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""" import numpy as np import matplotlib.pyplot as plt from scipy.integrate import solve_ivp from scipy import optimize # Santillan's model delayed differential equation def Santillan_E_coli_Tryptophan(time, state_all, parameters, T_ext): state_org = state_all state_all[state_all<0] = 0.0 #------------------------------------------------ # Uncertain parameters #------------------------------------------------ tau_p = parameters[0] tau_m = parameters[1] tau_rho = parameters[2] tau_e = parameters[3] R = parameters[4] n_H = parameters[5] b = parameters[6] e = parameters[7] f = parameters[8] O = parameters[9] k_mr = parameters[10] k_pr = parameters[11] k_mi = parameters[12] k_pi = parameters[13] k_mt = parameters[14] k_pt = parameters[15] c = parameters[16] d = parameters[17] gama = parameters[18] T_consume_rate = parameters[19] P = parameters[20] rho = parameters[21] mu = parameters[22] #---------------------------------- # Dependent variables #---------------------------------- K_i = k_mi/k_pi K_t = k_mt/k_pt K_r = k_mr/k_pr k_rho = 1/(tau_rho*rho) k_p = 1/(tau_p*P) kdD = rho*k_rho/30 #----------------------------------- # Steady-state #----------------------------------- T_SS = K_i K_g = T_SS/10/2 # < (steady-state Tryptophan concentration = 4.1)/10[/2(?)] g_SS = T_consume_rate*(K_i + K_g)/K_i G_SS = g_SS*K_i/(K_i+K_g) R_A_SS = T_SS/(T_SS+K_t)*R O_F_SS = (K_r*mu*O)/(K_r*k_p*(1-np.exp(-mu*tau_p))+mu*(K_r+R_A_SS)) M_F_SS = k_p*P*O_F_SS*np.exp(-mu*tau_m)*(1-b*(1-np.exp(-K_i/c))) \ /(k_rho*rho*(1-np.exp(-mu*tau_rho))+kdD+mu) E_SS = (k_rho*rho*M_F_SS*np.exp(-mu*tau_e))/(2*(gama+mu)) K = 2*(G_SS + mu*K_i)/E_SS # state O_F = state_all[0] M_F = state_all[1] E = state_all[2] T = state_all[3] # delayed state state_tau_p = state_all[4:6]; state_tau_p.resize((2,1)) state_tau_m = state_all[6:8]; state_tau_m.resize((2,1)) state_tau_rho = state_all[8:10]; state_tau_rho.resize((2,1)) state_tau_e = state_all[10::]; state_tau_e.resize((2,1)) A_tau_p = np.array([[0,1], [-12/tau_p**2, -6/tau_p]]) A_tau_m = np.array([[0,1], [-12/tau_m**2, -6/tau_m]]) A_tau_rho = np.array([[0,1], [-12/tau_rho**2, -6/tau_rho]]) A_tau_e = np.array([[0,1], [-12/tau_e**2, -6/tau_e]]) B_tau = np.array([[0], [1]]) C_tau_p = np.array([[0, -12/tau_p]]) C_tau_m = np.array([[0, -12/tau_m]]) C_tau_rho = np.array([[0, -12/tau_rho]]) C_tau_e = np.array([[0, -12/tau_e]]) D_tau = np.array([[1]]) # dxdt = Ax + Bu dO_F_tau_p = A_tau_p@state_tau_p + B_tau@np.array([[O_F]]) dO_F_tau_m = A_tau_m@state_tau_m + B_tau@np.array([[O_F]]) dM_F_tau_rho = A_tau_rho@state_tau_rho + B_tau@np.array([[M_F]]) dM_F_tau_e = A_tau_e@state_tau_e + B_tau@np.array([[M_F]]) # y = Cx + Du O_F_tau_p = C_tau_p@state_tau_p + D_tau@np.array([[O_F]]) O_F_tau_m = C_tau_m@state_tau_m + D_tau@np.array([[O_F]]) M_F_tau_rho = C_tau_rho@state_tau_rho + D_tau@np.array([[M_F]]) M_F_tau_e = C_tau_e@state_tau_e + D_tau@np.array([[M_F]]) # make 1x1 array to scalar O_F_tau_p=O_F_tau_p[0][0] O_F_tau_m=O_F_tau_m[0][0] M_F_tau_rho=M_F_tau_rho[0][0] M_F_tau_e=M_F_tau_e[0][0] d_delay_dt = np.vstack((dO_F_tau_p,dO_F_tau_m,dM_F_tau_rho,dM_F_tau_e)) d_delay_dt = d_delay_dt.squeeze() # auxilary variables A_T = b*(1-np.exp(-T/c)) E_A = K_i**n_H/(K_i**n_H + T**n_H)*E R_A = T/(T+K_t)*R G = g_SS*T/(T+K_g) F = d*T_ext/(e + T_ext*(1+T/f)) # kinetics dOF_dt = K_r/(K_r + R_A)*(mu*O - k_p*P*(O_F - O_F_tau_p*np.exp(-mu*tau_p))) - mu*O_F dMF_dt = k_p*P*O_F_tau_m*np.exp(-mu*tau_m)*(1-A_T) \ - k_rho*rho*(M_F - M_F_tau_rho*np.exp(-mu*tau_rho)) - (kdD + mu)*M_F dE_dt = 0.5*k_rho*rho*M_F_tau_e*np.exp(-mu*tau_e) - (gama + mu)*E; dT_dt = K*E_A - G + F - mu*T; if state_org[0] < 0 and dOF_dt < 0: dOF_dt = 0 if state_org[1] < 0 and dMF_dt < 0: dMF_dt = 0; if state_org[2] < 0 and dE_dt < 0: dE_dt = 0 if state_org[3] < 0 and dT_dt < 0: dT_dt = 0 dOF_MF_E_T_dt = np.array([dOF_dt, dMF_dt, dE_dt, dT_dt]) # return