2b:null 2c:["$","title",null,{"children":"Interactive Cell Cycle Model - Yang Lab | Franco Tavella | Observable"}] 31:T5d3,

The cell cycle is a fundamental biological process that ensures accurate cell divisions. In a [2018 eLife article](https://elifesciences.org/articles/33549), Guan et al. led a study that established an experimental and computational platform to study the early embryonic cell cycles of *Xenopus laevis* frogs. Here, we present the mathematical model developed in that article through an interactive application. This application allows the user to explore the model's behavior by changing parameters and initial conditions. The model is composed of two species, Total Cyclin B1 and Active CyclinB1:Cdk1 complex, which provides a simple representation of the cell cycle.

The model is described by the following system of ordinary differential equations: ${tex.block` \frac{dB}{dt} = k_{S} - DB `} ${tex.block` \frac{dC}{dt} = k_{S} + \frac{1}{\sqrt{r_{WT}}} T (B - C) - \sqrt{r_{WT}} W C - DC `} where ${tex.block` D = a_{D} + b_{D} \frac{C^{n_{D}}}{C^{n_{D}} + K_D^{n_{D}}} `} ${tex.block` T = a_{T} + b_{T} \frac{C^{n_{T}}}{C^{n_{T}} + K_T^{n_{T}}} `} ${tex.block` W = a_{W} + b_{W} \frac{K^{n_{W}}}{C^{n_{W}} + K_W^{n_{W}}} `}

where ${tex`B`} represents the total amount of CyclinB1 and ${tex`C`} the concentration of active cyclinB1:Cdk1 complex. Additionally, subscripts ${tex`D`}, ${tex`T`}, and ${tex`E`} represent protein degradation, Cdc25, and Wee1 interactions respectively.

32:[] 2f:{"id":"3ad2343ac1f69afb","slug":null,"title":"Interactive Cell Cycle Model - Yang Lab","thumbnail":"a5d59b551d3831dbd702a0c33dbf31d07a80185b9f8677b6dd1ca73270a1db3f","access_level":"public","create_time":"$D2023-10-25T20:17:05.302Z","update_time":"$D2023-10-31T13:54:10.144Z","view_time":null,"first_public_time":"$D2023-10-25T20:17:05.302Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"21c1147f0671f3ed","login":"frantavella","name":"Franco Tavella","avatar_url":"https://avatars-next.observableusercontent.com/000c981d1c181862b1675cb9e705e7304d2ae1a8b467bd7fe862a27a7fbd4979","tier":"starter_2024"},"version":1203,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"31dc1692ab688bc0","login":"frantavella","name":"Franco Tavella","avatar_url":"https://avatars-next.observableusercontent.com/000c981d1c181862b1675cb9e705e7304d2ae1a8b467bd7fe862a27a7fbd4979","tier":null}],"members":[{"type":"individual","id":"31dc1692ab688bc0","login":"frantavella","name":"Franco Tavella","avatar_url":"https://avatars-next.observableusercontent.com/000c981d1c181862b1675cb9e705e7304d2ae1a8b467bd7fe862a27a7fbd4979","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Interactive Cell Cycle Model - Yang Lab","cells":[{"id":1077,"value":"# Interactive Cell Cycle Model - [Yang Lab](http://www-personal.umich.edu/~qiongy/)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1085,"value":"$31","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":586,"value":"parameterSliders = html` \n

\n${viewof B0} ${viewof C0} ${viewof kS} ${viewof rWT}\n${viewof aD} ${viewof bD} ${viewof KD} ${viewof nD}\n${viewof aT} ${viewof bT} ${viewof KT} ${viewof nT}\n${viewof aW} ${viewof bW} ${viewof KW} ${viewof nW}\n${viewof simulationPoints} ${viewof finalTime} \n

\n\n

\n${viewof reset} \n

\n\n

Left plot: — Total Cyclin B1 (B) — Active Cdk1:Cyclin B1 (C)

Right plot: — B nullcline — C nullcline — Trajectory

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":306,"value":"renderedPlots = html`\n \n \n \n

