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{% extends "layout.html" %}
{% block title %}Model{% endblock %}
{% block page_content %}
<div class="sidebar">
<ul>
<li><a href="#description">General Description of Modeling</a></li>
<li><a href="#topic1">Compartment Model for Muscone Inhalation</a></li>
<li><a href="#topic2">Topic2</a></li>
<li><a href="#topic3">ODE of MAPK pathway</a></li>
<li><a href="#topic4">Lactic Acid Absorption Model</a></li>
</ul>
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<h2 id="description">General Description of Modeling</h2>
<hr>
<p>Our model serves three main purposes:</p>
<ol>
<li><strong>Quantitative Description of Project Design</strong>: It aids readers in understanding the
processes involved in our project design, enabling a better narration of our story.</li>
<li><strong>Computational Methods for Project Engineering</strong>: If the project proceeds, the model
can assist in determining parameters during project implementation, reducing computational effort
during experimentation.</li>
<li><strong>Reduction of Wet Lab Experiment Scale</strong>: The model can predict the behavior of the
system under study, helping to eliminate unnecessary trials in wet lab experiments.</li>
</ol>
<p>Our model is divided into four interconnected parts, representing the inhalation of muscone, its binding
to receptors, intracellular signal transduction and lactic acid secretion triggered by receptor
activation, and the absorption of lactic acid. These models provide a comprehensive understanding of the
project and yield valuable computational results.</p>
</div>
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<h2 id="topic1">
<h2>Compartment Model for Muscone Inhalation</h2>
<hr>
<h3>Model Description</h3>
<p>The main focus of our project is the use of muscone as a signaling molecule to activate engineered
bacteria in the gut for therapeutic purposes. Therefore, it is crucial to provide a quantitative
description and computational support for the diffusion of muscone in the body. This model describes
the entire process from the inhalation of muscone to its increased concentration in the intestinal
tract. We will establish a multi-compartment model that includes the following main processes:</p>
<ol>
<li><strong>Inhalation Process</strong>: Muscone is inhaled in the form of an aerosol into the
lungs.</li>
<li><strong>Pulmonary Process</strong>: Muscone distributes in the alveoli and may be exhaled,
adhered to, or permeated into the microvessels.</li>
<li><strong>Adhesion Process</strong>: A portion of muscone adheres to the respiratory mucosa and
then diffuses into the systemic circulation.</li>
<li><strong>Alveolar Microvessel Process</strong>: Muscone permeates into the alveolar microvessels
and gradually enters the systemic circulation.</li>
<li><strong>Systemic Circulation Process</strong>: Muscone distributes in the systemic circulation
and is transported to various parts of the body through the bloodstream.</li>
<li><strong>Intestinal Process</strong>: Muscone enters the target intestine through the mesenteric
microvascular network, where its concentration begins to increase.</li>
</ol>
<p>TODO:Insert design diagram</p>
<p>Corresponding to the above processes, five compartments need to be established for simulation, where
$t$ represents the time variable:</p>
<li><strong>Compartment 0</strong> (Alveolar Space, \(A\)): \(Q_A(t)\) represents the amount of
muscone
in the alveoli (mg).</li>
<li><strong>Compartment 1</strong> (Respiratory Mucosa, \(M\)): \(Q_M(t)\) represents the amount of
muscone adhered to the respiratory mucosa (\(\text {mg}\)).</li>
<li><strong>Compartment 2</strong> (Alveolar Capillaries, \(L\)): \(Q_L(t)\) represents the amount of
muscone in the alveolar capillaries (\(\text{mg}\)).</li>
<li><strong>Compartment 3</strong> (Systemic Circulation, \(C\)): \(Q_C(t)\) represents the amount of
muscone in the systemic circulation(\(\text{mg}\)).</li>
<li><strong>Compartment 4</strong> (Target Intestine, \(I\)): \(Q_I(t)\) represents the amount of
muscone in the intestine(\(\text{mg}\)).</li>
<h3>Initial Settings and Assumptions</h3>
<p>At \(t=0\), the amount of muscone in all compartments is \(0\).</p>
<p>Assuming that the total amount of inhaled muscone is \(Q_{\text{inhale}}\) (\(\text{mg}\)), which is
