Page 1
1
Supporting Information for
2
3
Deducing the temporal order of cofactor function in ligand-regulated gene transcription:
4
theory and experimental verification
5
6
7
8
9
10
11
Edward J. Dougherty1, Chunhua Guo1, S. Stoney Simons, Jr.1*, and Carson C. Chow2*
12
13
14
From the 2Laboratory of Biological Modeling, NIDDK, and 1Steroid Hormones Section,
15
NIDDK/CEB, National Institutes of Health, Bethesda, MD
16
17
18
19
20
21
22
23
24
Table of Contents
25
- Derivation of the graphical method for analyzing the competitive actions of factors
26
- Correction for non-linear protein expression from transfected plasmids
27
28
Derivation of the graphical method for analyzing the competitive actions of factors: Our
Page 2
29
graphical method stems from our theory of steroid-regulated gene induction, which
30
calculates the dose-response curve for a sequence of n reactions [1]. Each reaction step
31
has the form
32
33
34
where Yi is the reaction product of step i, Xi is an activating cofactor or activator, Ii is an
35
inhibiting cofactor or inhibitor. We denote a = 0 competitive inhibition, g = 0 uncompetitive
36
inhibition, a = g noncompetitive inhibition, b = 0 linear inhibition, and b > 0 partial inhibition.
37
Partial inhibition can be activating if it diverts the output to a higher yielding pathway. The
38
dose response curve is obtained by solving the steady state equations for mass
39
conservation and mass action. In general, these equations cannot be solved to obtain a
40
closed form expression for the dose response function of the protein product as a function
41
of the steroid concentration. However, experimentally, it is well established that the dose
42
response curve is closely modeled by a first order Hill function (i.e. Hill coefficient of one).
43
We showed that the dose-response curve for a sequence of n such reactions is a first-order
44
Hill function if reaction products have low concentrations or are short-lived [1]. As shown in
45
Ong et al. (2010), this assumption renders the steady state equations for mass
46
conservation and mass action into a solvable form.
47
48
The first-order Hill constraint also makes the calculation of the dose-response curve
49
tractable and a formula can be written down explicitly in terms of all the parameters of all
50
reaction steps. The constraint also allows intermediate and unknown reaction steps to
51
“telescope” into effective reactions with effective parameters. Hence, for experiments
Page 3
52
involving a small number of added cofactors, a model for the dose-response curve can be
53
generated that includes just the action of those cofactors. Within a sequence of reactions,
54
there may be a distinguishing step we termed the Concentration Limiting Step (CLS). The
55
CLS is defined as a step where sets of reactions immediately following it have the property
56
that the amount of cofactor bound to a reaction product is negligible compared to the free
57
concentration. The action of a cofactor is different if it acts before, at or after the CLS and a
58
59
60
61
different parametric model exists for each of these cases.
62
The parametric models have the abstract form
GV1cls[S]
[P] =
1+ W1cls[S]
(S1)
63
where [S] is the steroid concentration, G = å
64
W bm = å wiVbi-1. The parameters i and i, which depend on the parameters of the
65
reactions and differ depending on where they act with respect to the CLS, are given in
66
Table S2, where XT is the total concentration of an added factor and the other
67
parameters come from the reaction given above. The details of the calculation are
68
given in Ong et al. (2010).
