](data:image/png;base64,#../Figs/1ptn_outline.png)
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# Outline

![](data:image/png;base64,#../Figs/3ptn_outline.png)

<!-----------------
  Script: 
  Here is an outline for my talk today. I will show how we obtained a single protein model and show the three methods that we used for searching for limit cycles in this system.

I will then show how we extended the model to include three coupled proteins with a shared resource pool and apply the same methods for analysis.
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# 1-Protein Model

- Ribosome Initiation: `$\mu(t)=f(p(t))R(t)$`

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- Hill Function: `$f(x)=\frac{x^n}{\kappa^n+x^n}$`

- `$\dot{p}(t) = B\mu(t-\tau) - Dp(t)$`

- `$R(t) = R_T - A\int_{t-\tau}^t\mu(s)ds$` `$^1$`
  
.footnote[1.L. M. y Ter ́an-Romero, M. Silber, and V. Hatzimanikatis, “The origins of time-delay in template biopolymeriza-
tion processes,” PLoS Computational Biology, vol. 6, p. e1000726, apr 2010]
--

- `$\dot{R}(t) = A(f(p(t-\tau))R(t-\tau) - f(p(t))R(t))$`

Single Protein System:
`\begin{align}
  \dot{p}(t) &= Bf(p(t-\tau))R(t-\tau) - Dp(t),\\
  \dot{R}(t) &= A(f(p(t-\tau))R(t-\tau) - f(p(t))R(t))
\end{align}`

<!-----------------
  Script: 
  We will now discuss how we obtained our model for this system. 
  The rate of ribosome initiation is the product of the activator the f(p(t)) term and the processing  molecule or resource R(t). The activator is a hill function where if the input is sufficiently large its value approaches one and it is close to zero if the input is close to zero. The hill function parameters change the rate of initiation for the activation.

The rate of protein production is then proportional to the initiation at some time tau seconds in the past where the delay is the required time for protein synthesis. The parameter B is the growth rate and we also include the decay term here with decay rate D.

The total resource equation has been suggested as this equation where the resource at any time t is given by the total resource minus the integral of the initiation over the production time tau.

Differentiating this expression and using the initiation definition gives the second equation in our model and we write the single protein system as these two equations. 
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# 1-Protein Equilibrium Conditions

`\begin{equation*}
	(1+A \tau){p^*}^{n+1}-\frac{BR_T}{D}{p^*}^{n}+\kappa^n{p^*}=0
\end{equation*}`

`\begin{equation*}
        R^*=\frac{R_T}{1+A\tau f(p^*)} ~~~~~~~~\left(R(t) = R_T - A\int_{t-\tau}^t\mu(s)ds\right)
\end{equation*}`

- Trivial Equilibrium: `$(p^*, R^*) = (0, R_T)$`

- *At least* 1 real solution `$\forall n \in \mathbb{Z}$`

--
- *At most* 3 real solutions `$\forall n \in \mathbb{Z}$`
--

- Descartes' rule of signs
  - Assuming all positive parameters

<!-----------------
  Script: 
  The next step was to compute equilibrium conditions for this model. We do this by setting the derivative terms to zero in the model. We obtain this polynomial expression in terms of p from the first equation, and the second condition is obtained from the total resource equation.

We see that these equations have a trivial solution at 0 production rate and the total resource. Further, by analyzing the coefficients of the polynomial, we see that this system will have at most 3 positive real solutions for all n and at least one solution at the trivial equilibrium.
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# 1-Protein Equilibrium Conditions

- For `$n=2$`:

<font color=blue>Trivial</font>

`$(p^*, R^*)=(0,R_T)$`,

<font color=red>Middle</font>
`$(p^*, R^*)=\left(\frac{B R_T-\sqrt{B^2 R_T^2-4 D^2 \kappa ^2 (A \tau +1)}}{2 D(A  \tau +1)}, \frac{B R_T \sqrt{B^2 R_T^2-4 D^2 \kappa ^2 (A \tau +1)}-2 A D^2 \kappa ^2 \tau  (A \tau +1)-B^2 R_T^2}{B (A \tau +1) \left(\sqrt{B^2 R_T^2-4 D^2 \kappa ^2 (A \tau +1)}-B R_T\right)}\right)$`

<font color=green>Top</font>
`$(p^*, R^*)=\left(\frac{B R_T+\sqrt{B^2 R_T^2-4 D^2 \kappa ^2 (A \tau +1)}}{2 D(A  \tau +1)}, \frac{B R_T \sqrt{B^2 R_T^2-4 D^2 \kappa ^2 (A \tau +1)}+2 A D^2 \kappa ^2 \tau  (A \tau +1)+B^2 R_T^2}{B (A \tau +1) \left(\sqrt{B^2 R_T^2-4 D^2 \kappa ^2 (A \tau +1)}+B R_T\right)}\right)$`

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# 1-Protein Linearization

`$$\dot{\vec{y}}\approx\begin{bmatrix}
      -D & 0\\
      -A f'(p^*)R^* & -A f(p^*)
    \end{bmatrix} \vec{y}+\begin{bmatrix}
      B f'(p^*)R^* & B f(p^*)\\
      Af'(p^*)R^* & Af(p^*)
    \end{bmatrix}\vec{y}_\tau$$`

`$$\vec{y} = \begin{bmatrix}p(t)-p^*\\ R(t)-R^*\end{bmatrix},~~~
        \vec{y}_\tau = \begin{bmatrix}p(t-\tau)-p^*\\ R(t-\tau)-R^*\end{bmatrix}$$`

- Trivial Equilibrium `$(0,R_T)$`:

`$$\dot{\vec{y}}\approx\begin{bmatrix}
		-D & 0\\
		0 & 0
	\end{bmatrix} \vec{y}$$`

<font color=red>Nonhyperbolic Equilibrium</font>

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# Featurized Stability Analysis

- Compute a feature `$M: \mathbb{R}^n\rightarrow \mathbb{R}$` on the nonlinear simulations.

- Color a 2D projection of the parameter space using the feature.

</br>
![](data:image/png;base64,#../figs/features.png)