Combinatorially Complex Equilibrium Model Selection
description
Transcript of Combinatorially Complex Equilibrium Model Selection
Combinatorially Complex Combinatorially Complex Equilibrium Model Equilibrium Model
SelectionSelection
Tom RadivoyevitchAssistant ProfessorEpidemiology and BiostatisticsCase Western Reserve University
Email: [email protected]: http://epbi-radivot.cwru.edu/
Ultimate GoalUltimate Goal
EXPERIMENTALBIOLOGY
COMPUTERMODELING
CONTROLTHEORY
models
control lawsdata
hypotheses
proposed clinical trial
validated process model development
control system design methods development
Present Future
• Safer flying airplanes with autopilotsSafer flying airplanes with autopilots• Ultimate Goal: individualized, state feedback based Ultimate Goal: individualized, state feedback based
clinical trialsclinical trials
• Better understanding => better controlBetter understanding => better control• Conceptual models help trial designs today Conceptual models help trial designs today • Computer models of airplanes help train pilots and Computer models of airplanes help train pilots and
autopilotsautopilots
Radivoyevitch et al. (2006) BMC Cancer 6:104
Systems Cancer BiologySystems Cancer Biology
dNTP Supply System
Figure 1. dNTP supply. Many anticancer agents act on or through this system to kill cells. The most central enzyme of this system is RNR.
UDP
CDP
GDP
ADP
dTTP
dCTP
dGTP
dATP
dT
dC
dG
dA
DNA
dUMP
dU
TS
CDA
dCK
DN
A p
olym
eras
eTK1
cytosol
mitochondria
dT
dC
dG
dA
TK
2dG
K
dTMP
dCMP
dGMP
dAMP
dTTP
dCTP
dGTP
dATP
5NT
NT2
cytosol
nucleus
dUDP
dUTP
U-DNA
dUTPasedN
dN
dCK
flux activation inhibition
ATP RN
R
dCK
R1
R2 R2
R1 R1
R1 R1
R1 R1
R1
R1
R1
R1
R1 R1
R1
R1
R1
R1
R2 R2
UDP, CDP, GDP, ADP bind to catalytic site
dTTP, dGTP, dATP, ATP bind to selectivity site
dATP inhibits at activity site, ATP activates at activity site?
Selectivity site binding promotes R1 dimers. R2 is always a dimer.
ATP drives hexamer. Controversy: dATP drives inactive tetramer vs. inactive hexamer
Controversy: Hexamer binds one R22 vs. three R22
Total concentrations of R1, R22, dTTP, dGTP, dATP, ATP and NDPscontrol the distribution of R1-R2 complexes and this changes in S, G1-G2 and G0
ATP activates at hexamerization site??RNR LiteratureRNR Literature
R2 R2
Michaelis-Menten ModelMichaelis-Menten Model
With RNR: no NDP and no R2 dimer => kcat of complex is zero.Otherwise, many different R1-R2-NDP complexes can have many different kcat values.
)(0)(
][][
][0
][][
][
1/][
10
1/][
/][
][
][
00
00
00max
EPEESPEk
EES
EE
EES
ESEk
KSE
KS
KSEk
KS
SV
cat
cat
mm
mcat
m
E + S ES
0.005 0.010 0.020 0.050 0.100 0.200 0.500
02
04
06
08
01
00
Total [r] (uM)
Pe
rce
nt A
ctiv
ity
solid line = Eqs. (1-2) dotted = Eq. (3)
Data from Scott, C. P., Kashlan, O. B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry 40(6), 1651-166
Model Parameter Initial Value Optimal Value Confidence Interval
RRGGttr1.1.0 RRGGtt_r 0.020 0.012 (0.007, 0.024)
SSE 1070.252 823.793
AIC 45.006 42.650
MM Kd 0.020 0.033 (0.022, 0.049)
SSE 2016.335 1143.682
AIC 50.706 45.603
R=R1 r=R22
G=GDP t=dTTP
)2(]][[
][][0
)1(]][[
][][0