n
k= cls
k
, Vbm = Õ v i , and
ak-clsVcls+1
m
i= b
m
i= b
69
70
71
72
Table S2. Parameter expressions
Position
Before
CLS
i < cls
At CLS
i = cls
Activator
vi = qi X
T
i
wi = qiei
vi same as before
CLS
Activator with Inhibitor
qi X iT (1+ a ib iqi¢[Ii ])
vi =
1+ g iq¢i [Ii ]
wi =
qi (ei + a iq¢i [Ii ])
1+ g iq¢i [Ii ]
vi same as before CLS
k
æ n
ö
qiçå ek Õ
v + a clsq¢cls[Icls ]÷
j= i+1 j
è k= i
ø
wi =
1+ g clsq¢cls[Icls ]
Page 4
wi = qi å ek Õ
n
k
k= i
j= i+1
vj
After
CLS
vi same as before
CLS
i > cls
wi = 0
vi =
qi X iT
1+ g iq¢i [Ii ]
wi = 0
73
74
75
All combinations of cofactors are accounted for in the general model (S1). To render the
76
general model into a specific parametric form for a given case, we re-express (S1) such
77
that only the concentrations of the desired factors are explicitly visible. All other reactions
78
“telescope” into an effective reaction with a set of effective parameters. For example, if we
79
are interested in an activator acting somewhere before the CLS then we rewrite GV1cls as
80
BX T , where B is a positive parameter that depends on a complicated combination of
81
reaction parameters. The crucial point is that the mechanism of the cofactor can be
82
deduced without knowing the precise value of B. Likewise, we can write W1cls = C + DX T , for
83
positive constants C and D. Hence, the dose-response curve for a single activator acting
84
85
before the CLS has the form
86
[P] =
BX T [S]
1+ (C + DX T )[S]
87
88
which implies that Amax = BX T /(C + DX T ), and EC50 =1/(C + DX T ). Hence, for an activator,
89
the graph of Amax/EC50 vs XT is linear with a positive slope and zero y-intercept and the
90
graph of 1/EC50 vs. XT is a linear function with positive slope and nonzero y-intercept. This
91
can then be repeated for all positions and all co-factor types. The computations to isolate
92
93
94
95
cofactors are facilitated with the decomposition rule
96
The identical process can be repeated for two factors. In fact all the conclusions for a single
97
factor can be obtained from the parametric models for two cofactors. Table S3 gives the full
W ab = Wac + VacWcb+1
Page 5
98
99
100
101
parametric expressions for two cofactors in all the possible configurations.
Table S3. Decomposition rules
Position
s
g<h<cls
both
before
CLS
Decompositions
B parameters
cls
V1cls = V1g-1Vgh-1
+1 Vh +1v g v h
GV1cls = B2v gv h
W1cls = B3 + B4 wg + B5v g
g <h =cls
before, at
CLS
V1cls = V1g-1Vgcls-1
+1 v g v cls
W1cls = W1g-1
(
(
h-1
cls
+ V1g-1 wg + v g W gh-1
+1 + Vg +1 ( w h + v hW h +1 )
(
cls-1
W1cls = W1g-1 + V1g-1 wg + v g (Wgcls-1
+1 + Vg +1 w cls )
))
)
+ B6v g wh + B7v g v h
GV1cls = B2v gv h
W1cls = B3 + B4 wg + B5v g
+ B6v g wcls
g <cls< h
CLS
between
V1cls = V1g-1Vgcls+1v g
W1cls = W1g-1 + V1g-1( wg + v gWgcls+1)
GV1cls = B1v g + B2v g v h
W1cls = B3 + B4 wg + B5v g wcls
g=cls<h
at, after
CLS
V1cls = V1cls-1v cls