assumed to be \(100\text{mg}\). Only \(0.5\%\) of muscone enters the systemic circulation through
adhesion. In this model, since muscone only acts as a signaling molecule to activate yeast to
synthesize lactic acid, we only consider the metabolism and excretion of muscone in the systemic
circulation. We only focus on the short-term process of muscone appearing in the intestine from
scratch, and the subsequent process of reaching a certain concentration can be ignored.</p>
<h3>Model Equations</h3>
<h4>Inhalation Equation for Muscone</h4>
<p>
\( V_{\text{inhale}}(t) =\frac{Q_{\text{inhale}}}{5}(u(t)-u(t-5)) \)
</p>
<p><strong>Explanation</strong>：This describes the rate equation for inhaling muscone over five seconds,
where the total amount \( Q \) remains constant. The function \( u(t) \) is a step function, which
takes the value of \( \frac{Q_{\text{inhale}}}{5} \) from \( t=0s \) to \( t=5s \), and is \( 0 \)
otherwise, simulating the scenario of resting human respiration.</p>
<h4>Compartment 0: \( Q_A(t) \)</h4>
<p>
\( \frac{dQ_A(t)}{dt} = V_{\text{inhale}}(t) - \left( k_{\text{exhale}} + k_{\text{perm}} \right)
Q_A(t) \)
</p>
<p><strong>Explanation</strong>: The amount of muscone in the alveoli increases through inhalation and
decreases due to exhalation, adhesion to the respiratory mucosa, and permeation into the alveolar
capillaries.</p>
<p><strong>Parameters</strong>:</p>
<ul>
<li>
\( k_{\text{exhale}} \): Since most of the muscone is rapidly exhaled, this value is relatively
large, taken as \( 10 \ \text{min}^{-1} \)
</li>
<li>
\( k_{\text{perm}} \): The rate of muscone permeation into the capillaries, affected by its
physicochemical properties, is taken as \( 0.005 \ \text{min}^{-1} \)
</li>
</ul>
<h4>Compartment 1: \( Q_M(t) \)</h4>
<p>
\( \frac{dQ_M(t)}{dt} = 0.0005 \cdot k_{\text{adh}} V_{\text{inhale}}(t) - k_{\text{diffMC}} Q_M(t)
\)
</p>
<p><strong>Explanation</strong>: The increase in muscone on the mucosa comes from adhesion in the
alveoli, and the decrease is due to diffusion into the systemic circulation.</p>
<p><strong>Parameters</strong>:</p>
<ul>
<li>
\( k_{\text{adh}} \): The adhesion process is relatively slow, and only \( 0.5\% \) of muscone
enters the systemic circulation through this pathway, taken as \( 0.001 \ \text{min}^{-1} \)
</li>
<li>
\( k_{\text{diffMC}} \): Diffusion from the mucosa to the systemic circulation is slow, taken as
\( 0.01 \ \text{min}^{-1} \)
</li>
</ul>
<h4>Compartment 2: \( Q_L(t) \)</h4>
<p>
\( \frac{dQ_L(t)}{dt} = k_{\text{perm}} Q_A(t) - k_{\text{diffLC}} Q_L(t) \)
</p>
<p><strong>Explanation</strong>: The increase in muscone in the alveolar capillaries comes from
permeation in the alveoli, and the decrease is due to diffusion into the systemic circulation.</p>
<p><strong>Parameters</strong>:</p>
<ul>
<li>
\( k_{\text{perm}} \): Same as Compartment 0
</li>
<li>
\( k_{\text{diffLC}} \): The diffusion rate from alveolar capillaries to the systemic
circulation is relatively slow, taken as \( 0.05 \ \text{min}^{-1} \)
</li>
</ul>
<h4>Compartment 3: \( Q_C(t) \)</h4>
<p>
\( \frac{dQ_C(t)}{dt} = k_{\text{diffMC}} Q_M(t) + k_{\text{diffLC}} Q_L(t) - k_{\text{dist}}
Q_C(t) - k_{\text{excrete}} Q_C(t) \)
</p>
<p><strong>Explanation</strong>: The increase in muscone in the systemic circulation comes from the
input of mucosa and alveolar capillaries, and the decrease is due to distribution to the intestinal
mesenteric microvascular network and excretion through various routes.</p>
<p><strong>Parameters</strong>:</p>
<ul>
<li>
\( k_{\text{diffMC}} \): Same as Compartment 1
</li>
<li>
\( k_{\text{diffLC}} \): Same as Compartment 2
</li>
<li>
\( k_{\text{dist}} \): The rate constant of muscone distribution from the systemic circulation
to the intestinal mesenteric microvascular network, taken as \( 0.001 \ \text{min}^{-1} \)
</li>
<li>
\( k_{\text{excrete}} \): Muscone is excreted from the systemic circulation through epidermal
volatilization, urine, continuous respiration, etc., taken as \( 0.05 \ \text{min}^{-1} \)
</li>
</ul>
<h4>Compartment 4: \( Q_I(t) \)</h4>
<p>
\( \frac{dQ_I(t)}{dt} = k_{\text{dist}} Q_C(t) - k_{move}Q_I(t) \)
</p>
<p><strong>Explanation</strong>: The increase in muscone in the intestine comes from the distribution of
the systemic circulation, and the decrease is due to metabolism and excretion through intestinal
fluid and peristalsis.</p>
<p>
\( k_{\text{dist}} \): Same as above<br>