__
__
SEdT
SEdT
K
SESS
K
SEEE
SEd
T
SEd
T
T
SEd
SEdT
SEd
TSEd
T
KS
KS
EESversus
KS
KS
EES
KS
EEK
SEE
__
__
__
__
__
__ ][1
][
][][][
1
][
][][][
1
1][][
][1][][
Substitute this in here to get a quadratic in [S] which solves as
Bigger systems of higher polynomials cannot be solved algebraically => use ODEs (above)
][4][][(][][(5.][][ __2
____ TSEdTTSEdTTSEdT SKESKESKSES
0)0]([,0)0]([
]][[][][
][
]][[][][
][
__
__
SE
K
SESS
d
Sd
K
SEEE
d
Ed
SEdT
SEdT
Michaelis-Menten ModelMichaelis-Menten Model [S] vs. [S[S] vs. [STT] ]
(3)
IESdSEdIEdT
IESdSEdSEdT
IESdSEdIEdSEdT
KK
ISE
K
IEII
KK
ISE
K
SESS
KK
ISE
K
IE
K
SEEE
______
______
________
]][][[]][[][][0
]][][[]][[][][0
]][][[]][[]][[][][0
E ES
EI ESI
E ES
EI ESI
E ES
EI ESI
SEIdIEdIEdT
SEIdIEdSEdT
SEIdIEdIEdSEdT
KK
SIE
K
IEII
KK
SIE
K
SESS
KK
SIE
K
IE
K
SEEE
______
______
________
]][][[]][[][][0
]][][[]][[][][0
]][][[]][[]][[][][0
E ES
EI
EIT
EST
EIEST
K
IEII
K
SESS
K
IE
K
SEEE
]][[][][0
]][[][][0
]][[]][[][][0
ESIEIT
ESIEST
ESIEIEST
K
ISE
K
IEII
K
ISE
K
SESS
K
ISE
K
IE
K
SEEE
]][][[]][[][][0
]][][[]][[][][0
]][][[]][[]][[][][0
E ES
EI ESI
E
EI ESI
E ES
ESI
E
EI ESI
E ES
ESI
=
=E
EI
E
ESI
E ES E
Competitive inhibition
uncompetitive inhibition if kcat_ESI=0
E | ES
EI | ESI
noncompetitive inhibition Example of Kd=Kd’ Model
==
Let p be the probability that an E molecule is undamaged.Then in each model [ET] can be replace with p[ET] to double the number of models to 2*(23+3+1)=24.
E
EI
E
ESI
E ESKj=0 Models
][][0
]][[][][0
]][[][][0
II
K
SESS
K
SEEE
T
EST
EST
0][ and ][][][ ],[][ else ,0][ and ][][][ ],[][ )][][( if EESSEESSSEESESSE TTTTTTTT
as KES approaches 0
Enzyme, Substrate and InhibitorEnzyme, Substrate and Inhibitor
Total number of spur graph models is 16+4=20 Radivoyevitch, (2008) BMC Systems Biology 2:15
Rt Spur Graph ModelsRt Spur Graph Models
RRttRRtRt
T
RRttRRtRRRtT
K
tR
K
tR
K
tRtt=
K
tR
K
tR
K
R
K
tRRRp=
222
2222
20
2220
.0)0(;0)0(
2
222
222
2222
tR
K
tR
K
tR
K
tRtt=
d
td
K
tR
K
tR
K
R
K
tRRRp=
d
Rd
RRttRRtRtT
RRttRRtRRRtT
R RR
RRtt
RRt Rt
R
RRtt
RRt Rt
R RR
RRtt
Rt
R RR
RRt Rt
R RR
RRtt
RRt
R
RRtt
Rt
R
RRt Rt
R RR
Rt
3B 3C 3D 3E
3F 3G 3H
3A
R
RRtt
RRt
R RR
RRtt
R RR
RRt
R
Rt
R
RRtt
R
RRt
3J3I 3K
R RR R
3L 3M 3N 3O 3P
R
Rt
R
RRtt
R
RRt
R RR
3Q 3R 3S 3T
R = R1 t = dTTP
for dTTP induced R1 dimerization
R Rt t
RRt t
Rt R t
RRt t
Rt Rt RRtt
Kd_R_R
Kd_Rt_R
Kd_Rt_Rt
Kd_R_t
Kd_R_tKd_RRt_t
Kd_RR_t=
=
=
=
2A
R Rt t
RRt t
Rt R t
RRt t
Rt Rt RRtt
Kd_R_R
Kd_Rt_R
Kd_Rt_Rt
Kd_R_t
Kd_R_tKd_RRt_t
Kd_RR_t==
=
2B
R Rt t
RRt t
Rt R t
RRt t
Rt Rt RRtt
Kd_R_R
Kd_Rt_R
Kd_Rt_Rt
Kd_R_t
Kd_R_tKd_RRt_t
Kd_RR_t
2C
|
|
|
|
|
=
=
=
|
|
R Rt t
RRt t
Rt R t
RRt t
Rt Rt RRtt
Kd_R_R
Kd_Rt_R
Kd_Rt_Rt
Kd_R_t
Kd_R_tKd_RRt_t
Kd_RR_t
2D
|
|
R Rt t
RRt t
Rt R t
RRt t
Rt Rt RRtt
Kd_R_R
Kd_Rt_R
Kd_Rt_Rt
Kd_R_t
Kd_R_tKd_RRt_t
Kd_RR_t
2E
=
=
2A 2B 2C 2D 2E 2G 2I 2K
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
=
= =
=
2F 2H 2J 2L 2M 2N
Figure 3. Spur graph models. The following models are equivalent: 3A=2F, 3B=2H, 3C=2J, 3D=2L, 3E=2N
Acyclic spanning subgraphs are reparameterizations of equilvalent models
2F0
Figure 2. Grid graph models.
