PMC:1570349 / 23432-27769 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/1570349","sourcedb":"PMC","sourceid":"1570349","source_url":"https://www.ncbi.nlm.nih.gov/pmc/1570349","text":"4.3 Engineering bistable switch in G1/S transition\nWe consider the reverse engineering of check points in mammalian cell phase transition. In particular, we use a 9 state, 40 parameter differential equation model [28]. In the publication describing the model, parameter scanning is done to find out parameter values that achieve certain behaviors, such as changing the threshold for the onset of synthesis (S) phase from the G1 state and conditions ensuring irreversibility of the gene switch. Using our inverse bifurcation analysis, specifying a desired behavior on the transition points gives rise to conditions on the parameters in a more systematic and efficient manner.\nFigure 11 shows the bifurcation diagram for the regulatory module of the mammalian G1/S transition. The bifurcation parameter is the strength of mitogenic stimulation, Fm . The quantity of primary interest is the level of E2F1, which is a transcription factor targeting genes underlying cell cycle progression. For sufficiently low level of Fm, the level of E2F1 stays at a constant low value. At a critical value of Fm a transcritical point is reached, above which the level of E2F1 starts to increase with Fm . As the level of Fm increases further, a saddle-node (SN1) is reached and the level of E2F1 undergoes a discontinuous jump upwards, thereby triggering the S phase of the cell cycle.\nFigure 11 Bifurcation diagram of cell phase transition.\n\n4.3.1 Example 1: engineer irreversibility\nFirst, we mention a number of main species and their interactions. The regulatory system consists of a core double inhibitor-activator module as well as several positive-feedback loops. The transcription factor E2F1 activates itself as well as pRB, a tumor suppressor. In turn, pRB inhibits itself as well as E2F1. The following equations describe this core module:\nAnother important collection of species is AP-1, which is used to denote a family of transcription factors that mediate the mitogenic signal Fm . The family AP-1 is also activated by E2F1, with feedback strength k25. The dynamics for the concentration of the family [AP-1] is modelled as:\nIn the first example, we consider the construction of an irreversible gene switch via parameter changes. It has been demonstrated [28] that for some value of k25 in (8), the G1/S transition becomes irreversible. That is, if Fm is increased beyond a certain point, a subsequent decrease to zero will not lead to a G0 state with a low level of E2F1.\nGeometrically, the inverse problem is to find the value of k25 such that the x-abscissa of the upper saddle node (SN2) is as close to zero as possible. As inequality constraints, we take 0.1 ≤ k25 ≤ 1.5. The result of the inverse analysis shows that the feedback strength should increase from its initial value of k25 = 0.9 to k25 = 1.099, resulting in a change of bifurcation diagram shown in Figure 12. Row 1 of Table 3 shows the number of optimization iterations as well as the number of functional evaluations (which in this case equals the number of one-parameter continuations) required to reach a tolerance of 10-3 on the function value.\nFigure 12 Example 1: initial and optimized gene switches.\nTable 3 Iteration summary\nOptim. iter. Func. eval.\nExample 1: irreversibility 6 14\nExample 2: irreversibility, fixed G1/S 9 22\n\n4.3.2 Example 2: engineer irreversibility, fixed G1/S transition\nIn this example, we consider an extension of the previous problem. Here, we would like to engineer an irreversible switch with the additional constraint that the x-abscissa of the G1/S transition point SN1 should be fixed at the value of the nominal system. In addition to allowing 0.1 ≤ k25 ≤ 1.5 as before, we also modify the stability of E2F1 and AP-1. That is, we relax the values of φE2F1 and φAP-1 in (7) and (8) from from their fixed nominal values 0.1 and 0.01 respectively, to allow searching within the ranges,\n0.08 ≤ φE2F1 ≤ 0.12\n0.008 ≤ φAP-1 ≤ 0.012.\nMathematically, the inverse problem is formulated as an nonlinear optimization problem with equality constraint on the x-abscissa of the G1/S point, SN1. The bifurcation diagram corresponding to the solution (k25 = 1.182, φE2F1 = 0.0992, φAP-1 = 0.0109) is given in Figure 13. Row 2 of Table 3 summarizes the number of iterations required.\nFigure 13 Example 2: initial and optimized gene switches.","divisions":[{"label":"title","span":{"begin":0,"end":50}},{"label":"p","span":{"begin":51,"end":674}},{"label":"p","span":{"begin":675,"end":1368}},{"label":"figure","span":{"begin":1369,"end":1425}},{"label":"label","span":{"begin":1369,"end":1378}},{"label":"caption","span":{"begin":1380,"end":1425}},{"label":"p","span":{"begin":1380,"end":1425}},{"label":"sec","span":{"begin":1427,"end":3308}},{"label":"title","span":{"begin":1427,"end":1468}},{"label":"p","span":{"begin":1469,"end":1834}},{"label":"p","span":{"begin":1835,"end":2123}},{"label":"p","span":{"begin":2124,"end":2471}},{"label":"p","span":{"begin":2472,"end":3116}},{"label":"figure","span":{"begin":3117,"end":3175}},{"label":"label","span":{"begin":3117,"end":3126}},{"label":"caption","span":{"begin":3128,"end":3175}},{"label":"p","span":{"begin":3128,"end":3175}},{"label":"table-wrap","span":{"begin":3176,"end":3308}},{"label":"label","span":{"begin":3176,"end":3183}},{"label":"caption","span":{"begin":3185,"end":3202}},{"label":"p","span":{"begin":3185,"end":3202}},{"label":"table","span":{"begin":3203,"end":3308}},{"label":"tr","span":{"begin":3203,"end":3228}},{"label":"td","span":{"begin":3203,"end":3215}},{"label":"td","span":{"begin":3217,"end":3228}},{"label":"tr","span":{"begin":3229,"end":3262}},{"label":"td","span":{"begin":3229,"end":3255}},{"label":"td","span":{"begin":3257,"end":3258}},{"label":"td","span":{"begin":3260,"end":3262}},{"label":"tr","span":{"begin":3263,"end":3308}},{"label":"td","span":{"begin":3263,"end":3301}},{"label":"td","span":{"begin":3303,"end":3304}},{"label":"td","span":{"begin":3306,"end":3308}},{"label":"title","span":{"begin":3310,"end":3374}},{"label":"p","span":{"begin":3375,"end":3895}},{"label":"p","span":{"begin":3896,"end":3915}},{"label":"p","span":{"begin":3916,"end":3938}},{"label":"p","span":{"begin":3939,"end":4278}},{"label":"label","span":{"begin":4279,"end":4288}}],"tracks":[{"project":"2_test","denotations":[{"id":"16859561-15231543-1688456","span":{"begin":214,"end":216},"obj":"15231543"},{"id":"16859561-15231543-1688457","span":{"begin":2255,"end":2257},"obj":"15231543"}],"attributes":[{"subj":"16859561-15231543-1688456","pred":"source","obj":"2_test"},{"subj":"16859561-15231543-1688457","pred":"source","obj":"2_test"}]}],"config":{"attribute types":[{"pred":"source","value type":"selection","values":[{"id":"2_test","color":"#99ec93","default":true}]}]}}