\( k_{move} \): The metabolism and excretion of muscone in the intestine, taken as \( 0.02 \
\text{min}^{-1} \)
</p>
<h3>System of Equations:</h3>
<p>In summary, we can write a system of ordinary differential equations and import it into MATLAB for
simulation:</p>
<p>
\( Q_{\text{inhale}}(t)=100(mg)(Assumption) \)
</p>
<p>
\( V_{\text{inhale}}(t) =\frac{Q_{\text{inhale}}}{5}(u(t)-u(t-5)) \)
</p>
<p>
\( \frac{dQ_A(t)}{dt} = V_{\text{inhale}}(t) -\left( k_{\text{exhale}} + k_{\text{perm}} \right)
Q_A(t) \)
</p>
<p>
\( \frac{dQ_L(t)}{dt} = k_{\text{perm}} Q_A(t) - k_{\text{diffLC}} Q_L(t) \)
</p>
<p>
\( \frac{dQ_M(t)}{dt} = 0.0005\cdot k_{\text{adh}} V_{\text{inhale}}(t) - k_{\text{diffMC}} Q_M(t)
\)
</p>
<p>
\( \frac{dQ_C(t)}{dt} = k_{\text{diffMC}} Q_M(t) + k_{\text{diffLC}} Q_L(t) - k_{\text{dist}}
Q_C(t) - k_{\text{excrete}} Q_C(t) \)
</p>
<p>
\( \frac{dQ_I(t)}{dt} = k_{\text{dist}} Q_C(t)-k_{move}Q_I(t) \)
</p>
<p>TODO：插入结果图</p>
<p>We simulated the distribution of muscone in the systemic circulation and obtained the concentration
change curve of muscone in the systemic circulation. According to the model, after one breath,
traces of muskone can spread into the intestine, similarly, the concentration change caused by
continuous muskone is simulated by changing the inhalation equation, and the concentration of
muskone in the intestine can be obtained in combination with experimental determination. Because
there is no animal experimental support, the data are manually drafted, and the calculation method
is more meaningful than the calculation results.</p>
<button id="Button1" onclick="toggleCodeSnippet()">Expand the code</button>
<div id="codeSnippet" class="code-snippet">
% Define parameters
Q_inhale = 100; % mg
k_exhale = 10;
k_perm = 0.005;
k_adh = 0.001;
k_diffMC = 0.01;
k_diffLC = 0.05;
k_dist = 0.001;
k_excrete = 0.05;
k_move = 0.02;
% Define the time range
tspan = [0 300]; % From 0 to 5 minutes
initial_conditions = [0 0 0 0 0]; % The initial condition is 0
% solve ODE
[t, y] = ode45(@(t,y) odefun(t, y, Q_inhale, k_exhale, k_perm, k_adh, k_diffMC, k_diffLC, k_dist,
k_excrete, k_move), tspan, initial_conditions);
% calculate V_inhale
V_inhale = Q_inhale / 5 * (heaviside(t) - heaviside(t-5));
figure('Position', [100, 100, 1200, 1000]);
% V_inhale(t)
subplot(3,2,1)
plot(t, V_inhale)
title('V_{inhale}(t)')
xlabel('Time (s)')
ylabel('V_{inhale}')
% Q_A(t)
subplot(3,2,2)
plot(t, y(:,1))
title('Q_A(t)')
xlabel('Time (s)')
ylabel('Q_A')
% Q_L(t)
subplot(3,2,3)
plot(t, y(:,2))
title('Q_L(t)')
xlabel('Time (s)')
ylabel('Q_L')
% Q_M(t)
subplot(3,2,4)
plot(t, y(:,3))
title('Q_M(t)')
xlabel('Time (s)')
ylabel('Q_M')
% Q_C(t)
subplot(3,2,5)
plot(t, y(:,4))
title('Q_C(t)')
xlabel('Time (s)')
ylabel('Q_C')
% Q_I(t)
subplot(3,2,6)
plot(t, y(:,5))
title('Q_I(t)')
xlabel('Time (s)')
ylabel('Q_I')
sgtitle('Simulation Results')
% ODE
function dydt = odefun(t, y, Q_inhale, k_exhale, k_perm, k_adh, k_diffMC, k_diffLC, k_dist,
k_excrete, k_move)
V_inhale = Q_inhale / 5 * (heaviside(t) - heaviside(t-5));
dydt = zeros(5,1);
dydt(1) = V_inhale - (k_exhale + k_perm) * y(1); % dQ_A/dt
dydt(2) = k_perm * y(1) - k_diffLC * y(2); % dQ_L/dt
dydt(3) = 0.0005 * k_adh * V_inhale - k_diffMC * y(3); % dQ_M/dt
dydt(4) = k_diffMC * y(3) + k_diffLC * y(2) - k_dist * y(4) - k_excrete * y(4); % dQ_C/dt
dydt(5) = k_dist * y(4) - k_move * y(5); % dQ_I/dt
end
</div>
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<h2 id="topic2">
<h2>Topic2</h2>
<hr>
<p>Tsinghua University engages in extensive research and offers 51 bachelor's degree programs, 139
master's degree programs, and 107 doctoral programs through 20 colleges and 57 departments covering
a broad range of subjects, including science, engineering, arts and literature, social sciences,
law, medicine. Along with its membership in the C9 League, Tsinghua University affiliations include
the Association of Pacific Rim Universities, a group of 50 leading Asian and American universities,
Washington University in St. Louis's McDonnell International Scholars Academy, a group of 35 premier
global universities, and the Association of East Asian Research Universities, a 17-member research
collaboration network of top regional institutions. Tsinghua is an associate member of the
Consortium Linking Universities of Science and Technology for Education and Research (CLUSTER).