2F1 2F2 2F3 2F4 2F5 2F6 2F7 2F8Standardize: take E-shapes and sub E-shapes as defaults
Use n-shapes if needed. Other shapes are possible
3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T3A
Rt Grid Graph ModelsRt Grid Graph Models
5 10 15
10
01
20
14
01
60
18
0
Total [dTTP] (uM)
Ave
rag
e M
ass
(kD
a)
AICc = 2P+N*log(SSE/N)+2P(P+1)/(N-P-1)
Data and fit from Scott, C. P., Kashlan, O. B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry 40(6), 1651-166
~10-fold deviations from Scott et al. (initial values)
titration model has lowest AIC
Radivoyevitch, (2008) BMC Systems Biology 2:15
Application to DataApplication to Data
][
][2][2][2180
][
)1]([][90
TT
Ta R
RRttRRtRR
R
pRRM
2E
=
=
R
RRtt
3Rp
Infinitely tight binding situation wherein free molecule annihilation (the initial linear ramp) continues in a one-to-one fashion with increasing [dTTP]T until [dTTP]T equals [R1]T
=7.6 µM, the plateau point where R exists solely as RRtt.
Experiment becomes a titration scan of [tT] to estimate [RT],
but [RT]=7.6 µM was already known.
Model Space Fit with New DataModel Space Fit with New Data
Radivoyevitch, (2008) BMC Systems Biology 2:15
Yeast R1 structure. Chris Dealwis’ Lab, PNAS 102, 4022-4027, 2006
One additional data point here would reject 3Rp
If so, new data here would be logical next
No need to constrain data collection to orthogonal profiles
Model Space PredictionsModel Space Predictions
Best next 10 measurements if 3Rp is Best next 10 measurements if 3Rp is rejected rejected
Show new Show new datadata+ R Code+ R Code
Fast Total Concentration Constraint (TCC; i.e. g=0) solvers are critical to model
estimation/selection. TCC ODEs (#ODEs = #reactants) solve TCCs faster than kon =1 and koff = Kd systems (#ODEs = #species = high # in combinatorially complex situations)
Semi-exhaustive approach = fit all models with same number of parameters as parallel batch, then fit next batch only if current shows AIC improvement over previous batch. This reduces Rt model space fitting times by a factor of 5.
Comments on Methods
12
1
68
1
44
1
22
1
12
0
68
0
44
0
22
1
642
642
0
6420
i aR
i
i aR
i
i aR
i
i Ra
i
T
i aR
i
i aR
i
i aR
i
i Ra
i
T
iiii
iiii
K
aRi
K
aRi
K
aRi
K
aRiaa=
K
aR
K
aR
K
aR
K
aRRRp=
2+5+9+13 = 28 parameters => 228=2.5x108 spur graph models via Kj=∞ hypotheses
(if average fit = 60 sec, then need 60 cores running for 8 straight years)
28 models with 1 parameter, 428 models with 2, 3278 models with 3, 20475 with 4
[dATP] uM
a = dATP
Kashlan et al. Biochemistry 2002 41:462
)2(][
1
2
0
31
2
21
31
2
32
21
1
E
KK
S
K
S
K
S
KK
Sk
K
Sk
K
Sk
v
Human Thymidine Kinase 1Human Thymidine Kinase 1
Assumptions
(1)binding of enzyme monomers to form enzyme dimers is approximately infinitely tight across the enzyme concentrations of interest
(2)behavior of any higher order enzyme structures (if they exist) is dominated by the behavior of dimers within such structures
(3)enzyme concentrations are low enough that free substrate concentrations approximately equal total substrate concentrations.