W1cls = W1cls-1 + V1cls-1wcls
h-1
æ h-1 n
k
G = å ak-clsVcls+1
+ çVcls+1
åk= h ak-clsVhk+1ö÷øvh
k= cls
è
V1cls and W1cls are unaffected
g-1
æ g-1 h-1
k
G = å ak-clsVcls+1
+ çVcls+1
åk= g ak-clsVgk+1ö÷øv g
k= cls
è
æ g-1 h-1 n
ö
+ çVcls+1
Vg +1 å ak-clsVhk+1 ÷v g v h
k= h
è
ø
GV1cls = B1v cls + B2v clsv h
W1cls = B3 + B4 wcls
cls<g<h
both after
CLS
æ h-1 n
k
ak-clsVcls+1
+ çVcls+1
åk= h ak-clsVhk+1ö÷øvh
k= cls
è
G=å
h-1
GV1cls = B0 + B1v g + B2v g v h
W1cls = B3 + B4 wcls
102
103
Expressions for Amax and EC50 can then be obtained for all cases by inserting the
104
expressions from Table S3 into the general model (equation S1). The results can be
105
106
107
108
summarized in the following five cases:
109
110
1. k < l < CLS
Amax
= B2v kv l
EC50
1
= B3 + B4 wk + B5v k + B6v k wl + B7v kv l
EC50
Page 6
111
112
113
114
115
116
117
118
119
Amax
ql X lT (1+ a l b l q¢l [Il ]) qk X kT (1+ a kb k q¢k [Ik ])
= B2
EC50
1+ g l q¢l [Il ]
1+ g kq¢k [Ik ]
1
q (e + a k q¢k [Ik ]) æ
q (e + a l q¢l [Il ])
q X T (1+ a l b l q¢l [Il ]) ö qk X kT (1+ a kb kq¢k [Ik ])
= B3 + B4 k k
+ ç B5 + B6 l l
+ B7 l l
÷
EC50
1+ g k q¢k [Ik ]
1+ g l q¢l [Il ]
1+ g l q¢l [Il ]
1+ g kq¢k [Ik ]
è
ø
2. k < l = CLS
Amax
= B2v g v h
EC50
1
= B3 + B4 wk + B5v k + B6v k wcls
EC50
Amax
q X T (1+ a l b l q¢l [Il ]) qk X kT (1+ a kb k q¢k [Ik ])
= B2 l l
EC50
1+ g l q¢l [Il ]
1+ g kq¢k [Ik ]
120
ql ( B7 + a l q¢l [Il ]) ö qk X kT (1+ a k bk q¢k [Ik ])
1
q (e + a k q¢k [Ik ]) æ
÷
= B3 + B4 k k
+ çç B5 + B6
EC50
1+ g kq¢k [Ik ]
1+ g l q¢l [Il ] ÷ø
1+ g k q¢k [Ik ]
è
121
122
123
3. k< CLS < l
124
125
126
127
Amax
= B1v k + B2v k v l
EC50
1
= B3 + B4 wk + B5v k w cls
EC50
Amax æ
ql X lT ö qk X kT (1+ a k bkq¢k [Ik ])
= ç B1 + B2
÷
EC50 è
1+ g l q¢l [Il ] ø
1+ g kq¢k [Ik ]
128
1
qk (e k + a kq¢k [Ik ]) æ
ql X lT ö qk X kT (1+ a k bkq¢k [Ik ])
= B3 + B4
+ ç B5 + B6
÷
EC50
1+ g kq¢k [Ik ]
1+ g l q¢l [Il ]ø
1+ g kq¢k [Ik ]
è
129
130
131
4. k = CLS < l
132
133
134
135
136
137
138
Amax
= B1v k + B2v k v l
EC50
1
= B3 + B4 wk
EC50
Amax æ
ql X lT ö qk X kT (1+ a k bkq¢k [Ik ])
= çB + B
÷
EC50 è 1 2 1+ g l q¢l [Il ] ø
1+ g kq¢k [Ik ]
æ
ö
1
qk
ql X lT
= B3 + B4
+ a k q¢k [Ik ]÷
ç B5 + B6
EC50
1+ g kq¢k [Ik ] è
1+ g l q¢l [Il ]
ø
Page 7
139
140
141
142
143
5. CLS < k< l
Amax
= B0 + B1v k + B2v k v l
EC50
1
= B3 + B4 wcls
EC50
æ
Amax
ql X lT ö qk X kT
= B0 + ç B1 + B2
÷
EC50
1+ g l q¢l [Il ] ø 1+ g kq¢k [Ik ]
è
144
145
146
1
qk X kT
ql X lT
= B3 + B4
+ B5
EC50
1+ g kq¢k [Ik ]
1+ g l q¢l [Il ]
147
148
149
All of the B parameters are positive numbers. From these five cases we can then extract
150
the graphical behavior of Amax/EC50 vs cofactor and 1/EC50 vs cofactor for all cofactor
151
types, where X iT is the concentration of an activator at step i, [Ii ] is the concentration of the
152
inhibitor at step i, and the type of inhibitor is determined by the parameters a i, bi, and g i.
