Detailed ExampleΒΆ
This section illustrates the construction of a closed loop system for moderately complicated open loop system, the mTOR pathway as modeled in Biomodels 823 (Varusai2018). The figure below provides an overview of the pathway that is modelled.
Our objective is to control the concentration of the concentration of DEPTOR, the mTOR interacting proteint with the DEP dependendent domain. concentration of 100.
Below we display the model in the Antimony modeling language.
We have modified the model slightly by adding the chemical species E_v11.
In the spirit of Metabolic Control Analysis,
this species has an additive effect on the kinetics of reaction v11.
1NEW_MTOR = """
2// Created by libAntimony v2.13.2
3function Constant_flux__irreversible(v)
4 v;
5end
6
7Constant_flux__irreversible is "Constant flux (irreversible)"
8
9function Henri_Michaelis_Menten__irreversible(substrate, Km, V)
10 V*substrate/(Km + substrate);
11end
12
13Henri_Michaelis_Menten__irreversible is "Henri-Michaelis-Menten (irreversible)"
14
15function HMM_Mod(V, s, m, Km)
16 V*s*m/(Km + s);
17end
18
19HMM_Mod is "HMM_Mod"
20
21function Function_for_v11(k11ca, pmTORC1, DEPTOR, Km11a, pDEPTOR, k11cb, pmTORC2, Km11b)
22 k11ca*pmTORC1*DEPTOR/(Km11a + pDEPTOR) + k11cb*pmTORC2*DEPTOR/(Km11b + DEPTOR);
23end
24
25Function_for_v11 is "Function_for_v11"
26
27function Function_for_v5(k5ca, pIRS, Akt, Km5a, k5cb, pmTORC2, Km5b)
28 k5ca*pIRS*Akt/(Km5a + Akt) + k5cb*pmTORC2*Akt/(Km5b + Akt);
29end
30
31Function_for_v5 is "Function_for_v5"
32
33
34model *Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR()
35
36 // Compartments and Species:
37 compartment compartment_;
38 species IR in compartment_, pIR in compartment_, IRS in compartment_, pIRS in compartment_;
39 species iIRS in compartment_, Akt in compartment_, pAkt in compartment_;
40 species mTORC1 in compartment_, pmTORC1 in compartment_, mTORC2 in compartment_;
41 species pmTORC2 in compartment_, imTORC2 in compartment_, mTORC1_DEPTOR in compartment_;
42 species mTORC2_DEPTOR in compartment_, DEPTOR in compartment_, pDEPTOR in compartment_;
43 E_v11 in compartment_;
44
45 // Reactions:
46 v1: IR => pIR; compartment_*Henri_Michaelis_Menten__irreversible(IR, Km1, V1);
47 v2: pIR => IR; compartment_*Henri_Michaelis_Menten__irreversible(pIR, Km2, V2);
48 v3: IRS => pIRS; compartment_*HMM_Mod(k3c, IRS, pIR, Km3);
49 v4: pIRS => IRS; compartment_*Henri_Michaelis_Menten__irreversible(pIRS, Km4, V4);
50 v5: Akt => pAkt; compartment_*Function_for_v5(k5ca, pIRS, Akt, Km5a, k5cb, pmTORC2, Km5b);
51 v6: pAkt => Akt; compartment_*Henri_Michaelis_Menten__irreversible(pAkt, Km6, V6);
52 v7: mTORC1 => pmTORC1; compartment_*HMM_Mod(k7c, mTORC1, pAkt, Km7);
53 v8: pmTORC1 => mTORC1; compartment_*Henri_Michaelis_Menten__irreversible(pmTORC1, Km8, V8);
54 v9: mTORC2 => pmTORC2; compartment_*HMM_Mod(k9c, mTORC2, pIR, Km9);
55 v10: pmTORC2 => mTORC2; compartment_*Henri_Michaelis_Menten__irreversible(pmTORC2, Km10, V10);
56 v11: DEPTOR => pDEPTOR; E_v11 + compartment_*Function_for_v11(k11ca, pmTORC1, DEPTOR, Km11a, pDEPTOR, k11cb, pmTORC2, Km11b);
57 v12: pDEPTOR => DEPTOR; compartment_*Henri_Michaelis_Menten__irreversible(pDEPTOR, Km12, V12);
