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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
FYTN05/TEK267Chemical Forces and Self Assembly
Victor Olariu
CBBP - [email protected]
Victor Olariu CBBP - [email protected]
FYTN05/TEK267 Chemical Forces and Self Assembly 1
Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten (10.3,10.4)
I M-M formalism can be used in many contexts, e.g. generegulation, protein - protein interaction etc.
I In many cases reactions do not occur spontaneously
I The cells produce enzymes which, act as catalysts for reactions
I During reactions the enzymes do not get used up
S + E −→k1 P + E
I This is unrealistic because of obvious limitations e.g. volumeof the cell
Victor Olariu CBBP - [email protected]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten (10.3,10.4)I We consider M-M rule for describing a simple enzymatic
reactionI We employ transition state theory (see notes)
S + E �k1k2
SE −→k3 P + E
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten (10.3,10.4)I We assume SE to be in qwasi-equilibriumI We assume a fixed amount of enzyme CEtotal
= CE + CSE
I The goal is to describe P production by CS
S + E �k1k2
SE −→k3 P + E
dCS
dt= −k1CSCE + k2CSE
dCE
dt= −k1CSCE + (k2 + k3)CSE
dCSE
dt= k1CSCE − (k2 + k3)CSE
dCP
dt= k3CSE
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten S + E �k1
k2SE −→k3 P + E
dCSE
dt= k1CSCE − (k2 + k3)CSE = 0 (1)
CEtotal= CE + CSE (2)
from(1), (2) =⇒ CSE =k1CSCEtotal
k2 + k3 + k1CS(3)
dCP
dt= k3CSE (4)
from(3), (4) =⇒ dCP
dt=
k3CSCEtotal
k2+k3k1
+ CS
(5)
Vmax = k3CEtotal,K =
k2 + k3
k1(6)
from(5), (6)dCP
dt=
VmaxCS
K + CS(7)
(8)Victor Olariu CBBP - [email protected]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten (10.3,10.4)
d [P]
dt=
Vmax [R]
K + [R]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten - Example Gene Regulation
TF + DNA� DNAb −→ X + DNAb
Probability of TF bound: Pb =TF
K + TF
Probability of TF unbound: Pu =K
K + TF
Activator −→ If transcription takes place when TF is bound
d [X ]
dt= V · Pb =
V [TF ]
K + [TF ]
Repressor −→ If transcription takes place when TF is unbound
d [X ]
dt= V · Pu =
VK
K + [TF ]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Michaelis-Menten - Example Gene Regulation
Full Equation for activating [X ] −→ d [X ]dt = V [TF ]
K+[TF ] −[TF ]τTF
where τTF is the TF half-lifeVictor Olariu CBBP - [email protected]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Induced Pluripotent Stem (iPS) cells
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
The FrameworkOCT4 NANOG
MEF
ES
iPS
Methylation
Probabilitydensity
0 0.5 1
ES
iPS−MEF4−7
MEF
Methylation
0 0.5 1
ES
iPS−MEF4−7
MEF
Methylation
Probabilitydensity
0 0.5 1
iPS−MEF4−7
MEF
Methylation
0 0.5 1
iPS−MEF4−7
MEF
u h mβ1
μ1
σ2
μ2Tet1
σ1 σ3
β2
Tet1
FractionofunmethylatedCpG
sites
0 0.5 1
00.5
1
OCT4
NANOG TET1
N T
&
&
OCT4 NANOG OCT4 NANOG
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FYTN05/TEK267 Chemical Forces and Self Assembly 10
Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Simplified Gene Regulatory Network Topology
I Young R.A. Cell(2011)I Costa et al. Nature(2014)I Koh et al. Cell Stem Cell(2011)
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Fast Complex Formation
[Nfree] + [Tfree] [N|T ]
Kd =[Nfree] · [Tfree]
[N|T ]
[Ntotal] = [Nfree] + [N|T ]
[Ttotal] = [Tfree] + [N|T ]
[N|T ] =Kd + [Ntotal] + [Ttotal]
2−
√√√√(Kd + [Ntotal] + [Ttotal]
2
)2
− [Ntotal] · [Ttotal]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Slow Gene Regulation
