FYTN05/TEK267 Chemical Forces and Self Assembly

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1 FYTN05/TEK267 Chemical Forces and Self Assembly FYTN05/TEK267 Chemical Forces and Self Assembly 1

2 Michaelis-Menten (10.3,10.4) M-M formalism can be used in many contexts, e.g. gene regulation, protein - protein interaction etc. In many cases reactions do not occur spontaneously The cells produce enzymes which, act as catalysts for reactions During reactions the enzymes do not get used up S + E k 1 P + E This is unrealistic because of obvious limitations e.g. volume of the cell FYTN05/TEK267 Chemical Forces and Self Assembly 2

3 Michaelis-Menten (10.3,10.4) We consider M-M rule for describing a simple enzymatic reaction We employ transition state theory (see notes) S + E k 1 k 2 SE k 3 P + E FYTN05/TEK267 Chemical Forces and Self Assembly 3

4 Michaelis-Menten (10.3,10.4) We assume SE to be in qwasi-equilibrium We assume a fixed amount of enzyme C Etotal = C E + C SE The goal is to describe P production by C S S + E k 1 k 2 SE k 3 P + E dc S dc E dc SE dc P = k 1 C S C E + k 2 C SE = k 1 C S C E + (k 2 + k 3 )C SE = k 1 C S C E (k 2 + k 3 )C SE = k 3 C SE FYTN05/TEK267 Chemical Forces and Self Assembly 4

5 Michaelis-Menten S + E k 1 k 2 SE k 3 P + E dc SE = k 1 C S C E (k 2 + k 3 )C SE = 0 (1) C Etotal = C E + C SE (2) from(1), (2) = C SE = k 1 C S C Etotal k 2 + k 3 + k 1 C S (3) dc P = k 3 C SE (4) from(3), (4) = dc P from(5), (6) dc P = k 3C S C Etotal k 2 +k 3 k 1 + C S (5) V max = k 3 C Etotal, K = k 2 + k 3 k 1 (6) = V maxc S K + C S (7) (8) FYTN05/TEK267 Chemical Forces and Self Assembly 5

6 Michaelis-Menten (10.3,10.4) d[p] = V max[r] K + [R] FYTN05/TEK267 Chemical Forces and Self Assembly 6

7 Michaelis-Menten - Example Gene Regulation TF + DNA DNA b X + DNA b Probability of TF bound: P b = Probability of TF unbound: P u = TF K + TF K K + TF Activator If transcription takes place when TF is bound d[x ] = V P b = V [TF ] K + [TF ] Repressor If transcription takes place when TF is unbound d[x ] = V P u = VK K + [TF ] FYTN05/TEK267 Chemical Forces and Self Assembly 7

8 Michaelis-Menten - Example Gene Regulation Full Equation for activating [X ] d[x ] where τ TF is the TF half-life = V [TF ] K+[TF ] [TF ] τ TF FYTN05/TEK267 Chemical Forces and Self Assembly 8

9 Induced Pluripotent Stem (ips) cells FYTN05/TEK267 Chemical Forces and Self Assembly 9

10 MEF The Framework OCT4 NANOG MEF σ1 σ3 σ2 ES ips β 1 u h m μ1 Tet1 β 2 μ 2 OCT4 NANOG OCT4 NANOG Fraction of unmethylated CpG sites Tet1 Probability density ips MEF4 7 MEF ips MEF4 7 Probability density MEF ips MEF4 7 MEF NANOG N & T TET1 ES ES ips MEF4 7 & Methylation Methylation Methylation Methylation OCT4 FYTN05/TEK267 Chemical Forces and Self Assembly 10

11 Simplified Gene Regulatory Network Topology Young R.A. Cell(2011) Costa et al. Nature(2014) Koh et al. Cell Stem Cell(2011) FYTN05/TEK267 Chemical Forces and Self Assembly 11

12 Fast Complex Formation [N free ] + [T free ] [N T ] K d = [N free] [T free ] [N T ] [N total ] = [N free ] + [N T ] [T total ] = [T free ] + [N T ] [N T ] = K d + [N total ] + [T total ] 2 ( ) 2 K d + [N total ] + [T total ] [N total] [T total] 2 FYTN05/TEK267 Chemical Forces and Self Assembly 12

