Our model serves two main purposes:
Our model can be divided into four interconnected parts, representing the inhalation of muscone, its binding to receptors, intracellular signal transduction and lactate secretion triggered by receptor activation, and the absorption of lactate. These models provide a comprehensive understanding of the project and yield valuable computational results.
The main focus of our project is the use of muscone as a signaling molecule to activate engineered yeast 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:
TODO:Insert design diagram
Corresponding to the above processes, five compartments need to be established for simulation, where \(t\) represents the time variable:
At \(t=0\), the amount of muscone in all compartments is \(0\).
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 lactate, 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.
\[ V_{\text{inhale}}(t) =\frac{Q_{\text{inhale}}}{5}(u(t)-u(t-5)) \]
Explanation: 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.
\[ \frac{dQ_A(t)}{dt} = V_{\text{inhale}}(t) - \left( k_{\text{exhale}} + k_{\text{perm}} \right) Q_A(t) \]
Explanation: 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.
Parameters:
\[ \frac{dQ_M(t)}{dt} = 0.0005 \cdot k_{\text{adh}} V_{\text{inhale}}(t) - k_{\text{diffMC}} Q_M(t) \]
Explanation: The increase in muscone on the mucosa comes from adhesion in the alveoli, and the decrease is due to diffusion into the systemic circulation.
Parameters:
\[ \frac{dQ_L(t)}{dt} = k_{\text{perm}} Q_A(t) - k_{\text{diffLC}} Q_L(t) \]
Explanation: 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.
Parameters:
\[ \frac{dQ_C(t)}{dt} = k_{\text{diffMC}} Q_M(t) + k_{\text{diffLC}} Q_L(t) - k_{\text{dist}} Q_C(t) - k_{\text{excrete}} Q_C(t) \]
Explanation: 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.
Parameters:
\[ \frac{dQ_I(t)}{dt} = k_{\text{dist}} Q_C(t) - k_{move}Q_I(t) \]
Explanation: 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.
\( k_{\text{dist}} \): Same as Compartment 3
\( k_{move} \): The metabolism and excretion of muscone in the intestine, taken as \( 0.02 \
\text{min}^{-1} \)
In summary, we can write a system of ordinary differential equations and import it into MATLAB for simulation:
\[ \begin{align*} Q_{\text{inhale}}(t) & = 100(mg)(Assumption) \\ V_{\text{inhale}}(t) & =\frac{Q_{\text{inhale}}}{5}(u(t)-u(t-5)) \\ \frac{dQ_A(t)}{dt} & = V_{\text{inhale}}(t) -\left( k_{\text{exhale}} + k_{\text{perm}} \right) Q_A(t) \\ \frac{dQ_L(t)}{dt} & = k_{\text{perm}} Q_A(t) - k_{\text{diffLC}} Q_L(t) \\ \frac{dQ_M(t)}{dt} & = 0.0005\cdot k_{\text{adh}} V_{\text{inhale}}(t) - k_{\text{diffMC}} Q_M(t) \\ \frac{dQ_C(t)}{dt} & = k_{\text{diffMC}} Q_M(t) + k_{\text{diffLC}} Q_L(t) - k_{\text{dist}} Q_C(t) - k_{\text{excrete}} Q_C(t) \\ \frac{dQ_I(t)}{dt} & = k_{\text{dist}} Q_C(t)-k_{move}Q_I(t) \\ \end{align*} \]
TODO:插入结果图
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 muscone can spread into the intestine, similarly, the concentration change caused by continuous muscone is simulated by changing the inhalation equation, and the concentration of muscone in the intestine can be obtained in combination with experiment. Because there is no animal experimental support, the data are manually drafted, and the calculation method is more meaningful than the calculation results.
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).
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).
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:
\[ \begin{align*} \text{Pheromone} + \text{Ste2} & \rightarrow \text{PheromoneSte2} \\ \text{PheromoneSte2} & \rightarrow \text{Pheromone} + \text{Ste2} \\ \text{PheromoneSte2} + \text{Gpa1Ste4Ste18} & \rightarrow \text{PheromoneSte2Gpa1Ste4Ste18} \\ \text{PheromoneSte2Gpa1Ste4Ste18} & \rightarrow \text{PheromoneSte2Gpa1} + \text{Ste4Ste18} \\ \text{PheromoneSte2Gpa1} & \rightarrow \text{PheromoneSte2} + \text{Gpa1} \\ \text{Gpa1} + \text{Ste4Ste18} & \rightarrow \text{Gpa1Ste4Ste18} \end{align*} \]
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.