∂[Ntotal]
∂t= Nover + LIF + pN ·
[Ototal]
KO
1 +[Ototal]
KO
− [Ntotal]
∂[Ototal]
∂t= Oover + LIF + pO ·
[Ototal]
KO
1 +[Ototal]
KO
·
( [N|T ]
KNT
)n1 +
( [N|T ]
KNT
)n − [Ototal]
∂[Ttotal]
∂t= Tover + pT ·
[Ototal]
KO
1 +[Ototal]
KO
·
( [N|T ]
KNT
)n1 +
( [N|T ]
KNT
)n − [Ttotal]
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Reprogramming Simulation Results
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Exp
ress
ion
Leve
l
TET1OCT4 NANOG
Oct
4 ON
Oct
4 OFF
Nan
og O
NNan
og O
FF
Tet1
ON
Tet1
OFF
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
pre-iPS and Differentiation Simulation Results
0
0.2
0.4
0.6
0.8
1
ExpressionLevel
TET1 OCT4 NANOG
0
0.2
0.4
0.6
0.8
1
ExpressionLevel
0
0.2
0.4
0.6
0.8
1
ExpressionLevel
Oc4ON
NanogON
Tet1ON
Oct4OFF
NanogOFF
Tet1OFF
Over-expression=0.1
Over-expression=0.2
Over-expression=0.3
0
0.2
0.4
0.6
0.8
1
ExpressionLevel
OCT4 NANOGTET1
Oct4ON
Oct4OFF
Oct4+
NanogON
OFF
Oct4+
Nanog
LIF LIF Withdrawal0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
]
ExpressionLevel
TET1OCT4NANOG
A B
C
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Shea-Akers Formalism - Multiple TF
Depending on presence or absence of TF and/or RNAp the operator canbe in various states denoted s. Each state has a statistical weight Z (s)depending on binding rates. The partition sum is the sum of all weights:
Z =∑s
Z (s)
After normalization leading to the weight of the state with nothingbound to be 1 i.e. Z0 = 1 the expression of states become
Z (s) = e− ∆G(s)
kbT
∏i
[Ti ]
[Ti ]-concentration of bound regulators, e− ∆G(s)
kbT - parameter for binding
affinity, related to loss of free energy at binding.
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
The bound fraction is:
PT = αZ (bound)
Z
There are 5 possible states of the system thus partition sum is:
Z = 1 +p
kp+
A
kA+
Ap
kAp+
R
kR
all k are dissociation constants.
PT = α
pkp
+ ApkAp
1 + pkp
+ AkA
+ ApkAp
+ RkR
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
The Core Gene Regulatory Network for ES cells
LIF
BMP4
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Deterministic approach. Shea-Ackers equation
LIF
BMP4
d [N]
dt=
k0[OS](c0 + c1[N]2 + k0[OS] + c2LIF )
(1 + (k0[OS](c1[N]2 + k0[OS] + c2LIF + c3[FGF ]2)) + c4[OS][G ]2)
− γ[N],
d [OS]
dt= α+
(e0 + e1[OS])
(1 + e1[OS] + e2[G ]2)− γ[OS], (9)
d [FGF ]
dt=
(a0 + a1[OS])
(1 + a1[OS] + a2I3)− γ[FGF ],
d [G ]
dt=
(b0 + b1[G ]2 + b3[OS])
(1 + b1[G ]2 + b2[N]2 + b3[OS])− γ[G ],
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Deterministic approach
LIF
BMP4
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Stochastic Simulation Results -Time Series
LIF
BMP4
0 0.5 1 1.5 2 2.5 3 3.5
x 104
0
50
100
150
Time
Co
ncen
trati
on
LIF+BMP4
OCT4−SOX2
NANOG
0 0.5 1 1.5 2
x 104
0
50
100
150
Time
2i
OCT4−SOX2
NANOG
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
Stochastic Simulation Results - Distributions
0 50 100 1500
0.25
0.5
0.75
1
Concentration
Den
sit
y
LIF+BMP4
Nanog
Oct4−Sox2
0 50 100 1500
0.25
0.5
0.75
1
Concentration
2i/3i
Nanog
Oct4−Sox2
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
ES data Wray et al.
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Michaelis-Menten - Enzymes Michaelis-Menten - Example Gene Regulation Michaelis-Menten - Stem Cell Example Shea-Ackers Formalism Shea-Ackers Stem Cell Example
THANK YOU!
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FYTN05/TEK267 Chemical Forces and Self Assembly 24