all state dxdt = np.hstack((dOF_MF_E_T_dt,d_delay_dt)) return dxdt # uncertain parameters & initial conditions # # [Ref] Moisés Santillán and Michael C. Mackey. Dynamic reguiation of the tryptophan # operon: A modeling study and comparison with experimental data. # Proceedings of the National Academy of Sciences, 98(4):1364–1369, February 2001. # def Santillans_Tryptophan_Model_constants(delta): #------------------------------------------------ # Uncertain ranges without experimental evidences #------------------------------------------------ Santillan_tau_p = 0.1*(1 + delta[0]) # 1 Santillan_tau_m = 0.1*(1 + delta[1]) # 2 Santillan_tau_rho = 0.05*(1 + delta[2]) # 3 Santillan_tau_e = 0.66*(1 + delta[3]) # 4 Santillan_R = 0.8*(1 + delta[4]) # 5 Santillan_n_H = 2 + delta[5] # 6 # nominal = 1.2 # delta_nominal = -0.8 Santillan_b = 0.65 + 0.35*delta[6] # 7 [0.3, 1.0] # nominal = 0.85 # delta_nominal = 0.5714 Santillan_e = 0.9*(1 + delta[7]) # 8 Santillan_f = 380*(1 + delta[8]) # 9 Santillan_O = 3.32e-3*(1 + delta[9]) # 10 Santillan_k_mr = 1.2e-2*(1 + delta[10]) # 11 value in [Ref] & its supplementary is different Santillan_k_pr = 4.6*(1 + delta[11]) # 12 value in [Ref] & its supplementary is different # but the ratio, kmr/kpr is the same Santillan_k_mi = 7.2e-2*(1 + delta[12]) # 13 Santillan_k_pi = 1.76e-2*(1 + delta[13]) # 14 Santillan_k_mt = 2.1e4*(1 + delta[14]) # 15 Santillan_k_pt = 348*(1 + delta[15]) # 16 Santillan_c = 4e-2*(1 + delta[16]) # 17 Santillan_d = 23.5*(1 + delta[17]) # 18 Santillan_gama = 0.01*(1 + delta[18]) # 19 # nominal value 0 # delta nominal = -1 #---------------------------------- # Uncertain ranges from experiments #---------------------------------- Santillan_T_consume_rate = 21.5 + 7.5*delta[19] # 20 # range 14 ~ 29 # nominal 22.7 -> 0.16 Santillan_P = 2.785 + 0.675*delta[20] # 21 # range 2.11 - 3.46 micro-Molar, # nominal 2.6 -> -0.2741 # 1250 molecule per cell, cell average volume 6.0e-16 - 9.8e-16 # liters, average volumn = (6.0 + 9.8)/2*1e-16 = 7.9e-16 liters # 1250 molecule = 1250/6.022e23 = 2.0757e-21 mole # 2.0757e-21/7.9e-16 = 2.62e-6 Molar = 2.62 micro-Molar Santillan_rho = 3.12 + 0.75*delta[21] # 21 # range 2.37 - 3.87 micro-Molar, # nominal 2.9 -> -0.2933 # 1400 molecule per cell, cell average volume 6.0e-16 - 9.8e-16 # liters, average volumn = (6.0 + 9.8)/2*1e-16 = 7.9e-16 liters # 1400 molecule = 1400/6.022e23 = 2.3248e-21 mole # 2.3248e-21/7.9e-16 = 2.94e-6 Molar = 2.94 micro-Molar Santillan_mu = 0.0259 + 0.0159*delta[22] # 22 # range 0.01 ~ 0.0417 [min^-1], # nominal 0.01 -> -1 # actual range from 0.6 h^-1 ~ 2.5 h^-1 # return values num_para = 23 perturbed_model_para = np.zeros(num_para) perturbed_model_para[0] = Santillan_tau_p perturbed_model_para[1] = Santillan_tau_m perturbed_model_para[2] = Santillan_tau_rho perturbed_model_para[3] = Santillan_tau_e perturbed_model_para[4] = Santillan_R perturbed_model_para[5] = Santillan_n_H perturbed_model_para[6] = Santillan_b perturbed_model_para[7] = Santillan_e perturbed_model_para[8] = Santillan_f perturbed_model_para[9] = Santillan_O perturbed_model_para[10] = Santillan_k_mr perturbed_model_para[11] = Santillan_k_pr perturbed_model_para[12] = Santillan_k_mi perturbed_model_para[13] = Santillan_k_pi perturbed_model_para[14] = Santillan_k_mt perturbed_model_para[15] = Santillan_k_pt perturbed_model_para[16] = Santillan_c perturbed_model_para[17] = Santillan_d perturbed_model_para[18] = Santillan_gama perturbed_model_para[19] = Santillan_T_consume_rate perturbed_model_para[20] = Santillan_P perturbed_model_para[21] = Santillan_rho perturbed_model_para[22] = Santillan_mu return perturbed_model_para # check negative states to stop the integrator def negativeConcentration(time,state,parameters, T_ext): tol = -1e-1; OF_MF_E_T = state[0:4] delay_output = state[4::2] all_positive_state = np.hstack((OF_MF_E_T,delay_output)) value = 1-float(any(all_positive_state