Tsinghua is a member of a Low Carbon Energy University Alliance (LCEUA), together with the
University of Cambridge and the Massachusetts Institute of Technology (MIT).</pp>
<p>School of Life Sciences was first established in 1926 under the name Department of Biology. Botanist
Qian Chongshu took up the first dean.During the nationwide reorganization of universities in the
early 1950s, the Department of Biology was merged into other universities, namely Peking University
etc., resulting in a vacancy in the field of biological research in Tsinghua for almost 30 years.In
June 1984, decisions were made about the reestablishment of the Department of Biology, and the
department officially reopened in September. During the reestablishment the Department of Biology of
Peking University, the Institute of Biophysics of Chinese Academy of Sciences, and many other
institutes as well as biologists provided valuable support and help. The department changed its name
to the current name in September 2009. As of 2013, structural biologist and foreign associate of
National Academy of Sciences of United States Dr. Wang Hongwei (王宏伟) is the current dean of School
of Life Sciences. The school currently has 129 professors and employees, around 600 undergraduates
(including the candidates of Tsinghua University – Peking Union Medical College joint MD program).
</p>
</div>
</div>
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<h2 id="topic3">
<h2>Ordinary Differential Equation of the signal transduction of the yeast MAPK pathway</h2>
<hr>
<h3>Model Description</h3>
<p>In our project, we express the muscone receptor (GPCR) on the yeast cell membrane. After a
certain concentration of muscone diffuses into the intestine and binds to the receptor, it
activates the receptor, which in turn activates the G protein. The G protein dissociates into α and
βγ subunits, with the βγ subunit releasing and activating Ste20 and the scaffold protein Ste5. Ste5
can undergo oligomerization and other behaviors, recruiting Ste11, Ste7, and Fus3 near the plasma
membrane. The cascade reaction is initiated by Ste20, and the signal is transmitted along the
Ste11-Ste7-Fus3 cascade. Fus3 activates the transcription factor pFUS1, and the downstream gene is
LahA, which expresses lactate dehydrogenase LDH, catalyzing the conversion of pyruvate to lactate.
This model simulates the changes in the concentrations and phosphorylation states of molecules in
the signaling transduction pathway by writing out chemical reactions and converting them into
ordinary differential equations, in order to obtain the quantitative relationship between muscone
activation and lactate secretion. The model includes the following main processes:</p>
<ol>
<li><strong>Activation of Muscone Receptor</strong>: The muscone receptor Ste2, derived from
mouse olfactory epithelium, is a G protein-coupled receptor (GPCR) that is expressed on the cell
membrane and receives signals. Its domains consist of α, β, and γ, where the Gα subunit is
called Gpa1, and the Gα and Gγ subunits are Ste4 and Ste18, respectively, both anchored in the
cell membrane, without discussing the scenario of their separation. After binding with muscone,
Gpa1 will release Ste4-Ste18.</li>
<li><strong>Formation of Scaffold</strong>: The released Ste4-Ste18 can bind to Ste5, and the Ste5
protein can undergo dimerization, oligomerization, and other behaviors, forming a scaffold near
the cell membrane and recruiting proteins related to the cascade phosphorylation.</li>
<li><strong>Cascade Reaction</strong>: The scaffold composed of Ste5 can recruit Ste11 (MAPKKK),
Ste7 (MAPKK), and Fus3 (MAPK). Each of these three proteins has multiple phosphorylation
modification sites, and the efficiency of catalyzing phosphorylation varies under different
modification scenarios. Furthermore, the three proteins independently bind to Ste5, and a
reaction can only occur when two adjacent proteins are simultaneously present on the scaffold,
making this signaling pathway highly specific.</li>
<li><strong>Activation of pFUS1</strong>: The transcription factor pFUS1 is activated by Fus3, and
the downstream gene is LahA, which expresses lactate dehydrogenase to produce lactate.</li>
</ol>
<h3>Basic Assumptions</h3>
<ol>
<li>Since the model only simulates the signal transduction shortly after muscone activation, it
does not consider protein synthesis and degradation, assuming that the concentrations of each
protein remain stable during this time.</li>
<li>It is assumed that all proteins involved in the cascade reaction have the same dephosphorylation
rate, denoted by \(k_{cat_{dephosph}}\).</li>
<li>The behavior of all molecules in the system is random and not influenced by environmental
factors.</li>
</ol>
<h3>Model Equations</h3>
<h4>Activation of muscone Receptor</h4>
<strong>Reactions</strong>:
<div>
<p>
\[
\begin{align*}
\text{Pheromone} + \text{Ste2} & \rightarrow \text{PheromoneSte2} \\
\text{PheromoneSte2} & \rightarrow \text{Pheromone} + \text{Ste2} \\
\text{PheromoneSte2} + \text{Gpa1Ste4Ste18} & \rightarrow \text{PheromoneSte2Gpa1Ste4Ste18} \\
\text{PheromoneSte2Gpa1Ste4Ste18} & \rightarrow \text{PheromoneSte2Gpa1} + \text{Ste4Ste18} \\
\text{PheromoneSte2Gpa1} & \rightarrow \text{PheromoneSte2} + \text{Gpa1} \\
\text{Gpa1} + \text{Ste4Ste18} & \rightarrow \text{Gpa1Ste4Ste18}
\end{align*}
\]
</p>
</div>
<strong>Explanation</strong>
<p>
After Ste2 binds with muscone, it interacts with the G protein, causing the exchange of GDP
bound to the G protein with GTP in the cytoplasm, releasing Ste4 and Ste18. After Gpa1 catalyzes the
conversion of GTP to GDP, it can return to the cytoplasm and rebind, forming a G protein trimer.