Ordinary Differential Equations\[ \begin{align*} \frac{d{P}}{dt} & = k_{off_{PS}}{PS} - k_{on_{PS}}{P}*{S} \\ \frac{d{S}}{dt} & = k_{off_{PS}}{PS} - k_{on_{PS}}{P}*{S} \\ \frac{d{PS}}{dt} & = k_{on_{PS}}{P}*{S} + k_{off_{SG}} {PSG} \\ & \quad - k_{off_{PS}}{PS} - k_{on_{SG}}{PS} * {GSS} \\ \frac{d{GSS}}{dt} & = k_{on_{GS}}{SS} * {G} - k_{on_{SG}}{PS} * {GSS} \\ \frac{d{PSGSS}}{dt} & = k_{on_{SG}}{PS} * {GSS} - k_{on_{GS}}{PSGSS} \\ \frac{d{PSG}}{dt} & = k_{on_{GS}}{PSGSS} - k_{off_{SG}} {PSG} \\ \frac{d{SS}}{dt} & = k_{on_{GS}}{PSGSS} - k_{on_{GS}}{SS} * {G} \\ \frac{d{G}}{dt} & = k_{off_{SG}} {PSG} - k_{on_{GS}}{SS} * {G} \\ \end{align*} \]
Table 1: Variables of Receptor Activation Model
Variable | Represents Molecule | Concentration (\(\mu M\)) |
---|---|---|
\(P\) | Pheromone | - |
\(S\) | Ste2 | \(0.287\) |
\(PS\) | PheromoneSte2 | - |
\(GSS\) | Gpa1Ste4Ste18 | - |
\(PSGSS\) | PheromoneSte2Gpa1Ste4Ste18 | - |
\(PSG\) | PheromoneSte2Gpa1 | - |
\(SS\) | Ste4Ste18 | \(2\times 10^{-4}\) |
\(G\) | Gpa1 | \(2\times 10^{-4}\) |
Table 2: Parameters of Receptor Activation Model
Parameter | Meaning | Value | Unit |
---|---|---|---|
\(k_{on_{PS}}\) | Binding rate of Pheromone to Ste2 | \(0.185\) | \({\mu M}^{-1} \cdot s^{-1}\) |
\(k_{off_{PS}}\) | Dissociation rate of PheromoneSte2 | \(1 \times 10^{-3}\) | \(s^{-1}\) |
\(k_{on_{SG}}\) | Binding rate of PheromoneSte2 to Gpa1Ste4Ste18 | - | \({\mu M}^{-1} \cdot s^{-1}\) |
\(k_{off_{SG}}\) | Dissociation rate of PheromoneSte2Gpa1 | - | \(s^{-1}\) |
\(k_{on_{GS}}\) | Binding rate of Gpa1 to Ste4Ste18 | - | \({\mu M}^{-1} \cdot s^{-1}\) |
\(k_{off_{GS}}\) | Dissociation rate of PheromoneGpa1Ste4Ste18 | - | \(s^{-1}\) |
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.
TODO:插入结果图
Explanation: 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.
Ordinary Differential Equations:Table 3: Variables of Scaffold Formation Model
Variable | Represents Molecule |
---|---|
\(S5\) | Ste5 |
\(S55\) | Ste5Ste5 |
\(S45\) | Ste4Ste18Ste5 |
\(S455\) | Ste4Ste18Ste5Ste5 |
\(S4554\) | Ste4Ste18Ste5Ste5Ste4Ste18 |
\(S4\) | Ste4Ste18 |
Table 4: Parameters of Scaffold Formation Model
Parameter | Meaning |
---|---|
\(k_{on_{S5:S5}}\) | Binding rate of Ste5 and Ste5 |
\(k_{off_{S5:S5}}\) | Dissociation rate of Ste5:Ste5 |
\(k_{on_{S4:S5}}\) | Binding rate of Ste4Ste18 and Ste5 |
\(k_{off_{S4:S5}}\) | Dissociation rate of Ste4Ste18:Ste5 |
\(k_{on_{S4S5:S5}}\) | Binding rate of Ste4Ste18Ste5 and Ste5 |
\(k_{off_{S4S5:S5}}\) | Dissociation rate of Ste4Ste18Ste5:Ste5 |
\(k_{on_{S4:S5S5}}\) | Binding rate of Ste4Ste18 and Ste5Ste5 |
\(k_{off_{S4:S5S5}}\) | Dissociation rate of Ste4Ste18:Ste5Ste5 |
\(k_{on_{S4:S5S5S4}}\) | Binding rate of Ste4Ste18Ste5Ste5 and Ste4Ste18 |
\(k_{off_{S4:S5S5S4}}\) | Dissociation rate of Ste4Ste18Ste5Ste5:Ste4Ste18 |
\(k_{on_{S4S5:S5S4}}\) | Binding rate of Ste4Ste18Ste5 and Ste4Ste18Ste5 |
\(k_{off_{S4S5:S5S4}}\) | Dissociation rate of Ste4Ste18Ste5:Ste5Ste4Ste18 |
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.
