PMC:1570349 / 18222-21290 JSONTXT

{"project":"2_test","denotations":[{"id":"16859561-10659856-1688452","span":{"begin":232,"end":234},"obj":"10659856"},{"id":"16859561-10659857-1688453","span":{"begin":235,"end":237},"obj":"10659857"},{"id":"16859561-12787501-1688454","span":{"begin":428,"end":430},"obj":"12787501"},{"id":"16859561-12787501-1688455","span":{"begin":1065,"end":1067},"obj":"12787501"}],"text":"4.1 Finding distance to oscillation in a 4 gene model\nTo understand design principles underlying biological systems, simple oscillatory and switch-like systems have been constructed experimentally. Several systems based on E. coli [24,25] have been successfully demonstrated. Recently, components of the Lac and Ntr systems have been used to construct a genetic clock that upon change in connectivity, exhibits switch behavior [26]. After ignoring the dynamics of the read-out gene, the system of three equations for the concentrations of mRNA (x1, x3, x5) and proteins (x2, x4, x6) is given as,\nwhere βi are rate constants. The rates of transcription are represented by the following tri-phasic functions,\nwhere gjk are kinetic-order parameters, typically less than 4.\nGiven a stable, steady solution for a gene system with a set of nominal parameters, a relevant engineering question is how to construct a genetic oscillator. For the current system, the analytical solutions for the bifurcation manifolds are available via the Routh-Hurwitz stability criterion ([26], supplementary data). However, such analysis cannot be easily carried out for more complex models. Instead, computational methods have to be used.\nSuppose the stable system has rate parameter β1 = β3 = 22.2, β2 = β4 = 1.39, kinetic-order parameters g32 = -2, g12 = g14 = g54 = 1. If in practice only the rate parameters βi can be varied, the question is how best to construct an oscillatory system. The bifurcation diagram for the model is shown in Figure 7, where it can be seen that for the nominal values of gjk the system is stable and far away from the line of Hopf bifurcation. The inverse bifurcation question is: do there exist parameters βi such that the system is oscillatory? We consider the problem with the following parameter constraints:\nFigure 7 Initial bifurcation diagram for 4 gene model: β1 = β3 = 222, β2 = β4 = 1.39. 0.1 ≤ β1, β3 ≤ 25\n0.1 ≤ β2, β4 ≤ 4.\nBy minimizing the distance from the nominal parameter point (g12 = 1, g32g14 = -2) to the Hopf bifurcation line, the system can be made to lie on the boundary of the stable regime; see Figure 8. Table 1 shows the iteration counts required to obtain the result, using Sequential Quadratic Programming (SQP) with line-search [27]. Owing to the use of line-search, each optimization iteration entails a number of functional evaluations, this in turn requiring a number of one-parameter continuation iterations to find the (locally) closest point on Σ(ps). The optimality tolerance on the objective is set as 10-3 relative to its initial value and the relative parameter tolerance for approximating F(p) is set to 10-4. Clearly, the number of one-parameter continuations is significantly higher than the optimization iterations. Thus, for high dimensional examples it is important that the former can be carried out efficiently.\nFigure 8 Optimized bifurcation diagram for 4 gene model: β1 = 22.186, β2 = 4, β3 = 22.145, β4 = 0.1.\nTable 1 Iteration summary\nOptim. iter. Func. eval. One-param contin.\n4 gene system 7 15 60"}