153
These conclusions are listed in Tables 1 and 2. Note that decreasing curves are always
154
first order decay plots.
155
156
Correction for non-linear protein expression from transfected plasmids: Western blots
157
reveal a non-linear relationship between the optical density of scanned protein band and
158
the amount of transfected plasmid at constant levels either of total cellular protein or of an
159
internal standard, such as b -actin. All of the plots of this study require a linear relationship
160
for the amount of factor given on the x-axis. To determine the linear equivalent of
161
expressed plasmid, the non-linear plot of OD vs. ng of transfected plasmid is first fit to a
162
Michaelis-Menten plot (Fig. S1A) of
163
Amax = m1*plasmid/(m2 + plasmid)
164
The functional equivalent of the transfected plasmid that gives a linear OD vs. plasmid plot
165
is then obtained from the formula of
166
167
Plasmid (linear) = m2*plasmid/(m2 + plasmid)
The amount of plasmid plotted in the various graphs is then this “corrected plasmid” value
Page 8
168
(Fig. S1B).
169
170
Matching graphs to graphical descriptions: The following algorithms are used to determine
171
which graphical description in Table 1 best describes a given plot.
172
173
1) Examination of the underlying equations (see above) indicates that, for all plots, a
negative slope can only be fit to a non-linear MIchaelis-Menten decay curve.
174
2) Whether a plot with a positive slope is linear or non-linear (and curving
175
downward) is based upon the reproducibility of the fit to a straight line, both for all four
176
lines within a given graph and for all such graphs for a given competition experiment.
177
Random fluctuations will occasionally yield data points for one line that may appear to
178
be best described by a downward bending curve. However, if this behavior is not
179
consistently observed, both for the other three plots of the same graph and for the plots
180
of the other experiments under the same conditions, the plot is deemed to be best fit by
181
a straight line.
182
3) In order to decide whether the nonlinear, decreasing curves for Amax/EC50 go to
183
zero (entries 32 or 34 of Table S1) or to a positive plateau value (entries 33 or 35 of
184
Table 1b), one looks at the reciprocal plots (EC50/Amax vs. F1) (2). If the reciprocal plots
185
are straight lines, then the curves of the Amax/EC50 graph go to zero at infinite F1. If the
186
reciprocal plots curve downward, but give linear plots when first subtracting a probable
187
asymptote from the value of Amax/EC50 and then plotting the reciprocal this new value (=
188
1/[(Amax/EC50) – asymptote]), then the Amax/EC50 plot goes to that asymptote at infinite
189
F1
190
4) A final criterion for classifying the nature of a given plot comes from the analysis
191
of the graphical interpretations of the 4-6 plots utilized for a given competition
192
experiment. The final interpretation must be consistent with the partial interpretation
193
derived from each type of graph. No one graph can uniquely define all of the
194
relationships of the two factors and their kinetic mechanism. Rather the assembly of
195
graphs, and their partial interpretations, are used to construct the final mechanism. In
Page 9
196
this respect, the final mechanism is that area of common overlap of the possible
197
scenarios of each type of graph. This is pictorially represented in Fig. S4. Each circle
198
represents one type of graph, with sectors of each circle representing the different
199
possible mechanisms associated with the graph. By identifying that scenario of each
200
graph that is consistent with all of the other graphs (indicated by the filled area where all
201
six circles overlap), one usually arrives at a unique mechanistic explanation. If two
202
descriptions of one graph appear to be equally possible but only one gives “overlap”
203
with the other 3-5 plots, then this “overlapping” plot is selected.
204
205
References
206
1.
Ong KM, Blackford Jr JA, Kagan BL, Simons Jr SS, Chow CC (2010) A new
207
theoretical framework for gene induction and experimental comparisons. Proc Natl
208
Acad Sci U S A 107: 7107-7112.
209
210
211
212
213
214
2.
Chow CC, Ong KM, Dougherty EJ, Simons, Jr. SS (2011) Inferring mechanisms from
dose-response curves. Methods in Enzymology 487: 465-483.