. 2016 Dec 5;7:590. doi: 10.3389/fphys.2016.00590

Table 1.

Summary of the 16 PD fixed effects and mixed effects models for which the structural identifiability was investigated.

N	Model equations	I/O	ICs	Fixed effects models	Mixed effects models
				Fixed effect parameters	Fixed effect parameters^a	Random effect parameters^b
1	$E = k_{e} \frac{R_{t o t} C_{p}}{K_{d} + C}$	C_p/E		R_tot, k_e, K_d	k_e, K_d	η_ke, η_Kd
2	$E = \frac{E_{m} {(R_{t o t} C_{p})}^{n}}{{(K_{d} + C_{p})}^{n} R C_{50}^{n} + {(R_{t o t} C_{p})}^{n}}$	C_p/E		R_tot, E_m, RC₅₀, n, K_d	E_m, RC₅₀, n, K_d	η_Em, η_RC50, η_n, η_Kd
3	$Ė = k_{i n} (1 + k_{e} \frac{R_{t o t} C_{p}}{K_{d} + C_{p}}) - k_{o u t} E$	C_p/E	E(0) = k_out/k_in	R_tot, k_in, k_out, k_e, K_d	k_in, k_out, k_e, K_d	η_kin, η_kout, η_ke, η_Kd
4	$Ė = k_{i n} - k_{o u t} (1 + k_{e} \frac{R_{t o t} C_{p}}{K_{d} + C_{p}}) E$	C_p/E	E(0) = k_out/k_in	R_tot, k_in, k_out, k_e, K_d	k_in, k_out, k_e, K_d	η_kin, η_kout, η_ke, η_Kd
5	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{p} - k_{o f f} R C$	C_p/E	RC(0) = 0	R_tot, k_on, k_off, k_e	k_on, k_off, k_e	η_kon, η_koff, η_ke
	E = k_eRC
6	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{p} - k_{o f f} R C$	C_p/E	RC(0) = 0	R_tot, k_on, k_off, E_m, RC₅₀, n	k_on, k_off, E_m, RC₅₀, n	η_RC50, η_kon, η_koff, η_Em, η_n
	$E = \frac{E_{m} R C^{n}}{R C_{50}^{n} + R C^{n}}$
7	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{p} - k_{o f f} R C$	C_p/E	RC(0) = 0	R_tot, k_on, k_off, k_in, k_out, k_e	k_on, k_off, k_in, k_out, k_e	η_kon, η_koff, η_kin, η_kout, η_ke
	Ė = k_in(1+k_eRC)−k_outE		E(0) = k_out/k_in
8	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{p} - k_{o f f} R C$	C_p/E	RC(0) = 0	R_tot, k_on, k_off, k_in, k_out, k_e	k_on, k_off, k_in, k_out, k_e	η_kon, η_koff, η_kin, η_kout, η_ke
	Ė = k_in − k_out(1+k_eRC)E		E(0) = k_out/k_in
9	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, k_e, K_d	k_e0, k_e, K_d	η_ke0, η_ke, η_Kd
	$E = k_{e} \frac{R_{t o t} C_{e}}{K_{d} + C_{e}}$
10	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, E_m, RC₅₀, n, K_d	k_e0, E_m, RC₅₀, n, K_d	η_RC50, η_ke0, η_Em, η_n, η_Kd
	$E = \frac{E_{m} {(R_{t o t} C_{e})}^{n}}{{(K_{d} + C_{e})}^{n} R C_{50}^{n} + {(R_{t o t} C_{e})}^{n}}$
11	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, k_in, k_out, k_e, K_d	k_e0, k_in, k_out, k_e, K_d	η_ke0, η_kin, η_kout, η_ke, η_Kd
	$Ė = k_{i n} (1 + k_{e} \frac{R_{t o t} C_{e}}{K_{d} + C_{e}}) - k_{o u t} E$		E(0) = k_out/k_in
12	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, k_in, k_out, k_e, K_d	k_e0, k_in, k_out, k_e, K_d	η_ke0, η_kin, η_kout, η_ke, η_Kd
	$Ė = k_{i n} - k_{o u t} (1 + k_{e} \frac{R_{t o t} C_{e}}{K_{d} + C}) E$		E(0) = k_out/k_in
13	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, k_on, k_off, k_e	k_e0, k_on, k_off, k_e	η_ke0, η_kon, η_koff, η_ke
	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{e} - k_{o f f} R C$		RC(0) = 0
	E = k_eRC
14	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, RC₅₀, k_on, k_off, E_m, n	k_e0, RC₅₀, k_on, k_off, E_m, n	η_ke0, η_RC50, η_kon, η_koff, η_Em, η_n
	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{e} - k_{o f f} R C$		RC(0) = 0
	$E = \frac{E_{m} R C^{n}}{R C_{50}^{n} + R C^{n}}$
15	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, k_on, k_off, k_in, k_out, k_e	k_e0, k_on, k_off, k_in, k_out, k_e	η_ke0, η_kon, η_koff, η_kin, η_kout, η_ke
	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{e} - k_{o f f} R C$		RC(0) = 0
	Ė = k_in(1+k_eRC)−k_outE		E(0) = k_out/k_in
16	$\dot{C_{e}} = k_{e 0} * (C_{p} - C_{e})$	C_p/E	C_e(0) = 0	k_e0, R_tot, k_on, k_off, k_in, k_out, k_e	k_e0, k_on, k_off, k_in, k_out, k_e	η_ke0, η_kon, η_koff, η_kin, η_kout, η_ke
	$\dot{R C} = k_{o n} (R_{t o t} - R C) C_{e} - k_{o f f} R C$		RC(0) = 0
	Ė = k_in − k_out(1+k_eRC)E		E(0) = k_out/k_in

N, Model number; I/O, Model inputs/outputs; ICs, Initial conditions.^aR_tot was fixed at 100 when analysing the mixed effects models.^bEach mixed-effects model was assumed to have a diagonal covariance matrix Ω with lognormally distributed random effects.