\n ${plt}\n

\n ${plt2}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":400,"value":"# Parameter sliders","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":264,"value":"### Initial condition, synthesis rate, and Wee1 to Cdc25 balance","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":431,"value":"viewof B0 = Inputs.range([0.0, 120.0], {\n step: 0.1,\n value: defaultParams.B0,\n label: \"B0: Cyclin B1 initial value (nM)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":433,"value":"viewof C0 = Inputs.range([0.0, 120.0], {\n step: 0.1,\n value: defaultParams.C0,\n label: \"C0: CyclinB1:Cdk1 initial value (nM)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":435,"value":"viewof kS = Inputs.range([0.0, 2.0], {\n step: 0.001,\n value: defaultParams.kS,\n label: \"kS: Cyclin B1 synthesis rate (nM/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":437,"value":"viewof rWT = Inputs.range([0.00001, 5.0], {\n step: 0.001,\n value: defaultParams.rWT,\n label: \"rWT: Wee1 to Cdc25 activity ratio (a.u.)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":255,"value":"### Protein degradation parameters","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":439,"value":"viewof aD = Inputs.range([0.000001, 0.1], {\n step: 0.000001,\n value: defaultParams.aD,\n label: \"aD: Basal degradation rate (1/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":441,"value":"viewof bD = Inputs.range([0.000001, 5.0], {\n step: 0.001,\n value: defaultParams.bD,\n label: \"bD: Cdk1-based degradation rate (1/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":443,"value":"viewof KD = Inputs.range([1.0, 120.0], {\n step: 0.1,\n value: defaultParams.KD,\n label: \"KD: Half-max degradation rate (nM)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":447,"value":"viewof nD = Inputs.range([1.0, 25.0], {\n step: 0.1,\n value: defaultParams.nD,\n label: \"nD: Hill exponent for degradation (a.u.)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":257,"value":"### Cdc25 activation parameters","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":449,"value":"viewof aT = Inputs.range([0.000001, 10.0], {\n step: 0.000001,\n value: defaultParams.aT,\n label: \"aT: Basal Cdc25 activation rate (1/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"viewof bT = Inputs.range([0.000001, 10.0], {\n step: 0.001,\n value: defaultParams.bT,\n label: \"bT: Cdk1-based Cdc25 rate (1/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":453,"value":"viewof KT = Inputs.range([1.0, 120.0], {\n step: 0.1,\n value: defaultParams.KT,\n label: \"KT: Half-max Cdc25 rate (nM)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":455,"value":"viewof nT = Inputs.range([1.0, 25.0], {\n step: 0.1,\n value: defaultParams.nT,\n label: \"nT: Hill exponent for Cdc25 (a.u.)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":259,"value":"### Wee1 inhibition parameters","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":458,"value":"viewof aW = Inputs.range([0.000001, 10.0], {\n step: 0.000001,\n value: defaultParams.aW,\n label: \"aW: Basal Wee1 activation rate (1/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":460,"value":"viewof bW = Inputs.range([0.000001, 10.0], {\n step: 0.001,\n value: defaultParams.bW,\n label: \"bW: Cdk1-based Wee1 rate (1/min)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":462,"value":"viewof KW = Inputs.range([1.0, 120.0], {\n step: 0.1,\n value: defaultParams.KW,\n label: \"KW: Half-max Wee1 rate (nM)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":464,"value":"viewof nW = Inputs.range([1.0, 25.0], {\n step: 0.1,\n value: defaultParams.nW,\n label: \"nW: Hill exponent for Wee1 (a.u.)\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":266,"value":"### Simulation setup","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":466,"value":"viewof simulationPoints = Inputs.range([10, 10000], {\n step: 1,\n value: defaultParams.simulationPoints,\n label: \"Number of datapoints to simulate\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":468,"value":"viewof finalTime = Inputs.range([0.0, 5000.0], {\n step: 1,\n value: defaultParams.finalTime,\n label: \"Final simulation time\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":863,"value":"## Reset parameters","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":530,"value":"viewof reset = Inputs.button([[\"Reset parameters\", resetParameters]])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":332,"value":"# Plots","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":294,"value":"plt = Plot.plot({\n height: 320,\n width: 900,\n marginLeft: 60,\n marginBottom: 60,\n marginRight: 60,\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.lineY(solution, { x: \"t\", y: \"0\", stroke: \"#6495ED\" }),\n Plot.lineY(solution, { x: \"t\", y: \"1\", stroke: \"#B22222\" }),\n Plot.axisX({\n labelAnchor: \"center\",\n labelArrow: \"none\",\n label: \"Time (min)\",\n labelOffset: 55,\n fontSize: 20\n }),\n Plot.axisY({\n labelAnchor: \"center\",\n labelArrow: \"none\",\n label: \"Concentration (nM)\",\n labelOffset: 60,\n fontSize: 20\n })\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":300,"value":"plt2 = Plot.plot({\n height: 320,\n width: 320,\n marginLeft: 65,\n marginBottom: 60,\n marginRight: 60,\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.lineY(solution, {\n x: \"0\",\n y: \"1\",\n stroke: \"#A9A9A9\",\n strokeWidth: 5\n }),\n Plot.lineY(BNullclineCurve, { x: \"1\", y: \"0\", stroke: \"#6495ED\" }),\n Plot.lineY(CNullclineCurve, { x: \"1\", y: \"0\", stroke: \"#B22222\" }),\n Plot.axisX({\n labelAnchor: \"center\",\n labelArrow: \"none\",\n label: \"Cyclin