58 v13: mTORC1 + DEPTOR -> mTORC1_DEPTOR; compartment_*(k13f*mTORC1*DEPTOR - k13r*mTORC1_DEPTOR);
59 v14: mTORC2 + DEPTOR -> mTORC2_DEPTOR; compartment_*(k14f*mTORC2*DEPTOR - k14r*mTORC2_DEPTOR);
60 v15: IRS => iIRS; compartment_*HMM_Mod(k15c, IRS, pmTORC1, Km15);
61 v16: iIRS => IRS; compartment_*Henri_Michaelis_Menten__irreversible(iIRS, Km16, V16);
62 v17: => DEPTOR; compartment_*Constant_flux__irreversible(ks17);
63 v18: pDEPTOR => ; compartment_*kd18*pDEPTOR;
64
65 // Species initializations:
66 IR = 50;
67 pIR = 0;
68 IRS = 100;
69 pIRS = 0;
70 iIRS = 0;
71 Akt = 100;
72 pAkt = 0;
73 mTORC1 = 250;
74 pmTORC1 = 0;
75 mTORC2 = 200;
76 pmTORC2 = 0;
77 imTORC2 = 0;
78 mTORC1_DEPTOR = 0;
79 mTORC2_DEPTOR = 0;
80 DEPTOR = 350;
81 pDEPTOR = 0;
82 // Added to model
83 E_v11 = 1;
84
85 // Compartment initializations:
86 compartment_ = 1;
87
88 // Variable initializations:
89 V1 = 1;
90 Km1 = 95;
91 V2 = 1;
92 Km2 = 35;
93 k3c = 0.1;
94 Km3 = 50;
95 V4 = 1;
96 Km4 = 50;
97 k5ca = 0.05;
98 Km5a = 7;
99 k5cb = 1.5;
100 Km5b = 4;
101 V6 = 2;
102 Km6 = 34;
103 k7c = 0.1;
104 Km7 = 2;
105 V8 = 6;
106 Km8 = 1;
107 k9c = 0.3;
108 Km9 = 160;
109 V10 = 3;
110 Km10 = 7;
111 k11ca = 0.1;
112 Km11a = 120;
113 k11cb = 0.13;
114 Km11b = 11;
115 V12 = 4;
116 Km12 = 7;
117 k13f = 0.001;
118 k13r = 0.006;
119 k14f = 0.007;
120 k14r = 0.006;
121 k15c = 0.1;
122 Km15 = 50;
123 V16 = 1;
124 Km16 = 50;
125 ks17 = 0;
126 kd18 = 0;
127
128 // Other declarations:
129 const compartment_, V1, Km1, V2, Km2, k3c, Km3, V4, Km4, k5ca, Km5a, k5cb;
130 const Km5b, V6, Km6, k7c, Km7, V8, Km8, k9c, Km9, V10, Km10, k11ca, Km11a;
131 const k11cb, Km11b, V12, Km12, k13f, k13r, k14f, k14r, k15c, Km15, V16;
132 const Km16, ks17, kd18;
133
134 // Unit definitions:
135 unit volume = 1e-3 litre;
136 unit substance = 1e-3 mole;
137
138 // Display Names:
139 compartment_ is "compartment";
140
141 // CV terms:
142 compartment_ hypernym "http://identifiers.org/ncit/C48694"
143 IR identity "http://identifiers.org/pr/PR:000009064"
144 pIR hypernym "http://identifiers.org/pr/PR:000009064"
145 IRS identity "http://identifiers.org/ncit/C28474"
146 pIRS hypernym "http://identifiers.org/ncit/C28474"
147 iIRS hypernym "http://identifiers.org/ncit/C28474"
148 Akt identity "http://identifiers.org/pr/PR:000029189"
149 pAkt hypernym "http://identifiers.org/pr/PR:000029189"
150 mTORC1 hypernym "http://identifiers.org/ncit/C96314"
151 pmTORC1 hypernym "http://identifiers.org/ncit/C96314"
152 mTORC2 identity "http://identifiers.org/ncit/C96315"
153 pmTORC2 hypernym "http://identifiers.org/ncit/C96315"
154 imTORC2 hypernym "http://identifiers.org/ncit/C96315"
155 mTORC1_DEPTOR part "http://identifiers.org/ncit/C96314"
156 mTORC1_DEPTOR part "http://identifiers.org/ncit/C101595"
157 mTORC2_DEPTOR part "http://identifiers.org/ncit/C101595"
158 mTORC2_DEPTOR part "http://identifiers.org/ncit/mTORC2"
159 DEPTOR identity "http://identifiers.org/ncit/C101595"
160 pDEPTOR hypernym "http://identifiers.org/ncit/C101595"
161 v1 hypernym "http://identifiers.org/go/GO:0016310"
162 v2 hypernym "http://identifiers.org/go/GO:0016311"
163 v3 hypernym "http://identifiers.org/go/GO:0016310"
164 v4 hypernym "http://identifiers.org/go/GO:0016311"
165 v5 hypernym "http://identifiers.org/go/GO:0016310"