Since the original signaling pathway is the yeast pheromone signaling pathway, with the ligand being
the pheromone, this section uses Pheromone to represent the molecules that activate the receptor.
</p>
<strong>Ordinary Differential Equations</strong>
<div>
<p>
\[
\begin{align*}
\frac{d{P}}{dt} & = k_{off_{PS}}{PS} - k_{on_{PS}}{P}*{S} \\
\frac{d{S}}{dt} & = k_{off_{PS}}{PS} - k_{on_{PS}}{P}*{S} \\
\frac{d{PS}}{dt} & = k_{on_{PS}}{P}*{S} + k_{off_{SG}} {PSG} \\
& \quad - k_{off_{PS}}{PS} - k_{on_{SG}}{PS} * {GSS} \\
\frac{d{GSS}}{dt} & = k_{on_{GS}}{SS} * {G} - k_{on_{SG}}{PS} * {GSS} \\
\frac{d{PSGSS}}{dt} & = k_{on_{SG}}{PS} * {GSS} - k_{on_{GS}}{PSGSS} \\
\frac{d{PSG}}{dt} & = k_{on_{GS}}{PSGSS} - k_{off_{SG}} {PSG} \\
\frac{d{SS}}{dt} & = k_{on_{GS}}{PSGSS} - k_{on_{GS}}{SS} * {G} \\
\frac{d{G}}{dt} & = k_{off_{SG}} {PSG} - k_{on_{GS}}{SS} * {G} \\
\end{align*}
\]
</p>
</div>
<strong>Variables</strong>
<table>
<thead>
<tr>
<th>Variable</th>
<th>Represents Molecule</th>
<th>Concentration (\(\mu M\))</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(P\)</td>
<td>Pheromone</td>
<td>-</td>
</tr>
<tr>
<td>\(S^*\)</td>
<td>Ste2</td>
<td>0.287</td>
</tr>
<tr>
<td>\(PS\)</td>
<td>PheromoneSte2</td>
<td>-</td>
</tr>
<tr>
<td>\(GSS\)</td>
<td>Gpa1Ste4Ste18</td>
<td>-</td>
</tr>
<tr>
<td>\(PSGSS\)</td>
<td>PheromoneSte2Gpa1Ste4Ste18</td>
<td>-</td>
</tr>
<tr>
<td>\(PSG\)</td>
<td>PheromoneSte2Gpa1</td>
<td>-</td>
</tr>
<tr>
<td>\(SS^*\)</td>
<td>Ste4Ste18</td>
<td>\(2\times 10^{-4}\)</td>
</tr>
<tr>
<td>\(G^*\)</td>
<td>Gpa1</td>
<td>\(2\times 10^{-4}\)</td>
</tr>
</tbody>
</table>
<strong>Parameters</strong>
<table>
<thead>
<tr>
<th>Parameter</th>
<th>Meaning</th>
<th>Value</th>
<th>Unit</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(k_{on_{PS}}^*\)</td>
<td>Binding rate of Pheromone to Ste2</td>
<td>\(0.185\)</td>
<td>\({\mu M}^{-1} \cdot s^{-1}\)</td>
</tr>
<tr>
<td>\(k_{off_{PS}}^*\)</td>
<td>Dissociation rate of PheromoneSte2</td>
<td>\(1 \times 10^{-3}\)</td>
<td>\(s^{-1}\)</td>
</tr>
<tr>
<td>\(k_{on_{SG}}\)</td>
<td>Binding rate of PheromoneSte2 to Gpa1Ste4Ste18</td>
<td>-</td>
<td>\({\mu M}^{-1} \cdot s^{-1}\)</td>
</tr>
<tr>
<td>\(k_{off_{SG}}\)</td>
<td>Dissociation rate of PheromoneSte2Gpa1</td>
<td>-</td>
<td>\(s^{-1}\)</td>
</tr>
<tr>
<td>\(k_{on_{GS}}\)</td>
<td>Binding rate of Gpa1 to Ste4Ste18</td>
<td>-</td>
<td>\({\mu M}^{-1} \cdot s^{-1}\)</td>
</tr>
<tr>
<td>\(k_{off_{GS}}\)</td>
<td>Dissociation rate of PheromoneGpa1Ste4Ste18</td>
<td>-</td>
<td>\(s^{-1}\)</td>
</tr>
</tbody>
</table>
<strong>Initial Conditions</strong>
<p>
There are \(1{\mu M}\) of Pheromone and \(1{\mu M}\) of inactive G proteins. Known variables are
entered, other variables are set to zero, and unknown parameters are defined. After starting the
simulation, reactions occur according to the equations listed.