B1 (nM)\",\n labelOffset: 55,\n fontSize: 20\n }),\n Plot.axisY({\n labelAnchor: \"center\",\n labelArrow: \"none\",\n label: \"Active Cdk1:CyclinB1 (nM)\",\n labelOffset: 65,\n fontSize: 20\n })\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":131,"value":"# Solve the differential equation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":412,"value":"params = ({\n kS: kS,\n rWT: rWT,\n B0: B0,\n C0: C0,\n aD: aD,\n bD: bD,\n KD: KD,\n nD: nD,\n aT: aT,\n bT: bT,\n KT: KT,\n nT: nT,\n aW: aW,\n bW: bW,\n KW: KW,\n nW: nW,\n finalTime: finalTime,\n simulationPoints: simulationPoints\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"function solve_ode(params) {\n let simulationPoints = parseInt(params.simulationPoints);\n let initialTime = 0.0;\n let finalTime = params.finalTime;\n // Unpack parameters\n let kS = params.kS;\n let rWT = params.rWT;\n let aD = params.aD;\n let bD = params.bD;\n let KD = params.KD;\n let nD = params.nD;\n let aT = params.aT;\n let bT = params.bT;\n let KT = params.KT;\n let nT = params.nT;\n let aW = params.aW;\n let bW = params.bW;\n let KW = params.KW;\n let nW = params.nW;\n // Solve\n let sol = rk4(\n guan2008(kS, rWT, aD, bD, KD, nD, aT, bT, KT, nT, aW, bW, KW, nW),\n [params.B0, params.C0],\n initialTime,\n initialTime,\n finalTime,\n simulationPoints\n );\n return sol;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":59,"value":"solution = solve_ode(params);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":992,"value":"valuesC = linspace(0, 120, 1000)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1013,"value":"BNullclineCurve = d3.transpose([\n valuesC,\n valuesC.map((x) => nullclineB(x, params))\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1063,"value":"CNullclineCurve = d3.transpose([\n valuesC,\n valuesC.map((x) => nullclineC(x, params))\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"# Default parameters","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":31,"value":"defaultParams = ({\n kS: 1.5,\n rWT: 1.5,\n B0: 75,\n C0: 35,\n aD: 0.01,\n bD: 0.04,\n KD: 32,\n nD: 17,\n aT: 0.16,\n bT: 0.8,\n KT: 30,\n nT: 11,\n aW: 0.08,\n bW: 0.4,\n KW: 35,\n nW: 3.5,\n finalTime: 2000,\n simulationPoints: 5000\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"# Model","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":21,"value":"function actHill(x, a, b, K, n){\n return a + b * ((x ** n) / (x ** n + K ** n))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":25,"value":"function inhHill(x, a, b, K, n){\n return a + b * ((K ** n) / (x ** n + K ** n))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"function guan2008(kS, rWT, aD, bD, KD, nD, aT, bT, KT, nT, aW, bW, KW, nW) {\n return ([B, C], t) => [\n kS - actHill(C, aD, bD, KD, nD) * B,\n kS + (1.0 / Math.sqrt(rWT)) * actHill(C, aT, bT, KT, nT) * (B - C) - Math.sqrt(rWT) * inhHill(C, aW, bW, KW, nW) * C - actHill(C, aD, bD, KD, nD) * C\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":964,"value":"function nullclineB(C, params) {\n return params.kS / actHill(C, params.aD, params.bD, params.KD, params.nD);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1054,"value":"function nullclineC(C, params) {\n let HD = actHill(C, params.aD, params.bD, params.KD, params.nD);\n let W = inhHill(C, params.aW, params.bW, params.KW, params.nW);\n let T = actHill(C, params.aT, params.bT, params.KT, params.nT);\n return (\n C +\n (Math.sqrt(params.rWT) *\n (HD * C + Math.sqrt(params.rWT) * W * C - params.kS)) /\n T\n );\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":419,"value":"# Utilities","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":423,"value":"function set(input, value) {\n input.value = value;\n input.dispatchEvent(new Event(\"input\", { bubbles: true }));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":426,"value":"function resetParameters() {\n // Initial condition\n set(viewof B0, defaultParams.B0);\n set(viewof C0, defaultParams.C0);\n // Synthesis rate\n set(viewof kS, defaultParams.kS);\n // Feedback balance\n set(viewof rWT, defaultParams.rWT);\n // Degradation\n set(viewof aD, defaultParams.aD);\n set(viewof bD, defaultParams.bD);\n set(viewof KD, defaultParams.KD);\n set(viewof nD, defaultParams.nD);\n // Cdc25 interaction\n set(viewof aT, defaultParams.aT);\n set(viewof bT, defaultParams.bT);\n set(viewof KT, defaultParams.KT);\n set(viewof nT, defaultParams.nT);\n // Wee1 interaction\n set(viewof aW, defaultParams.aW);\n set(viewof bW, defaultParams.bW);\n set(viewof KW, defaultParams.KW);\n set(viewof nW, defaultParams.nW);\n // Simulation parameters\n set(viewof simulationPoints, defaultParams.simulationPoints);\n set(viewof finalTime, defaultParams.finalTime);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":854,"value":"# Styling","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":783,"value":"ns = Inputs.range().classList[0] // Get class of inputs","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":785,"value":"htmlStyling = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"# Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":9,"value":"import {rk4} from \"@mcmcclur/runge-kutta-for-systems-of-odes\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":95,"value":"import {slider} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":879,"value":"import { view } from \"@tomlarkworthy/view\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":973,"value":"import { linspace } from \"@dkirkby/jumpy\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1028,"value":"d3 = require(\"d3-array@3\", \"d3-fetch@3\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"