1. Question 1
2. Question 2
a.
b.
N1' = 0.1 (K1 N1 - a N1 N2 - (N1)^2)
N2' = 0.2 (K2 N2 - (N2)^2)
K1 = 40
K2 = 10
a = 2.5
N1 = 40
N2 = 0
@ 10 N1 = 35 + rand[10]
@ 10 N2 = rand[10]
t0 = 0
tf = 100
hr = 0.1
count1 = 1
script
write "
N1
", "
N2", "
N1 Final", "N2 Final"
setting count1 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
run
write N1@10.000001, N2@10.000001, N1@last, N2@last
next
end script
Below is a phase plot of each stability point with multiple simulations run. The
simulations run were done using the sample PLAS code above, where the initial
values were changed to be the steady-state values and the lines with the rand
functions were changed slightly for each analysis of the steady-states to perturb
the values of N1 and N2 slightly and observe the effects and determine the
stability of each point.
(0,0) = unstable:
(0,10) = unstable:
(15,10) = stable:
(40,0) = unstable:
c. Changing K2 from 10 to 20 causes two changes: decreases the number of
steady-states from 4 to 3 and changes the stable node from (15,10) to (0,20).
The new steady-states when K2 = 20 is (0,0), (0,20), and (40,0). Shown below,
when the graph is perturbed slightly from each of the steady-states, it only
returns back to (0,20). No other changes were made to the system besides that.
(0,0) = unstable:
(0,20) = stable:
Note: the x-axis and y-axis begin at -10, but the phase plots do converge to
(0,20) on the graph above.
(40,0) = unstable:
Note: the x-axis and y-axis begin at -10, but the phase plots do converge to
(0,20) on the graph above.
3. Question 3
a. Below are the elementary modes of the system:
b. In order to consume only Gext and produce only Fext, we need to ensure that the
following node is the only one possible:
This can be done by blocking r1 in order to prevent Aext from acting as an input
to the system. In addition, blocking r8 internally makes sure that the pathway
does not go from node F to node E, which will essentially block Dext and Eext
from being produced as an output. All the other reactions can be left alone, as
they will not affect the outcome. By blocking only r1 and r8, it can be assured that
the only input into the system is Gext and the only output out of the system is
Fext.
4. Question 4
Below is the following ODE and parameters that we used to model the release of the drug from
the pill in the stomach to the bloodstream.
S' = - 0.5 (Vmax S / (Km+S))
B' = 0.5 (Vmax S / (Km+S)) + k0B + kLB L - (kB0 + kBL) B
L' = kBL B - (kL0+ kLB) L
B = 0
L = 0
S = 5
Vmax = 0.45
Km = .1
k0B = 0
kLB = .5
kB0 = .9
kBL = .8
kL0 = .3
t0 = 0
tf = 48
hr = .1
!! B L S
The main parameters changed were Vmax and Km, used to model the Michaelis-Menten
kinetics of the drug release. These parameters were found using a Monte-Carlo simulation to
test multiple parameters and optimize which would satisfy the conditions stated in the problem,
which were to sustain a concentration in the bloodstream at a level above 0.1 for at least 20 of
the 24 hours, and also have the pill metabolized in the stomach to less than 5% of the original
dose by 24 hours. Below is the graph of the plot, showing the dynamics of the drug in the
bloodstream (B), stomach (S), and liver (L). Note that time t = 0 signifies when the drug first
entered the stomach.
In addition, with the table below, we see that the bloodstream concentration does not drop
below 0.1 between t = 0 and 20 hours as well as the concentration in the stomach at t = 24
hours being less than 5% of the original dose (5% of 5 = 0.25).