</p>
<p>TODO：插入结果图</p>
<h4>Formation of the Scaffold</h4>
<strong>Reactions</strong>:
<div>
\[
\begin{align*}
Ste5 + Ste5 & \leftrightarrows Ste5Ste5 \\
Ste4Ste18Ste5 + Ste5 & \leftrightarrows Ste4Ste18Ste5Ste5 \\
Ste4Ste18Ste5 + Ste4Ste18Ste5 & \leftrightarrows Ste4Ste18Ste5Ste5Ste4Ste18 \\
Ste4Ste18 + Ste5 & \leftrightarrows Ste4Ste18Ste5 \\
Ste4Ste18 + Ste5Ste5 & \leftrightarrows Ste4Ste18Ste5Ste5 \\
Ste4Ste18 + Ste4Ste18Ste5Ste5 & \leftrightarrows Ste4Ste18Ste5Ste5Ste4Ste18 \\
\end{align*}
\]
</div>
<strong>Explanation</strong>: The binding of Ste4Ste18 with Ste5 and the oligomerization of Ste5 is a
process that is not completely independent. Many equations can be derived through combinations, but here
we only consider the dimerization process, and each reaction is reversible. Since Ste5 actually binds to
Ste4, we abbreviate Ste5 as S5 and Ste4 as S4 in the equations.
<strong>Ordinary Differential Equations</strong>:
<div>
\[
\begin{align*}
\frac{d{S5}}{dt} & = -2 k_{on_{S5:S5}}{S5}^2 + 2 k_{off_{S5:S5}}{S55} \\
& \quad -k_{on_{S4:S5}}{S5}*{S4} + k_{off_{S4:S5}}{S45} \\
& \quad -k_{on_{S4S5:S5}}{S5}*{S45}+k_{off_{S4S5:S5}}{S5}*{S455}\\
\frac{d{S55}}{dt} & = k_{on_{S5:S5}} {S5}^2- k_{off_{S5:S5}}{S55} \\
& \quad - k_{on_{S4:S5S5}} {S4}* {S55} + k_{off_{S4:S5S5}} {S455} \\
\frac{d{S45}}{dt} & = k_{on_{S4:S5}}{S5}*{S4}- k_{off_{S4:S5}} {S45} \\
& \quad -k_{on_{S4S5:S5}}{S5}*{S45}+k_{off_{S4S5:S5}}{S5}*{S455}\\
& \quad -2 k_{on_{S4S5:S5S4}}{S45}^2 + 2 k_{off_{S4S5:S5S4}}{S4554} \\
\frac{d{S455}}{dt} & = k_{on_{S4:S5S5}} {S4}* {S55} - k_{off_{S4:S5S5}} {S455} \\
& \quad +k_{on_{S4S5:S5}}{S5}*{S45}-k_{off_{S4S5:S5}}{S5}*{S455}\\
& \quad -k_{on_{S4:S5S5S4}}{S455}*{S4}+k_{off_{S4:S5S5S4}}{S4554}\\
\frac{d{S4554}}{dt} & = k_{on_{S4:S5S5S4}}{S455}*{S4}-k_{off_{S4:S5S5S4}}{S4554}\\
& \quad +k_{on_{S4S5:S5S4}}{S45}^2 - k_{off_{S4S5:S5S4}}{S4554} \\
\frac{d{S4}}{dt} & = -k_{on_{S4:S5}}{S5}*{S4}+ k_{off_{S4:S5}} {S45} \\
& \quad - k_{on_{S4:S5S5}} {S4}* {S55} + k_{off_{S4:S5S5}} {S455} \\
& \quad -k_{on_{S4:S5S5S4}}{S455}*{S4}+k_{off_{S4:S5S5S4}}{S4554}\\
\end{align*}
\]
</div>
<strong>Variables</strong>
<table>
<thead>
<tr>
<th>Variable</th>
<th>Represents Molecule</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(S5\)</td>
<td>Ste5</td>
</tr>
<tr>
<td>\(S55\)</td>
<td>Ste5Ste5</td>
</tr>
<tr>
<td>\(S45\)</td>
<td>Ste4Ste18Ste5</td>
</tr>
<tr>
<td>\(S455\)</td>
<td>Ste4Ste18Ste5Ste5</td>
</tr>
<tr>
<td>\(S4554\)</td>
<td>Ste4Ste18Ste5Ste5Ste4Ste18</td>
</tr>
<tr>
<td>\(S4\)</td>
<td>Ste4Ste18</td>
</tr>
</tbody>
</table>
<strong>Parameters</strong>
<table>
<thead>
<tr>
<th>Parameter</th>
<th>Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(k_{on_{S5:S5}}\)</td>
<td>Binding rate of Ste5 and Ste5</td>
</tr>
<tr>
<td>\(k_{off_{S5:S5}}\)</td>
<td>Dissociation rate of Ste5:Ste5</td>
</tr>
<tr>
<td>\(k_{on_{S4:S5}}\)</td>
<td>Binding rate of Ste4Ste18 and Ste5</td>
</tr>
<tr>
<td>\(k_{off_{S4:S5}}\)</td>
<td>Dissociation rate of Ste4Ste18:Ste5</td>
</tr>
<tr>
<td>\(k_{on_{S4S5:S5}}\)</td>
<td>Binding rate of Ste4Ste18Ste5 and Ste5</td>
</tr>
<tr>
<td>\(k_{off_{S4S5:S5}}\)</td>
<td>Dissociation rate of Ste4Ste18Ste5:Ste5</td>
</tr>
<tr>
<td>\(k_{on_{S4:S5S5}}\)</td>
<td>Binding rate of Ste4Ste18 and Ste5Ste5</td>
</tr>
<tr>
<td>\(k_{off_{S4:S5S5}}\)</td>
<td>Dissociation rate of Ste4Ste18:Ste5Ste5</td>
</tr>
<tr>
<td>\(k_{on_{S4:S5S5S4}}\)</td>
<td>Binding rate of Ste4Ste18Ste5Ste5 and Ste4Ste18</td>
</tr>
<tr>
<td>\(k_{off_{S4:S5S5S4}}\)</td>
<td>Dissociation rate of Ste4Ste18Ste5Ste5:Ste4Ste18</td>
</tr>
<tr>
<td>\(k_{on_{S4S5:S5S4}}\)</td>
<td>Binding rate of Ste4Ste18Ste5 and Ste4Ste18Ste5</td>
</tr>
<tr>
<td>\(k_{off_{S4S5:S5S4}}\)</td>
<td>Dissociation rate of Ste4Ste18Ste5:Ste5Ste4Ste18</td>
</tr>
</tbody>
</table>
<strong>Initial conditions</strong>