166 v6 hypernym "http://identifiers.org/go/GO:0016311"
167 v7 hypernym "http://identifiers.org/go/GO:0016310"
168 v8 hypernym "http://identifiers.org/go/GO:0016311"
169 v9 hypernym "http://identifiers.org/go/GO:0016310"
170 v10 hypernym "http://identifiers.org/go/GO:0016311"
171 v11 hypernym "http://identifiers.org/go/GO:0016310"
172 v12 hypernym "http://identifiers.org/go/GO:0016311"
173 v13 hypernym "http://identifiers.org/ncit/C18469"
174 v14 hypernym "http://identifiers.org/ncit/C18469"
175 v15 hypernym "http://identifiers.org/ncit/C16983"
176 v15 hypernym "http://identifiers.org/sbo/SBO:0000169"
177 v16 hypernym "http://identifiers.org/ncit/C21018"
178 v17 hypernym "http://identifiers.org/ncit/C80450"
179 v18 hypernym "http://identifiers.org/ncit/C61559"
180end
181
182Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR is "Varusai2018 - Dynamic modelling of the mTOR signalling network reveals complex emergent behaviours conferred by DEPTOR"
183
184Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR description "http://identifiers.org/pubmed/29330362"
185Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR model_entity_is "http://identifiers.org/biomodels.db/MODEL1909250003",
186 "http://identifiers.org/biomodels.db/BIOMD0000000823"
187Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR property "http://identifiers.org/mamo/MAMO_0000046"
188Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR property "http://identifiers.org/pw/PW:0000180"
189Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR property "http://identifiers.org/ncit/C101595"
190Varusai2018___Dynamic_modelling_of_the_mTOR_signalling_network_reveals_complex_emergent_behaviours_conferred_by_DEPTOR taxon "http://identifiers.org/taxonomy/9606"
191"""
To simplify the analysis, we make use of the ctl.Timeseries class (essentially a pandas.DataFrame whose index is time in milliseconds).
controlSBML has several methods that makes it easy to plot and manipulate ctl.Timeseries objects.
We begin by creating a control.NonlinearIOSystem for the mTOR model.
ctlsb = ctl.ControlSBML(NEW_MTOR,
input_names=["E_v11"],
output_names=["mTORC1_DEPTOR", "pAkt"])
mtor = ctlsb.makeNonlinearIOSystem("mtor")
Next, we plot the open loop behavior of the model.
1ctlsb = ctl.ControlSBML(NEW_MTOR,
2 input_names=["E_v11"],
3 output_names=["mTORC1_DEPTOR", "pAkt"])
4mtor = ctlsb.makeNonlinearIOSystem("mtor")
5ctl.plotOneTS(ctl.simulateSystem(mtor, end_time=500), figsize=(5,5))
1ref = 100 # Desired concentration for mTORC1_DEPTOR
2kP = 0.1
3kI = 0.01
4# Calculate derivative of state
5def updfcn(t, x, u, _):
6 # Accumulate the control error
7 # t: float (time)
8 # x: array-float (state)
9 # u: array-float (output from OLS)
10 # returns: array-float (derivative of state)
11 dx = ref - u[0]
12 #print(t, u, dx)
13 return dx
14
15# Calculate output value
16def outfcn(t, x, y, _):
17 # Calculate the output from the input
18 # t: float (time)
19 # x: array-float (state)
20 # e: array-float (inputs)
21 # returns: array (output)
22 new_err = ref - y[0]
23 output = -kI*x[0] - kP*new_err
24 #print(t, output)
25 return output
26
27# Create a PI Controller
28controller = control.NonlinearIOSystem(
29 updfcn,
30 outfcn,
31 states=1,
32 inputs=['in'],
33 outputs=['out'], name='controller')