-
Notifications
You must be signed in to change notification settings - Fork 0
/
ar_analytic.nb
312 lines (254 loc) · 11.3 KB
/
ar_analytic.nb
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
(************** Content-type: application/mathematica **************
CreatedBy='Mathematica 5.2'
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do
one of the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the
application;
* Copy the data starting with the line of stars above to the
clipboard, then use the Paste menu command inside the application.
Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.
For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
web: http://www.wolfram.com
email: [email protected]
phone: +1-217-398-0700 (U.S.)
Notebook reader applications are available free of charge from
Wolfram Research.
*******************************************************************)
(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 9566, 232]*)
(*NotebookOutlinePosition[ 10291, 257]*)
(* CellTagsIndexPosition[ 10247, 253]*)
(*WindowFrame->Normal*)
Notebook[{
Cell["\<\
We start with a simple model of activation/repression. The basal \
unregulated expression level is S. Furthermore, we assume that the induction \
is perfect, so the effect of a repressor can be removed completely. This \
function has two variables which range from 0 (full induced) to 1 (fully \
repressed) for each repressor. The three parameters account for the strength \
of repression (c3), and two activation parameters c1 and c2, where the \
activation is productive when c1 > c2.\
\>", "Text",
FontSize->18],
Cell[BoxData[{
\(P[A_, R_, c1_, c2_,
c3_] := \(S \((1 + c1*A)\)\)\/\(1 + c2*A + c3*R\)\), "\n",
\(r[c1_, c2_, c3_] :=
Log[10, P[1, 0, c1, c2, c3]/P[0, 1, c1, c2, c3]]\), "\n",
\(\[Alpha][c1_, c2_, c3_] :=
Log[10, P[1, 0, c1, c2, c3]/P[1, 1, c1, c2, c3]]/
r[c1, c2, c3]\), "\n",
\(\[Beta][c1_, c2_, c3_] :=
Log[10, P[1, 0, c1, c2, c3]/P[0, 0, c1, c2, c3]]/
r[c1, c2, c3]\), "\[IndentingNewLine]",
\(l[c1_, c2_,
c3_] := \(\[Alpha][c1, c2, c3] + \[Beta][c1, c2, c3]\)\/2\), "\
\[IndentingNewLine]",
\(a[c1_,
c2_, \[Omega]_] := \[Alpha][c1, c2, \[Omega]] - \[Beta][c1,
c2, \[Omega]]\)}], "Input",
CellFrame->{{0, 0}, {3, 0}},
FontSize->18],
Cell[CellGroupData[{
Cell[BoxData[
\(r[c1, c2, c3]\)], "Input"],
Cell[BoxData[
\(Log[\(\((1 + c1)\)\ \((1 + c3)\)\)\/\(1 + c2\)]\/Log[10]\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(Simplify[a[c1, c2, c3], c2 > 0]\)], "Input"],
Cell[BoxData[
\(\(\(-Log[1 + c1]\) + Log[1 + c2 + c3]\)\/Log[\(\((1 + c1)\)\ \((1 + c3)\
\)\)\/\(1 + c2\)]\)], "Output"]
}, Open ]],
Cell[BoxData[""], "Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(Simplify[l[c1, c2, c3], c2 > 0]\)], "Input"],
Cell[BoxData[
\(\(Log[1 + c1] - 2\ Log[1 + c2] + Log[1 + c2 + c3]\)\/\(2\ Log[\(\((1 + \
c1)\)\ \((1 + c3)\)\)\/\(1 + c2\)]\)\)], "Output"]
}, Open ]],
Cell["\<\
Since there are three microscopic (c1, c2, c3) and three logical \
(r,a,l) parameters, we can solve for the microscopic parameters in terms of \
the logic parameters explicitly. \
\>", "Text",
FontSize->18],
Cell[CellGroupData[{
Cell[BoxData[
\(Solve[r[c1, c2, c3] \[Equal] r, c2]\)], "Input"],
Cell[BoxData[
\({{c2 \[Rule] \(-10\^\(-r\)\)\ \((\(-1\) + 10\^r - c1 - c3 -
c1\ c3)\)}}\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(solnc3 =
Solve[a[c1, \(-10\^\(-r\)\)\ \((\(-1\) + 10\^r - c1 - c3 - c1\ c3)\),
c3] \[Equal] a, c3]\)], "Input"],
Cell[BoxData[
\({{c3 \[Rule] \(\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + \
c1)\)\)\/\(1 + 10\^r + c1\)}}\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(solnc1 =
Solve[l[c1, \(-10\^\(-r\)\)\ \((\(-1\) + 10\^r - c1 - c3 - c1\ c3)\),
c3] \[Equal] l, c1] /. solnc3\)], "Input"],
Cell[BoxData[
\({{{c1 \[Rule] \((10\^r - \((10\^r)\)\^\(2\ l\) - \(\((10\^r)\)\^\(2\ \
l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\^2\ \((1 + c1)\)\^2\)\/\((1 + \
10\^r + c1)\)\^2 + \(10\^r\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\
\)\)\/\(1 + 10\^r + c1\) + \(10\^\(2\ r\)\ \((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) - \(2\ \((10\^r)\)\^\(2\ l\)\ \
\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)/\
\((\((1 + \(\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + \
10\^r + c1\))\)\ \((\(-10\^r\) + \((10\^r)\)\^\(2\ l\) + \(\((10\^r)\)\^\(2\ \
l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + \
c1\))\))\)}}}\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(solnc3 =
Solve[a[c1, \(-10\^\(-r\)\)\ \((\(-1\) + 10\^r - c1 - c3 - c1\ c3)\),
c3] \[Equal] a, c3] /. solnc1\)], "Input"],
Cell[BoxData[
\({{{{c3 \[Rule] \((\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + \
\((10\^r - \((10\^r)\)\^\(2\ l\) - \(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\
\^r)\)\^\(1 + a\))\)\^2\ \((1 + c1)\)\^2\)\/\((1 + 10\^r + c1)\)\^2 + \(10\^r\
\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) + \