<p>Assume that before signal transduction starts, there are only free Ste5 and just released Ste4Ste18
in the cell, with concentrations both equal to 1, and parameters are assumed. After starting the
simulation, reactions occur according to the listed equations, and after a period of time, the
concentrations reach equilibrium.</p>
<p>TODO: Insert result graph</p>
<h4>Cascading Reactions</h4>
<p><strong>Reactions</strong>:</p>
<div>
<p>
\[
\begin{align*}
Ste5_{off_{Ste11}} + Ste11_{off} & \leftrightarrows Ste5Ste11 \\
Ste5_{off_{Ste7}} + Ste7_{off} & \leftrightarrows Ste5Ste7 \\
Ste5_{off_{Fus3}} + Fus3_{off} & \leftrightarrows Ste5Fus3 \\
\end{align*}
\]
</p>
<p>
\[
\begin{align*}
Ste11 & \xrightarrow {Ste20} Ste11_{pS} \\
Ste11_{pS} & \xrightarrow {Ste20} Ste11_{pSpS} \\
Ste11_{pSpS} & \xrightarrow {Ste20} Ste11_{pSpSpT} \\
\end{align*}
\]
</p>
<p>
\[
\begin{align*}
Ste7 & \xrightarrow {Ste11_{pS},Ste11_{pSpS},Ste11_{pSpSpT}} Ste7_{pS} \\
Ste7_{pS} & \xrightarrow {Ste11_{pS},Ste11_{pSpS},Ste11_{pSpSpT}} Ste7_{pSpT}\\
\end{align*}
\]
</p>
<p>
\[
\begin{align*}
Fus3 & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pY} \\
Fus3 & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pT} \\
Fus3_{pY} & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pYpT} \\
Fus3_{pT} & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pYpT} \\
\end{align*}
\]
</p>
</div>
<h2>Explanation</h2>
<p>Only the Ste5 bound to the scaffold has significance in recruiting Ste11, Ste7, and Fus3, and the
binding to these three proteins is independent. Therefore, the Ste5 on the scaffold can be treated
as three copies to calculate its binding with Ste11, Ste7, and Fus3 separately. The three proteins
are activated through cascading phosphorylation initiated by Ste20, and the conditions for the
reactions to occur are that the kinases are activated and bound to the scaffold. Each protein has
different forms of phosphorylation modifications, which may have different catalytic reaction rates;
thus, they need to be listed separately.</p>
<h2>Ordinary Differential Equations</h2>
<p>The forms of multiple reactions are similar; here, only a portion is selected for demonstration.</p>
<p>Taking Ste11 as an example to illustrate the binding of the kinase with Ste5:</p>
<div>
<p>
\[
\begin{align*}
\frac{dSte5_{off_{Ste11}}}{dt} & = k_{off_{Ste5Ste11}}Ste5Ste11 -
k_{on_{Ste5Ste11}}Ste5_{off_{Ste11}} * Ste11_{off} \\
\frac{dSte11_{off}}{dt} & = k_{off_{Ste5Ste11}}Ste5Ste11 - k_{on_{Ste5Ste11}}Ste5_{off_{Ste11}}
* Ste11_{off} \\
\frac{dSte5Ste11}{dt} & = - k_{off_{Ste5Ste11}}Ste5Ste11 + k_{on_{Ste5Ste11}}Ste5_{off_{Ste11}}
* Ste11_{off} \\
\end{align*}
\]
</p>
</div>
<h2>Variables</h2>
<table>
<thead>
<tr>
<th>Variable</th>
<th>Represents Molecule</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(Ste5_{off_{Ste11}}\)</td>
<td>Unbound kinase Ste5</td>
</tr>
<tr>
<td>\(Ste11_{off}\)</td>
<td>Unbound scaffold Ste11</td>
</tr>
<tr>
<td>\(Ste5Ste11\)</td>
<td>Bound Ste5 and Ste11</td>
</tr>
</tbody>
</table>
<h2>Parameters</h2>
<table>
<thead>
<tr>
<th>Parameter</th>
<th>Meaning</th>
<th>Units</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(k_{off_{Ste5Ste11}}\)</td>
<td>Dissociation rate of Ste5Ste11</td>
<td>\({s}^{-1}\)</td>
</tr>
<tr>
<td>\(k_{on_{Ste5Ste11}}\)</td>
<td>Association rate of Ste5 and Ste11</td>
<td>\({\mu M}^{-1}·s^{-1}\)</td>
</tr>
</tbody>
</table>
<p>Using Ste11 catalyzing the phosphorylation of Ste7 as an example to illustrate the phosphorylation
process:</p>
<div>
<p>
\[
\frac{dSte7_{pS}}{dt} =
kcat_{Ste11pS{Ste7_{pS}}}Ste11_{pS}*\frac{Ste5Ste11}{Ste11_{total}}*\frac{Ste5Ste7}{Ste7_{total}}*\frac{Ste7_{pS}}{Ste7_{total}}+\ldots
\]
</p>
</div>
<h2>Variables</h2>
<table>
<thead>
<tr>
<th>Variable</th>
<th>Represents Molecule</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(Ste7_{pS}\)</td>
<td>Phosphorylated Ste7 at S359</td>
