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+ }
+
+ .code-snippet {
+ display: none;
+ /* åˆå§‹æ—¶éšè—代ç æ®µè½ */
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+
+<body>
+ {% 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>
+ </div>
+
+ <div class="progress-container">
+ <svg class="progress-bar-circle" width="60" height="60">
+ <circle class="progress-circle" cx="30" cy="30" r="25" stroke-width="5" fill="transparent"></circle>
+ </svg>
+ <div class="progress-text">0%</div>
+ </div>
+
+ <div class="row mt-4">
+ <div class="col-lg-12">
+ <h2 id="description">General Description of Modeling</h2>
+ <hr>
+ <p>Our model serves two main purposes:</p>
+ <ol>
+ <li><strong>Quantitative Description of Project Design</strong>: Due to safety considerations, we were
+ unable to conduct animal experiments to demonstrate the processes occurring during the operation of
+ the project. Modeling can help in understanding therapeutic pathways, provide a quantitative
+ perspective, and better tell our story.</li>
+ <li><strong>Computational Methods for Project Engineering</strong>: If the project can be carried out,
+ the model can help determine the parameters in the implementation process of the project, reduce the
+ calculation amount in the experimental process, connect the wet experimental system independent of
+ each other, and make the design mathematically encapsulated as new components.</li>
+ </ol>
+ <p>Our model can be 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>
+ <div class="image-container">
+ <img src="https://static.igem.wiki/teams/5187/figure/ibd-figure.jpg" alt="ibd_figure"
+ class="shadowed-image">
+ </div>
+ </div>
+
+ <div class="row mt-4">
+ <div class="col-lg-12">
+ <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：æ’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>
+ <script>
+ function toggleCodeSnippet() {
+ var codeSnippet = document.getElementById("codeSnippet");
+ var button = document.getElementById("Button1"); // 注æ„å˜é‡å通常使用å°å†™å¼€å¤´
+ if (codeSnippet.style.display === "none") {
+ codeSnippet.style.display = "block";
+ button.textContent = "Collapse the code"; // 使用之å‰é€‰ä¸çš„按钮元ç´
+ } else {
+ codeSnippet.style.display = "none";
+ button.textContent = "Expand the code"; // 使用之å‰é€‰ä¸çš„按钮元ç´
+ }
+ }
+ </script>
+ </div>
+ </div>
+
+ <div class="row mt-4">
+ <div class="col-lg-12">
+ <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å®ä¼Ÿ) 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>
+
+ <div class="row mt-4">
+ <div class="col-lg-12">
+ <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：æ’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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+ background-color: #f0f0f0;
+ /* MATLAB类似的背景色 */
+ border: 1px solid #ccc;
+ padding: 10px;
+ margin-top: 10px;
+ font-family: 'Courier New', monospace;
+ /* 设置ç‰å®½å—体 */
+ font-size: 12px;
+ /* 设置å—ä½“å¤§å° */
+ color: #000;
+ /* 文本颜色 */
+ white-space: pre;
+ /* ä¿ç•™ä»£ç æ ¼å¼ */
+ overflow-x: auto;
+ /* å…许水平滚动 */
+ }
+</style>
</head>
<body>