DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems ...

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DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems Biology(ATP) concentration originally proposed in [21] and later slightly modified and solved in [28] is investigated. Wehave skipped the path constraints added in [28], because on the basis of the system behavior, these constraints werenever violated. The aim of the optimization is to minimize the intracellular oscillations behavior with the help oftwo binary control variables. The values of these variables and the time of the switching from one mode to anothertogether with the time-independent parameter are decision variables. Afterwards, the problem is formulated asminimization of the state variables deviations from the desired values (see Table 8.1) over the whole time intervalsubject to∫ tFmin J 0 =x,u i,p0⎛⎝4∑j=1⎞(w j xj (t)−x s ) 2j +w5 u 1 +w 6 u 2⎠dt (8.11)ẋ 1 = k 1 +k 2 x 1 − k 3x 1 x 2x 1 +K 4− k 5x 1 x 3x 1 +K 6(8.12)ẋ 2 = (1−u 2 )k 7 x 1 − k 8x 2x 2 +K 9(8.13)ẋ 3 = k 10x 2 x 3 x 4+k 12 x 2 +k 13 x 1 − k 16x 3+ x 4x 4 +K 11 x 3 +K 17 10 −u k 14 x 3 k 14 x 31 −(1−u 1 )p 1 x 3 +K 15 x 3 +K 15(8.14)ẋ 4 = − k 10x 2 x 3 x 4x 4 +K 11+ k 16x 3x 3 +K 17− x 410and the time-independent parameter(8.15)1 ≤ p 1 ≤ 1.3 (8.16)where state variables represent the concentration of activated G-protein (x 1 ), active phospholipase C (x 2 ), intracellularcalcium (x 3 ), and intra-ER calcium (x 4 ). The time-fixed parameters p = (k 1 ,...,K 17 ) together with theinitial concentrations, desired values of the state variables and weighted coefficients are described in detail in theTable 8.1. As the control variables are chosen binaries (u 1 ,u 2 ), which have an impact on the concentration of anuncompetitive inhibitor of the PMCA (plasma membrane Ca 2+ ) ion pump and on the inhibitor of PLC activationby the G-protein. The influence of the first inhibitor is modeled according to Michaelis-Menten kinetics and of thesecond inhibitor with the help of the term (1−u 2 ), where u 2 = 1 corresponds with the maximum amount of theinhibitor.For testing the toolbox applicability two cases were chosen: (i) scenario with 6 free time intervals and onecontrol variable, second one is set at the value of zero all the time; (ii) scenario with 5 free time intervals and twocontrol variables. One additional equality constraint was added to retain the total time at the fixed value (t F ). Allresults presented later were obtained with the help of MITS solver implemented directly into the toolbox. The costfunction in all scenarios neglects all weight parameters and therms: u 1 ,u 2 if those exist.3020x 1x 1x 218x 225x 316x 40 2 4 6 8 10 12 14 16 18 20 22x 3x 4State variables201510State variables1412108654200 2 4 6 8 10 12 14 16 18 20 22Time0TimeFigure 8.3 – Simulation of the system with no inhibition (left) and with the constant maximum inhibition of thePMCA (right) for the calcium oscillator problem.The complex oscillations of the state variables are shown in the Figures 8.3 where two cases were investigated.The first one, on the left side where no inhibition is considered (u 1 = 0, u 2 = 0, p 1 = 1) and the secondPage – 37
DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems Biologyone, on the right side, where full inhibition of the PMCA and no channel blocking is considered (u 1 = 1, u 2 = 0,p 1 = 1.3) during the whole simulation time. These figures are obtained directly with the help of the simulationmodule implemented in our toolbox.8.2.1 The scenario with free transition times and one control variableDOTcvp: midop_PhaseResettingOfCalciumOscillationsA.mIn the paper [28] the author reported that the system is extremely sensitive to the small perturbations in thestimulus. Previous authors used the multiple shooting method for solving this problem. For the first scenario withone control variable the value of the cost function 1604.13 was reported. We obtained a solution of 1620.55 whichis slightly worse that the one presented in the literature.2520x 1x 2x 310.90.8State variables1510x 40 2 4 6 8 10 12 14 16 18 20 220.7Control variable0.60.50.40.350.20.100 2 4 6 8 10 12 14 16 18 20 22Time0TimeFigure 8.4 – Optimal state trajectories (left) with corresponding control profile (right) for the calcium oscillatorproblem.The optimal trajectories are shown in the Figure 8.4 and the final value of the time-independent parameteris 1.14601340. Following the figures it is possible to say that the impact of the PMCA inhibitor is significant andsmooths the state trajectories considerably.8.2.2 The scenario with free transition times and two control variablesDOTcvp: midop_PhaseResettingOfCalciumOscillationsB.mFor the second scenario, where two control variables are active, the cost function value of 1538.00 wasreported from the afore mentioned literature. Our value is 1542.50 what is comparable but the total time of theuse of the first and the second inhibitor is during the simulation 13.3% less. As well not only the total time of thestimuli with the inhibitor is lower but also the amount of the inhibition of the PMCA ion pump. This improvementhas an influence on the total inhibitor price.The appropriate optimal trajectories are shown in the Figure 8.5 where oscillations are smoothed considerably.The dotted red lines always indicate the desired states. The optimal value of the time-independent parameteris 1.02430397. In the case that the stimuli will stop, the oscillations appear again (see Figure 8.6) due to theinstability of the steady states.Page – 38
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