</tr>
<tr>
<td>\(Ste11_{pS}\)</td>
<td>Phosphorylated Ste11 at S302</td>
</tr>
<tr>
<td>\(Ste5Ste11\)</td>
<td>Ste11 bound to Ste5</td>
</tr>
<tr>
<td>\(Ste5Ste7\)</td>
<td>Ste7 bound to Ste5</td>
</tr>
<tr>
<td>\(Ste7_{total}\)</td>
<td>Total amount of Ste7</td>
</tr>
</tbody>
</table>
<h2>Parameters</h2>
<p>\(kcat_{Ste11pS{Ste7_{pS}}}\): Represents the catalytic efficiency in this case.</p>
<h2>Initial Conditions</h2>
<p>The concentrations of the three kinases are known, assuming their initial state has not undergone
phosphorylation. Some enzyme activity parameters are known, and other parameters are roughly
estimated to the same order of magnitude.</p>
<p>TODO: Insert result figure</p>
</div>
</div>
<div class="row mt-4">
<div class="col-lg-12">
<h2 id="topic4">
<h2>Lactic Acid Absorption Model</h2>
<hr>
<h3>Model Description</h3>
<p>
Our project alleviates IBD symptoms by secreting lactic acid in the intestine to weaken
autoimmunity, but it may face two aspects of doubt: first, why can't lactic acid or lactic acid
bacteria probiotics be taken directly; second, will the considerable secretion of lactic acid cause
acidosis in the human body? We hope to model our project to describe how it has a better sustained
release effect compared to direct lactic acid consumption, more precise control compared to
probiotic intake, and to avoid adaptation of the immune system and gut microbiota. Additionally, we
need to develop a computational method to achieve precise control over lactic acid secretion to
regulate treatment time and prevent acidosis.
</p>
<h3>Basic Assumptions</h3>
<ol>
<li>Only the absorption process of lactic acid is described, without considering other effects of
lactic acid on the human body.</li>
<li>It is assumed that the location where lactic acid acts on immune cells is separated from the
intestinal environment.</li>
<li>It is assumed that the secretion rate of lactic acid is uniform, and activated yeast cells
secrete a total amount of lactic acid \(a\) within time \(t_0\), secreting \(\frac{a}{n}\) of
lactic acid in the time interval \(\frac{t_0}{n}\).</li>
</ol>
<h3>Model Equation</h3>
<h4>Direct Administration</h4>
<p>In the case of direct lactic acid intake, the content of lactic acid in the intestine can be
described by the following equation:</p>
<p>
\( Q_d = (Q_{d_0} + a)e^{-(k_1 + k_2)t} \)
</p>
<p><strong>Explanation</strong>: The absorption rate is proportional to the concentration of lactic
acid, and the concentration of lactic acid declines in an exponential form.</p>
<p><strong>Parameters</strong>:</p>
<ul>
<li>\( Q_d \): Remaining lactic acid content in the intestinal environment</li>
<li>\( Q_{d_0} \): Initial lactic acid content in the intestinal environment</li>
<li>\( a \): Total amount of lactic acid ingested</li>
<li>\( k_1 \): Absorption rate of lactic acid</li>
<li>\( k_2 \): Rate at which lactic acid is eliminated due to metabolism and excretion</li>
<li>\( t \): Time</li>
</ul>
<h4>Induced Secretion</h4>
<p>According to Fick's Law</p>
<p>The remaining lactic acid content in the intestinal environment has a recursive relationship over
time:</p>
<p>
\( Q_{d_i} = \left(Q_{d_{i-1}} + \frac{a}{n}\right)e^{-(k_1 + k_2)(t - (i-1)\frac{t_0}{n})} \)
</p>
<p>We can obtain the expression:</p>
<p>
\( Q_{d_i} = \frac{a}{n} \sum_{m=1}^{i-1} e^{-(k_1 + k_2)\left(mt - \left(j \frac{(m+2)(m+1)}{2}
\frac{t_0}{n}\right)\right)} \)
</p>
<p>TODO: Insert result graph</p>
<p>By simulating the absorption process of lactic acid, we can conclude that in the case of direct
administration, the concentration of lactic acid decreases exponentially over time, while in the
case of induced secretion, the concentration of lactic acid slowly increases over time and reaches
equilibrium after a certain period.</p>
</div>
</div>
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