TODO: Insert result graph
Reactions:
\[ \begin{align*} Ste5_{off_{Ste11}} + Ste11_{off} & \leftrightarrows Ste5Ste11 \\ Ste5_{off_{Ste7}} + Ste7_{off} & \leftrightarrows Ste5Ste7 \\ Ste5_{off_{Fus3}} + Fus3_{off} & \leftrightarrows Ste5Fus3 \\ \end{align*} \]
\[ \begin{align*} Ste11 & \xrightarrow {Ste20} Ste11_{pS} \\ Ste11_{pS} & \xrightarrow {Ste20} Ste11_{pSpS} \\ Ste11_{pSpS} & \xrightarrow {Ste20} Ste11_{pSpSpT} \\ \end{align*} \]
\[ \begin{align*} Ste7 & \xrightarrow {Ste11_{pS},Ste11_{pSpS},Ste11_{pSpSpT}} Ste7_{pS} \\ Ste7_{pS} & \xrightarrow {Ste11_{pS},Ste11_{pSpS},Ste11_{pSpSpT}} Ste7_{pSpT}\\ \end{align*} \]
\[ \begin{align*} Fus3 & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pY} \\ Fus3 & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pT} \\ Fus3_{pY} & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pYpT} \\ Fus3_{pT} & \xrightarrow {Ste7_{pS},Ste7_{pSpT}} Fus3_{pYpT} \\ \end{align*} \]
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.
The forms of multiple reactions are similar; here, only a portion is selected for demonstration.
Taking Ste11 as an example to illustrate the binding of the kinase with Ste5:
\[ \begin{align*} \frac{dSte5_{off_{Ste11}}}{dt} & = k_{off_{Ste5Ste11}}Ste5Ste11 - k_{on_{Ste5Ste11}}Ste5_{off_{Ste11}} * Ste11_{off} \\ \frac{dSte11_{off}}{dt} & = k_{off_{Ste5Ste11}}Ste5Ste11 - k_{on_{Ste5Ste11}}Ste5_{off_{Ste11}} * Ste11_{off} \\ \frac{dSte5Ste11}{dt} & = - k_{off_{Ste5Ste11}}Ste5Ste11 + k_{on_{Ste5Ste11}}Ste5_{off_{Ste11}} * Ste11_{off} \\ \end{align*} \]
Table 5: Variables of Ste11 Binding Model
Variable | Represents Molecule |
---|---|
\(Ste5_{off_{Ste11}}\) | Unbound kinase Ste5 |
\(Ste11_{off}\) | Unbound scaffold Ste11 |
\(Ste5Ste11\) | Bound Ste5 and Ste11 |
Table 6: Parameters of Ste11 Binding Model
Parameter | Meaning | Units |
---|---|---|
\(k_{off_{Ste5Ste11}}\) | Dissociation rate of Ste5Ste11 | \({s}^{-1}\) |
\(k_{on_{Ste5Ste11}}\) | Association rate of Ste5 and Ste11 | \({\mu M}^{-1}·s^{-1}\) |
Using Ste11 catalyzing the phosphorylation of Ste7 as an example to illustrate the phosphorylation process:
\[ \frac{dSte7_{pS}}{dt} = kcat_{Ste11pS{Ste7_{pS}}}Ste11_{pS}*\frac{Ste5Ste11}{Ste11_{total}}*\frac{Ste5Ste7}{Ste7_{total}}*\frac{Ste7_{pS}}{Ste7_{total}}+\ldots \]
Table 7: Variables of Ste7 Phosphorylation Model
Variable | Represents Molecule |
---|---|
\(Ste7_{pS}\) | Phosphorylated Ste7 at S359 |
\(Ste11_{pS}\) | Phosphorylated Ste11 at S302 |
\(Ste5Ste11\) | Ste11 bound to Ste5 |
\(Ste5Ste7\) | Ste7 bound to Ste5 |
\(Ste7_{total}\) | Total amount of Ste7 |
\(kcat_{Ste11pS{Ste7_{pS}}}\): Represents the catalytic efficiency in this case.
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.
TODO: Insert result figure
Our project alleviates IBD symptoms by secreting lactate in the intestine to weaken autoimmunity, but it may face two aspects of doubt: first, why can't lactate or lactate bacteria probiotics be taken directly; second, will the considerable secretion of lactate 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 lactate 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 lactate secretion to regulate treatment time and prevent acidosis.
According to Fick's law :
\[ \frac{dQd}{dt} = -D \frac{dC}{dx} \]
Because the distance between diffusion is very small, the concentration difference between the two sides of the system replaces the concentration gradient, so this formula can be simplified to:
\[ \frac{dQd}{dt} = K\times Qd \]
In the case of direct lactate intake, the content of lactate in the intestine can be described by the following equation:
\[ Q_d = (Q_{d_0} + a)e^{-(k_1 + k_2)t} \]
Explanation: The absorption rate is proportional to the concentration of lactic acid, and the concentration of lactate declines in an exponential form.
Parameters:
The remaining lactate content in the intestinal environment has a recursive relationship over time:
\[ Q_{d_i} = \left(Q_{d_{i-1}} + \frac{a}{n}\right)e^{-(k_1 + k_2)(t - (i-1)\frac{t_0}{n})} \]
We can obtain the expression:
\[ Q_{d_i} = \frac{a}{n} \sum_{m=1}^{i-1} e^{-(k_1 + k_2)\left(mt - \left(j \frac{(m+2)(m+1)}{2} \frac{t_0}{n}\right)\right)} \]
TODO: Insert result graph
By simulating the absorption process of lactate, we can conclude that in the case of direct administration, the concentration of lactate decreases exponentially over time, while in the case of induced secretion, the concentration of lactate slowly increases over time and reaches equilibrium after a certain period.