fytn05/tek267 chemical forces and self assemblyhome.thep.lu.se/~carl/fytn05/fytn05-c2.pdf · k +...

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 (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




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




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




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]



Michaelis-Menten (10.3,10.4)

d [P]

dt=

Vmax [R]

K + [R]




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 ]




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]



Induced Pluripotent Stem (iPS) cells




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




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)




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]




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




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




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.




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




The Core Gene Regulatory Network for ES cells

LIF

BMP4




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 ],




Deterministic approach

LIF

BMP4




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




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




ES data Wray et al.




THANK YOU!



fytn05/tek267 chemical forces and self assemblyhome.thep.lu.se/~carl/fytn05/fytn05-c2.pdf · k +...

Documents