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
B
L
0
0
0.02029805
0.03749223
0.05212304
0.06463315
0.07538534
0.08467704
0.0927523
0.09981149
0.1060193
0.1115112
0.1163988
0.120774
0.124713
0.1282787
0.1315231
0.1344896
0.1372144
0.1397277
0.1420546
0.1442165
0.1462314
0.1481145
0.1498788
0.1515354
0.1530939
0.1545626
0.1559486
0.1572584
0.1584974
0.1596707
0.1607825
0.1618369
0.1628374
0.1637872
0.1646893
0.1655464
0.1663609
0.1671351
S
5
0.0008125354 4.977942
0.002998463 4.955886
0.006235299 4.933832
0.01026297 4.91178
0.01487223 4.88973
0.01989517 4.867682
0.0251975
4.845636
0.03067229 4.823592
0.03623477 4.80155
0.04181819 4.77951
0.04737039 4.757472
0.052851
4.735436
0.05822919 4.713403
0.06348182 4.691371
0.06859193 4.669342
0.07354753 4.647315
0.07834057 4.62529
0.08296617 4.603267
0.08742197 4.581247
0.09170753 4.559228
0.09582399 4.537213
0.09977365 4.515199
0.1035597
4.493188
0.1071861
4.471179
0.1106571
4.449172
0.1139776
4.427168
0.1171524
4.405166
0.1201867
4.383167
0.1230856
4.36117
0.1258543
4.339175
0.1284979
4.317183
0.1310215
4.295194
0.13343
4.273207
0.1357283
4.251223
0.137921
4.229242
0.1400127
4.207263
0.1420078
4.185286
0.1439106
4.163313
3.9
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8
8.1
8.2
0.1678712
0.1685711
0.1692366
0.1698696
0.1704715
0.171044
0.1715884
0.1721062
0.1725985
0.1730666
0.1735116
0.1739346
0.1743367
0.1747188
0.1750818
0.1754267
0.1757542
0.1760651
0.1763603
0.1766404
0.1769061
0.1771581
0.177397
0.1776233
0.1778377
0.1780407
0.1782328
0.1784144
0.1785862
0.1787483
0.1789015
0.1790459
0.179182
0.1793101
0.1794307
0.179544
0.1796504
0.1797501
0.1798434
0.1799307
0.1800121
0.1800879
0.1801584
0.1802237
0.1457251
0.1474552
0.1491048
0.1506775
0.1521767
0.1536058
0.1549679
0.156266
0.1575032
0.1586822
0.1598056
0.160876
0.1618958
0.1628673
0.1637928
0.1646743
0.1655138
0.1663133
0.1670747
0.1677996
0.1684898
0.1691468
0.1697721
0.1703672
0.1709335
0.1714723
0.1719849
0.1724724
0.172936
0.1733768
0.1737958
0.1741941
0.1745724
0.1749319
0.1752733
0.1755974
0.175905
0.1761969
0.1764738
0.1767363
0.1769851
0.1772208
0.177444
0.1776553
4.141342
4.119374
4.097408
4.075446
4.053486
4.031529
4.009575
3.987624
3.965676
3.943731
3.921789
3.89985
3.877914
3.855981
3.834052
3.812125
3.790202
3.768282
3.746365
3.724452
3.702542
3.680635
3.658732
3.636832
3.614936
3.593044
3.571155
3.54927
3.527388
3.50551
3.483636
3.461766
3.4399
3.418037
3.396179
3.374324
3.352474
3.330628
3.308786
3.286948
3.265114
3.243285
3.22146
3.19964
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
11
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
12
12.1
12.2
12.3
12.4
12.5
12.6
0.1802841
0.1803398
0.180391
0.1804378
0.1804805
0.1805191
0.1805539
0.180585
0.1806126
0.1806367
0.1806576
0.1806752
0.1806898
0.1807015
0.1807103
0.1807163
0.1807197
0.1807205
0.1807188
0.1807147
0.1807083
0.1806996
0.1806887
0.1806756
0.1806604
0.1806433
0.1806241
0.180603
0.18058
0.1805551
0.1805284
0.1804999
0.1804697
0.1804377
0.180404
0.1803687
0.1803317
0.180293
0.1802527
0.1802109
0.1801674
0.1801223
0.1800756
0.1800274
0.1778552
0.1780441
0.1782226
0.1783912
0.1785502
0.1787002
0.1788414
0.1789742
0.1790992
0.1792165
0.1793265
0.1794295
0.1795259
0.1796159
0.1796997
0.1797776
0.17985
0.1799169
0.1799786
0.1800354
0.1800874
0.1801348
0.1801778
0.1802166
0.1802513
0.1802821
0.1803091
0.1803325
0.1803524
0.180369
0.1803822
0.1803924
0.1803995
0.1804036
0.180405
0.1804035
0.1803994
0.1803927
0.1803835
0.1803718
0.1803578
0.1803414
0.1803227
0.1803019
3.177824
3.156013
3.134206
3.112404
3.090607
3.068815
3.047027
3.025245
3.003467
2.981695
2.959927
2.938165
2.916409
2.894657
2.872911
2.851171
2.829436
2.807707
2.785984
2.764266
2.742555
2.720849
2.69915
2.677457
2.65577
2.63409
2.612416
2.590749
2.569089
2.547435
2.525788
2.504149
2.482517
2.460891
2.439274
2.417664
2.396061
2.374467
2.35288
2.331301
2.309731
