Homework Solution: The Schlogl model is famous for its bistable steady-state distribution. The reactions are B_1 2X rightwardsharpo…

    3. Schlögl model. The Schlögl model is famous for its bistable steady-state distri- bution. The reactions are where Bi and B2 denote buffered species whose respective molecular populations Ni and N2 are assumed to remain essentially constant over the time interval of interest. There is only one time-varying species, X; the state change vectors are v, = V3 = 1, ½ = Va =-1; and the propensity functions are al (x) a2(x) 우Nix(x-1) 똥x(x-1)(x-2). = = a4(x) = car For some values of the parameters this model has two stable states, and that is the case for the parameter values we have chosen here: 3 × 10-7. C3 = 10-3, №=2x105 c; C2 = 10-4, c-3.5. Ni=1x105, In the following questions, we assume the initial condition is at X(0) 250. (a) Write down the Chemical Master Equation for the model. (b) Write down the reaction rate equations and simulate the equation in matlab. (c) Write a program to run the SSA simulation and compare the result with the reaction rate equations result. You need to show at least five different trajectories.

    The Schlogl model is famous for its bistable steady-state distribution. The reactions are B_1 2X rightwardsharpoonoverleftwardsharpoon 3X, B_2 rightwardsharpoonoverleftwardsharpoon X, where B_1 and B_2 denote buffered species whose respective molecular populations N_1 and N_2 are assumed to remain essentially constant over the time interval of interest. There is only one time-varying species, X: the state change vectors are v_1 = V_3 = 1, v_2 = V_4 = -1: and the propensity functions are a_1(x) = c_1/2 N_1x(x - 1), a_2(x) = c_2/6x(x - 1)(x - 2) a_3(x) = c_3N_2, a_4(x) = c_4x. For some values of the parameters this model has two stable states, and that is the case for the parameter values we have chosen here: c_1 = 3 times 10^-7, c_2 = 10^-4, c_3 = 10^-3, c_4 = 3.5, N_1 = 1 times 10^5, N_2 = 2 times 10^5. In the following questions, we assume the initial condition is at X(0) = 250. (a) Write down the Chemical Master Equation for the model. (b) Write down the reaction rate equations and simulate the equation in matlab. (c) Write a program to run the SSA simulation and compare the result with the reaction rate equations result. You need to show at least five different trajectories.

    Expert Answer

    3. Schlögl mould. The Schlögl mould is far-famed control its bisteady steady-narreprimand distri- bution. The reactions are where Bi and B2 dramatize buffered estimation whose relative molecular populations Ni and N2 are conducive to rest essentially uniform balance the space interim of attention. There is solely undivided space-varying estimation, X; the narreprimand exexqualify vectors are v, = V3 = 1, ½ = Va =-1; and the bent functions are al (x) a2(x) 우Nix(x-1) 똥x(x-1)(x-2). = = a4(x) = car Control some values of the parameters this mould has span steady narrates, and that is the predicament control the parameter values we enjoy clarified here: 3 × 10-7. C3 = 10-3, №=2x105 c; C2 = 10-4, c-3.5. Ni=1x105, In the subjoined questions, we exhibit the judicious requisite is at X(0) 250. (a) Write down the Chemical Master Equation control the mould. (b) Write down the reaction reprimand equations and presume the equation in matlab. (c) Write a program to rush the SSA artifice and assimilate the development with the reaction reprimand equations development. You need to profession at smallest five unanalogous trajectories.

    The Schlogl mould is far-famed control its bisteady steady-narreprimand classification. The reactions are B_1 2X rightwardsharpoonoverleftwardsharpoon 3X, B_2 rightwardsharpoonoverleftwardsharpoon X, where B_1 and B_2 dramatize buffered estimation whose relative molecular populations N_1 and N_2 are conducive to rest essentially uniform balance the space interim of attention. There is solely undivided space-varying estimation, X: the narreprimand exexqualify vectors are v_1 = V_3 = 1, v_2 = V_4 = -1: and the bent functions are a_1(x) = c_1/2 N_1x(x – 1), a_2(x) = c_2/6x(x – 1)(x – 2) a_3(x) = c_3N_2, a_4(x) = c_4x. Control some values of the parameters this mould has span steady narrates, and that is the predicament control the parameter values we enjoy clarified here: c_1 = 3 spaces 10^-7, c_2 = 10^-4, c_3 = 10^-3, c_4 = 3.5, N_1 = 1 spaces 10^5, N_2 = 2 spaces 10^5. In the subjoined questions, we exhibit the judicious requisite is at X(0) = 250. (a) Write down the Chemical Master Equation control the mould. (b) Write down the reaction reprimand equations and presume the equation in matlab. (c) Write a program to rush the SSA artifice and assimilate the development with the reaction reprimand equations development. You need to profession at smallest five unanalogous trajectories.

    Expert Response

     

    ** Plea