Max(E0)=200 pMMin(ST)=50 nM => OK
OK
Questionable (but I need to start simple)
row Eq k1 k2 k3 K1 K2 K3 Numerator Denominator
1s 3 k1 k1 k3 K1 K1 K3 31
2
31
1 22KK
Sk
K
Sk
31
2
1
21KK
S
K
S
2 3 k3 k2 k3 K1 K1 K3 31
2
31
23 2KK
Sk
K
Skk
31
2
1
21KK
S
K
S
3 3 k1 k3 k3 K1 K1 K3 31
2
31
31 2KK
Sk
K
Skk
31
2
1
21KK
S
K
S
4 3 k1 k1 k3 K1 K2 K1 21
2
321
1 211
K
SkS
KKk
21
2
21
111
K
SS
KK
5 3 k3 k2 k3 K1 K2 K1 21
2
32
21
3 211
K
SkS
Kk
Kk
21
2
21
111
K
SS
KK
6 3 k1 k3 k3 K1 K2 K1 21
2
32
31
1 211
K
SkS
Kk
Kk
21
2
21
111
K
SS
KK
7 3 k1 k1 k3 K1 K2 K2 21
23
211
211
KK
SkS
KKk
21
2
21
111
KK
SS
KK
8 3 k3 k2 k3 K1 K2 K2 21
23
2
2
1
3 2
KK
SkS
K
k
K
k
21
2
21
111
KK
SS
KK
9 3 k1 k3 k3 K1 K2 K2 21
23
2
3
1
1 2
KK
SkS
K
k
K
k
21
2
21
111
KK
SS
KK
.10 4 k1+k2=2k3 K1 K1 K3
31
2
132
KK
S
K
Sk
31
2
1
21KK
S
K
S
11 11b k1+k2=2k3 K1 K2 K1 2
1
2
212
2
1
1
K
SkkS
K
k
K
k
21
2
21
111
K
SS
KK
12 11b k1+k2=2k3 K1 K2 K2 21
2
212
2
1
1
KK
SkkS
K
k
K
k
21
2
21
111
KK
SS
KK
13 4 k3 k3 k3 K1 K2 K3
31
2
213
112
KK
SS
KKk
31
2
21
111
KK
SS
KK
14s 4 k3 k3 k3 K1 K1 K3
31
2
132
KK
S
K
Sk
31
2
1
21KK
S
K
S
15 4 k3 k3 k3 K1 K2 K1
21
2
213
115.2
K
SS
KKk
21
2
21
111
K
SS
KK
16 4 k3 k3 k3 K1 K2 K2
21
2
213
115.2
KK
SS
KKk
21
2
21
111
KK
SS
KK
17 5 k1 k2 k3 K1 K1 K1 2
21
3
1
21 2S
K
kS
K
kk
2
1
1
K
S
18s 5 k1 k1 k3 K1 K1 K1 2
21
3
1
1 22S
K
kS
K
k
2
1
1
K
S
19 5 k3 k2 k3 K1 K1 K1 2
21
3
1
23 2S
K
kS
K
kk
2
1
1
K
S
20 5 k1 k3 k3 K1 K1 K1 2
21
3
1
31 2S
K
kS
K
kk
2
1
1
K
S
21 8 k3 k3 k3 K1 K1 K1 1
32K
Sk
1
1K
S
22 3 X k2 k3 ∞ K2 K3 2
31
3
2
2 2S
KK
kS
K
k
31
2
2
11
KK
SS
K
23 4 X k3 k3 ∞ K2 K3
2
3123
15.2 S
KKS
Kk
31
2
2
11
KK
SS
K
24 3 X k2 k3 ∞ K2 K2 2
21
3
2
2 2S
KK
kS
K
k
21
2
2
11
KK
SS
K
25 4 X k3 k3 ∞ K2 K2
2
2123
15.2 S
KKS
Kk
21
2
2
11
KK
SS
K
26 3 k1 X k3 K1 ∞ K3 2
31
3
1
1 2S
KK
kS
K
k
31
2
1
11
KK
SS
K
27 4 k3 X k3 K1 ∞ K3
2
3113
15.2 S
KKS
Kk
31
2
1
11
KK
SS
K
28 6 k1 X k3 K1 ∞ K1 2
21
3
1
1 2S
K
kS
K
k
21
2
1
11
K
SS
K
29 7 k3 X k3 K1 ∞ K1
2
211
3
15.2 S
KS
Kk
21
2
1
11
K
SS
K
30 8 k1 k2 X K1 K2 ∞ SK
k
K
k
2
2
1
1 SKK
21
111
31 8 k1 k1 X K1 K2 ∞ SKK
k
211
11 SKK
21
111
32 8 k1 k2 X K1 K1 ∞ SK
kk
1
21 SK1
21
33s 8 k1 k1 X K1 K1 ∞ SK
k1
1
2 SK1
21
34s 9 X X k3 ∞ ∞ K3 31
2
32KK
Sk
31
2
1KK
S
)2(][
1
2
0
31
2
21
31
2
32
21
1
E
KK
S
K
S
K
S
KK