13 Slow Gene Regulation [N total ] t [O total ] t [T total ] t [O total ] = N over + LIF + p N K O [N total ] 1 + [O total] K O = [O total ] ( [N T ] ) n K O over + LIF + p O O K 1 + [O NT ( total] [N T ] ) [O total ] n 1 + K O K NT [O total ] ( [N T ] = T over + p T K O 1 + [O total] K O 1 + ) n K NT ( [N T ] ) n K NT [T total ] FYTN05/TEK267 Chemical Forces and Self Assembly 13

14 Reprogramming Simulation Results 1 OCT4 NANOG TET Expression Level Oct4 ON Oct4 OFF Nanog ON Nanog OFF Tet1 ON Tet1 OFF FYTN05/TEK267 Chemical Forces and Self Assembly 14

15 Expression Level ] pre-ips and Differentiation Simulation Results A 1 TET1 OCT4 NANOG B 1 TET1 OCT4 NANOG Expression Level Expression Level Over-expression = 0.1 Over-expression = 0.2 Over-expression=0.3 C Expression Level Expression Level Oct4 ON LIF Oct4 OFF Oct4+ Nanog ON LIF Withdrawal Oct4+ OFF Nanog TET1 OCT4 NANOG 0 Oc4 ON Oct4 OFF Nanog ON Nanog OFF Tet1 ON Tet1 OFF FYTN05/TEK267 Chemical Forces and Self Assembly 15

16 Shea-Akers Formalism - Multiple TF Depending on presence or absence of TF and/or RNAp the operator can be 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 nothing bound to be 1 i.e. Z 0 = 1 the expression of states become Z(s) = e G(s) k b T [T i ] [T i ]-concentration of bound regulators, e G(s) k b T - parameter for binding affinity, related to loss of free energy at binding. i FYTN05/TEK267 Chemical Forces and Self Assembly 16

17 The bound fraction is: P T = α Z(bound) Z There are 5 possible states of the system thus partition sum is: all k are dissociation constants. Z = 1 + p k p + A k A + Ap k Ap + R k R P T = α p k p + Ap k Ap 1 + p k p + A k A + Ap k Ap + R k R FYTN05/TEK267 Chemical Forces and Self Assembly 17

18 The Core Gene Regulatory Network for ES cells LIF BMP4 FYTN05/TEK267 Chemical Forces and Self Assembly 18

19 LIF BMP4 Deterministic approach. Shea-Ackers equation d[n] d[os] d[fgf ] d[g] = k 0 [OS](c 0 + c 1 [N] 2 + k 0 [OS] + c 2 LIF ) (1 + (k 0 [OS](c 1 [N] 2 + k 0 [OS] + c 2 LIF + c 3 [FGF ] 2 )) + c 4 [OS][G] 2 ) γ[n], = α + = = (e 0 + e 1 [OS]) (1 + e 1 [OS] + e 2 [G] 2 γ[os], (9) ) (a 0 + a 1 [OS]) γ[fgf ], (1 + a 1 [OS] + a 2 I 3 ) (b 0 + b 1 [G] 2 + b 3 [OS]) (1 + b 1 [G] 2 + b 2 [N] 2 + b 3 [OS]) γ[g], FYTN05/TEK267 Chemical Forces and Self Assembly 19

20 LIF BMP4 Deterministic approach FYTN05/TEK267 Chemical Forces and Self Assembly 20

21 LIF BMP4 Stochastic Simulation Results -Time Series 150 OCT4 SOX2 NANOG LIF+BMP4 150 OCT4 SOX2 NANOG 2i Concentration Time x Time x 10 4 FYTN05/TEK267 Chemical Forces and Self Assembly 21

22 Stochastic Simulation Results - Distributions Nanog Oct4 Sox2 LIF+BMP Nanog Oct4 Sox2 2i/3i Density Concentration Concentration FYTN05/TEK267 Chemical Forces and Self Assembly 22

23 ES data Wray et al. FYTN05/TEK267 Chemical Forces and Self Assembly 23

24 THANK YOU! FYTN05/TEK267 Chemical Forces and Self Assembly 24