\(10\^\(2\ r\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + \
10\^r + c1\) - \(2\ \((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)/\((\((1 + \(\((\(-1\) + \
\((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)\ \((\(-10\^r\
\) + \((10\^r)\)\^\(2\ l\) + \(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\
\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\))\))\))\)/\((1 +
10\^r + \((10\^r - \((10\^r)\)\^\(2\ l\) - \
\(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\^2\ \((1 + c1)\
\)\^2\)\/\((1 + 10\^r + c1)\)\^2 + \(10\^r\ \((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) + \(10\^\(2\ r\)\ \((\(-1\) + \
\((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) - \(2\ \
\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + \
c1)\)\)\/\(1 + 10\^r + c1\))\)/\((\((1 + \(\((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)\ \((\(-10\^r\) + \
\((10\^r)\)\^\(2\ l\) + \(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 \
+ a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\))\))\)}}}}\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(Solve[
r[c1, c2, \(\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + \
c1)\)\)\/\(1 + 10\^r + c1\)] \[Equal] r, c2] /. solnc1\)], "Input"],
Cell[BoxData[
\({{{{c2 \[Rule] \((\(-10\^r\) + \((10\^r)\)\^a + \((\((10\^r)\)\^a\ \
\((10\^r - \((10\^r)\)\^\(2\ l\) - \(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\
\^r)\)\^\(1 + a\))\)\^2\ \((1 + c1)\)\^2\)\/\((1 + 10\^r + c1)\)\^2 + \(10\^r\
\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) + \
\(10\^\(2\ r\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + \
10\^r + c1\) - \(2\ \((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)\^2)\)/\((\((1 + \(\((\(-1\) + \
\((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)\^2\ \((\(-10\
\^r\) + \((10\^r)\)\^\(2\ l\) + \(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \
\((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)\^2)\) + \((2\
\ \((10\^r)\)\^a\ \((10\^r - \((10\^r)\)\^\(2\ l\) - \(\((10\^r)\)\^\(2\ l\)\ \
\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\^2\ \((1 + c1)\)\^2\)\/\((1 + 10\^r + \
c1)\)\^2 + \(10\^r\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + \
c1)\)\)\/\(1 + 10\^r + c1\) + \(10\^\(2\ r\)\ \((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) - \(2\ \((10\^r)\)\^\(2\ l\)\ \
\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\))\
\)/\((\((1 + \(\((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + \
10\^r + c1\))\)\ \((\(-10\^r\) + \((10\^r)\)\^\(2\ l\) + \(\((10\^r)\)\^\(2\ \
l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + \
c1\))\))\))\)/\((1 +
10\^r + \((10\^r - \((10\^r)\)\^\(2\ l\) - \
\(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\^2\ \((1 + c1)\
\)\^2\)\/\((1 + 10\^r + c1)\)\^2 + \(10\^r\ \((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) + \(10\^\(2\ r\)\ \((\(-1\) + \
\((10\^r)\)\^\(1 + a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\) - \(2\ \
\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 + a\))\)\ \((1 + \
c1)\)\)\/\(1 + 10\^r + c1\))\)/\((\((1 + \(\((\(-1\) + \((10\^r)\)\^\(1 + \
a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\)\ \((\(-10\^r\) + \
\((10\^r)\)\^\(2\ l\) + \(\((10\^r)\)\^\(2\ l\)\ \((\(-1\) + \((10\^r)\)\^\(1 \
+ a\))\)\ \((1 + c1)\)\)\/\(1 + 10\^r + c1\))\))\))\)}}}}\)], "Output"]
}, Open ]]
},
FrontEndVersion->"5.2 for Macintosh",
ScreenRectangle->{{0, 1440}, {0, 874}},
WindowSize->{520, 740},
WindowMargins->{{134, Automatic}, {Automatic, 66}},
PrintingCopies->1,
PrintingPageRange->{1, Automatic},
StyleDefinitions -> "Report.nb"
]
(*******************************************************************
Cached data follows. If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of the file. The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[1754, 51, 530, 9, 210, "Text"],
Cell[2287, 62, 743, 18, 417, "Input"],
Cell[CellGroupData[{
Cell[3055, 84, 46, 1, 32, "Input"],
Cell[3104, 87, 90, 1, 73, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[3231, 93, 64, 1, 32, "Input"],
Cell[3298, 96, 124, 2, 72, "Output"]
}, Open ]],
Cell[3437, 101, 26, 0, 32, "Input"],
Cell[CellGroupData[{
Cell[3488, 105, 64, 1, 32, "Input"],
Cell[3555, 108, 143, 2, 72, "Output"]
}, Open ]],
Cell[3713, 113, 219, 5, 78, "Text"],
Cell[CellGroupData[{
Cell[3957, 122, 68, 1, 32, "Input"],
Cell[4028, 125, 122, 2, 47, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[4187, 132, 151, 3, 64, "Input"],
Cell[4341, 137, 126, 2, 65, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[4504, 144, 163, 3, 64, "Input"],
Cell[4670, 149, 732, 10, 263, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[5439, 164, 163, 3, 64, "Input"],
Cell[5605, 169, 1510, 20, 590, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[7152, 194, 162, 3, 68, "Input"],
Cell[7317, 199, 2233, 30, 803, "Output"]
}, Open ]]
}
]
*)
(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)