Sk
K
Sk
K
Sk
v
)3(
21 221
221
SS
SVSVv
)1(
150
50max
n
n
S
S
S
SV
v
)3(21 2
21
221
SS
SVSVv
)4(
21 221
221
SS
SSVv
)5(
1 21
2211
S
SVSVv
)6(1 22
11
2211
SS
SVSVv
)7(
1
5.22
11
2211
SS
SSVv
)8(1 1
1
S
SVv
)9(1 2
1
21
S
SVv
3-parameter models
2-parameter models
4-parameter model
)18(21 2
21
1
SS
SVv
)19(21 2
21
22
SS
SVv
)20(
21
5.2
21
221
SS
SSVv
.
)21(21
)5(.2
21
222
SS
SSVv
)22()(1
)5(.2
221
221
SS
SSVv
)23()(1
)5(.2
2121
2211
SS
SSVv
)24(
1 21
1
S
SVv
)25(
1 21
221
S
SVv
)26(
1
5.2
1
2211
S
SVSVv
)27(1 22
11
1
SS
SVv
)28(1 22
11
221
SS
SVv
Eq. (25): k1 = k2 = 0, k3 > 0 and K1 = K2 = K3
)7(
1
5.22
11
2211
SS
SSVv
)25(
1 21
221
S
SVv
Fits to human thymidine kinase 1 data of Birringer et al. Protein Expr Purif 2006, 47(2):506-515
Eq. (7): k1 = k3 and K2 = ∞ and K1 = K3
EQ SSE AICc α1 V∞ V1 α2 neq25 0.00026 -57.67 3.765 0.151eq28 0.00028 -56.93 2.756 0.145eq7 0.00029 -56.86 1.658 0.145eq26 0.00034 -55.35 2.316 0.156eq8 0.00039 -54.1 1.37 0.158eq22 0.00026 -50.48 0.833 0.148 6.732eq20 0.00027 -50.23 1.293 0.148 4.635eq9 0.00061 -49.99 3.645 0.132eq1 0.00028 -49.88 1.941 0.144 1.297eq6 0.00028 -49.71 2.747 0.145 0eq4 0.00029 -49.6 0.815 0.146 2.704eq24 0.00065 -49.5 0.272 0.149eq21 3.00E-04 -49.33 1.459 0.149 3.996eq19 0.00032 -48.79 0.946 0.144 6.156eq23 0.00033 -48.49 2.145 0.155 3.006eq18 0.00037 -47.41 0.576 0.2 0.022eq5 0.00037 -47.25 0.96 0.157 0.207eq27 0.00105 -45.19 0.303 0.127eq3 0.00026 -38.57 2.5287 0.1496 0.0205 10.1832
Same top 4 models found by fits to
Frederiksen H, Berenstein D, Munch-Petersen B: Eur J Biochem 2004, 271(11):2248-2256.
1e-02 1e-01 1e+00 1e+01 1e+02
0.0
0.2
0.4
0.6
0.8
1.0
dT (uM)
no
rma
lize
d fl
ux
Eq. 25Eq. 7
Hill rises from 5th, 6th and worse to 2nd if older data is merged in
Conclusion: across the board Hill fits should be avoided since it is not biologically plausible at non-integer n and since it needs competition to guide subsequent experiments
AcknowledgementsAcknowledgements
Case Comprehensive Cancer CenterCase Comprehensive Cancer Center NIH (K25 CA104791)NIH (K25 CA104791) Chris Dealwis (CWRU Pharmacology)Chris Dealwis (CWRU Pharmacology) Sanath Wijerathna (CWRU Pharmacology)Sanath Wijerathna (CWRU Pharmacology) Anders Hofer (Umea) Anders Hofer (Umea) Thank you Thank you