2.288169
2.266615
2.24507
12.7
12.8
12.9
13
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
14
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
16
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
17
0.1799776
0.1799262
0.1798732
0.1798187
0.1797625
0.1797048
0.1796455
0.1795845
0.1795219
0.1794577
0.1793918
0.1793241
0.1792549
0.1791839
0.1791111
0.1790366
0.1789602
0.178882
0.1788019
0.1787198
0.1786358
0.1785499
0.1784618
0.1783717
0.1782794
0.1781849
0.1780882
0.1779892
0.1778877
0.1777838
0.1776774
0.1775684
0.177457
0.1773424
0.1772251
0.177105
0.1769819
0.1768557
0.1767263
0.1765935
0.1764572
0.1763174
0.1761738
0.1760264
0.1802788
0.1802537
0.1802264
0.1801972
0.1801659
0.1801326
0.1800974
0.1800603
0.1800213
0.1799804
0.1799377
0.1798931
0.1798467
0.1797984
0.1797483
0.1796964
0.1796427
0.1795872
0.1795299
0.1794707
0.1794097
0.1793468
0.1792821
0.1792155
0.179147
0.1790765
0.1790042
0.1789299
0.1788535
0.1787752
0.1786948
0.1786123
0.1785278
0.1784409
0.1783518
0.1782605
0.1781669
0.1780709
0.1779724
0.1778714
0.1777679
0.1776617
0.1775527
0.177441
2.223534
2.202007
2.180489
2.15898
2.137481
2.115991
2.094512
2.073042
2.051583
2.030134
2.008695
1.987269
1.965852
1.944446
1.923053
1.901671
1.880301
1.858944
1.837599
1.816267
1.794948
1.773642
1.75235
1.731072
1.709808
1.688559
1.667325
1.646106
1.624903
1.603716
1.582545
1.561391
1.540253
1.519135
1.498034
1.476951
1.455888
1.434843
1.413819
1.392816
1.371834
1.350874
1.329936
1.309021
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
18
18.1
18.2
18.3
18.4
18.5
18.6
18.7
18.8
18.9
19
19.1
19.2
19.3
19.4
19.5
19.6
19.7
19.8
19.9
20
20.1
20.2
20.3
20.4
20.5
20.6
20.7
20.8
20.9
21
21.1
21.2
21.3
21.4
0.1758751
0.1757197
0.17556
0.1753958
0.1752271
0.1750536
0.174875
0.1746912
0.174502
0.1743072
0.1741064
0.1738995
0.1736861
0.173466
0.1732388
0.1730043
0.172762
0.1725117
0.1722529
0.1719851
0.1717079
0.171421
0.1711236
0.1708153
0.1704955
0.1701636
0.1698188
0.1694604
0.1690877
0.1686998
0.1682958
0.1678746
0.1674354
0.1669769
0.1664978
0.1659968
0.1654725
0.1649233
0.1643474
0.163743
0.1631079
0.16244
0.1617369
0.1609959
0.1773264
0.1772088
0.1770881
0.1769642
0.1768371
0.1767065
0.1765725
0.1764348
0.1762934
0.1761481
0.1759987
0.1758452
0.1756873
0.1755249
0.1753578
0.1751857
0.1750086
0.1748261
0.1746381
0.1744443
0.1742444
0.1740382
0.1738254
0.1736057
0.1733787
0.1731441
0.1729016
0.1726506
0.1723909
0.1721219
0.1718431
0.171554
0.1712541
0.1709427
0.1706192
0.1702829
0.1699331
0.1695689
0.1691894
0.1687937
0.1683808
0.1679495
0.1674988
0.1670271
1.28813
1.267263
1.246422
1.225606
1.204817
1.184055
1.163322
1.142617
1.121943
1.101301
1.08069
1.060112
1.039569
1.019062
0.9985912
0.9781586
0.9577656
0.9374137
0.9171041
0.8968387
0.8766193
0.8564474
0.8363251
0.8162545
0.7962376
0.7762765
0.756374
0.7365323
0.7167544
0.6970432
0.6774016
0.6578332
0.6383412
0.6189295
0.5996023
0.5803636
0.5612183
0.5421712
0.5232279
0.504394
0.4856758
0.4670803
0.4486142
0.4302857
21.5
21.6
21.7
21.8
21.9
22
22.1
22.2
22.3
22.4
22.5
22.6
22.7
22.8
22.9
23
23.1
23.2
23.3
23.4
23.5
23.6
23.7
23.8
23.9
24
0.1602141
0.1593882
0.1585146
0.1575896
0.1566089
0.1555677
0.1544612
0.1532835
0.1520287
0.1506899
0.1492597
0.1477301
0.1460924
0.1443371
0.1424544
0.1404334
0.1382631
0.135932
0.1334287
0.1307423
0.127863
0.1247825
0.1214952
0.1179994
0.1142977
0.1103989
0.1665332
0.1660155
0.1654722
0.1649016
0.1643016
0.16367
0.1630044
0.1623021
0.1615603
0.1607759
0.1599453
0.1590648
0.1581302
0.1571371
0.1560804
0.154955
0.153755
0.1524742
0.151106
0.1496435
0.1480795
0.1464065
0.1446173
0.1427047
0.1406624
0.1384847
0.4121033
0.3940761
0.3762146
0.3585295
0.3410328
0.3237375
0.3066576
0.2898088
0.2732078
0.256873
0.2408245
0.225084
0.2096755
0.1946244
0.1799589
0.1657086
0.1519056
0.1385838
0.1257788
0.1135271
0.1018655
0.0908309
0.08045764
0.07077707
0.06181534
0.05359145