Oscillations in Michaelis-Menten Systems Hwai-Ray Tung Brown University July 18, 2017 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 1 / 22
Background A simple example of genetic oscillation Gene makes compound (transcription factor) Transcription factor binds to promoter Circadian Clocks Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 2 / 22
Background A simple example of genetic oscillation Gene makes compound (transcription factor) Transcription factor binds to promoter Circadian Clocks Mass Action Kinetics Write chemical reaction network as a system of differential equations Michaelis-Menten (MM) Approximation for Biochemical Systems Assume low concentration of intermediates Allows for elimination of variables, reducing differential equations from mass action Michaelis-Menten approximation has oscillations implies original has oscillations Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 2 / 22
Michaelis-Menten System (Dual Futile Cycle) k 1 k 4 k 3 k 6 S 0 + E − S 0 E → S 1 + E − S 1 E → S 2 + E − − ⇀ − ⇀ − ↽ ↽ Phosphorylation k 2 k 5 Dephosphorylation l 1 l 4 l 3 l 6 − − S 2 + F S 2 F − → S 1 + F S 1 F − → S 0 + F − − ⇀ ⇀ ↽ ↽ l 2 l 5 Oscillatory behavior is unknown. Wang and Sontag (2008) showed Michaelis-Menten approximation has no oscillations Bozeman and Morales (REU 2016) showed Michaelis-Menten approximation has no oscillations with more elementary techniques Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 3 / 22
Processive vs Distributive Distributive: k 1 k 4 k 3 k 6 − − S 0 + E S 0 E − → S 1 + E S 1 E − → S 2 + E ⇀ − ⇀ − ↽ ↽ k 2 k 5 l 1 l 4 l 3 l 6 S 2 + F − S 2 F → S 1 + F − S 1 F → S 0 + F − − ⇀ − ⇀ − ↽ ↽ l 2 l 5 Processive: k 1 k 7 k 6 S 0 + E − S 0 E → S 1 E → S 2 + E − − ⇀ − ↽ k 2 l 1 l 7 l 6 − S 2 + F S 2 F − → S 1 F − → S 0 + F ⇀ − ↽ l 2 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 4 / 22
This Year’s Project S 1 + F S 2 + F S 2 F S 1 F S 0 + F S 0 + E S 0 E S 1 E S 2 + E S 1 + E Does it have oscillations? Does its Michaelis-Menten approximation have oscillations? Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 5 / 22
Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [ S 0 E ] k 1 S 0 + E − → S 0 E gives Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 6 / 22
Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [ S 0 E ] k 1 S 0 + E − → S 0 E gives d [ S 0 E ] = k 1 [ S 0 ][ E ] dt Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 6 / 22
Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [ S 0 E ] k 1 S 0 + E − → S 0 E gives d [ S 0 E ] = k 1 [ S 0 ][ E ] dt k 2 S 0 + E − S 0 E gives ← Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 6 / 22
Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [ S 0 E ] k 1 S 0 + E − → S 0 E gives d [ S 0 E ] = k 1 [ S 0 ][ E ] dt k 2 S 0 + E − S 0 E gives ← d [ S 0 E ] = − k 2 [ S 0 E ] dt Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 6 / 22
Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [ S 0 E ] k 1 S 0 + E − → S 0 E gives d [ S 0 E ] = k 1 [ S 0 ][ E ] dt k 2 S 0 + E − S 0 E gives ← d [ S 0 E ] = − k 2 [ S 0 E ] dt k 1 − S 0 + E S 0 E gives ⇀ − ↽ k 2 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 6 / 22
Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [ S 0 E ] k 1 S 0 + E − → S 0 E gives d [ S 0 E ] = k 1 [ S 0 ][ E ] dt k 2 S 0 + E − S 0 E gives ← d [ S 0 E ] = − k 2 [ S 0 E ] dt k 1 − S 0 + E S 0 E gives ⇀ − ↽ k 2 d [ S 0 E ] = k 1 [ S 0 ][ E ] − k 2 [ S 0 E ] dt Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 6 / 22
Corresponding ODEs d [ S 0 ] = l 6 [ S 1 F ] − k 1 [ S 0 ][ E ] + k 2 [ S 0 E ] dt d [ S 2 ] = k 6 [ S 1 E ] − l 1 [ S 2 ][ F ] + l 2 [ S 2 F ] dt d [ S 1 ] = k 3 [ S 0 E ] − k 4 [ S 1 ][ E ] + k 5 [ S 1 E ] + l 3 [ S 2 F ] + l 5 [ S 1 F ] − l 4 [ S 1 ][ F ] dt d [ E ] = ( k 2 + k 3 )[ S 0 E ] + ( k 5 + k 6 )[ S 1 E ] − k 1 [ S 0 ][ E ] − k 4 [ S 1 ][ E ] dt d [ F ] = ( l 2 + l 3 )[ S 2 F ] + ( l 5 + l 6 )[ S 1 F ] − l 1 [ S 2 ][ F ] − l 4 [ S 1 ][ F ] dt d [ S 0 E ] = k 1 [ S 0 ][ E ] − ( k 2 + k 3 )[ S 0 E ] − k 7 [ S 0 E ] dt d [ S 1 E ] = k 4 [ S 1 ][ E ] − ( k 5 + k 6 )[ S 1 E ]+ k 7 [ S 0 E ] dt d [ S 2 F ] = l 1 [ S 2 ][ F ] − ( l 2 + l 3 )[ S 2 F ] − l 7 [ S 2 F ] dt d [ S 1 F ] = l 4 [ S 1 ][ F ] − ( l 5 + l 6 )[ S 1 F ]+ l 7 [ S 2 F ] dt Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 7 / 22
Michaelis-Menten Approximation of ODEs: Conservation Equations Conservation equations refer to conservation of mass. S T = [ S 0 ] + [ S 1 ] + [ S 2 ] + [ S 0 E ] + [ S 1 E ] + [ S 2 F ] + [ S 1 F ] E T = [ E ] + [ S 0 E ] + [ S 1 E ] F T = [ F ] + [ S 2 F ] + [ S 1 F ] S 1 + F S 2 + F S 2 F S 1 F S 0 + F S 0 + E S 0 E S 1 E S 2 + E S 1 + E Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 8 / 22
Michaelis-Menten Approximation of ODEs: Conservation Equations Conservation equations refer to conservation of mass. S T = [ S 0 ] + [ S 1 ] + [ S 2 ] + [ S 0 E ] + [ S 1 E ] + [ S 2 F ] + [ S 1 F ] E T = [ E ] + [ S 0 E ] + [ S 1 E ] F T = [ F ] + [ S 2 F ] + [ S 1 F ] d [ E ] = ( k 2 + k 3 )[ S 0 E ] + ( k 5 + k 6 )[ S 1 E ] − k 1 [ S 0 ][ E ] − k 4 [ S 1 ][ E ] dt d [ S 0 E ] = k 1 [ S 0 ][ E ] − ( k 2 + k 3 )[ S 0 E ] − k 7 [ S 0 E ] dt d [ S 1 E ] = k 4 [ S 1 ][ E ] − ( k 5 + k 6 )[ S 1 E ]+ k 7 [ S 0 E ] dt Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 9 / 22
Michaelis-Menten Approximation of ODEs: Assumption We assume enzyme concentrations are small: E T = ε � E T , F T = ε � F T , [ E ] = ε [ � E ] , [ F ] = ε [ � F ] , [ S 0 E ] = ε [ � S 0 E ] , [ S 1 E ] = ε [ � S 1 E ] , [ S 2 F ] = ε [ � S 2 F ] , [ S 1 F ] = ε [ � S 1 F ] , τ = ε t Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 10 / 22
Michaelis-Menten Approximation of ODEs: Plug in d [ S 0 ] = l 6 [ � S 1 F ] − k 1 [ S 0 ][ � E ] + k 2 [ � S 0 E ] d τ d [ S 2 ] = k 6 [ � S 1 E ] − l 1 [ S 2 ][ � F ] + l 2 [ � S 2 F ] d τ d [ S 1 ] X = k 3 [ S 0 E ] − k 4 [ S 1 ][ E ] + k 5 [ S 1 E ] + l 3 [ S 2 F ] + l 5 [ S 1 F ] − l 4 [ S 1 ][ F ] dt d [ E ] X = ( k 2 + k 3 )[ S 0 E ] + ( k 5 + k 6 )[ S 1 E ] − k 1 [ S 0 ][ E ] − k 4 [ S 1 ][ E ] dt d [ F ] X = ( l 2 + l 3 )[ S 2 F ] + ( l 5 + l 6 )[ S 1 F ] − l 1 [ S 2 ][ F ] − l 4 [ S 1 ][ F ] dt ε d [ � S 0 E ] = k 1 [ S 0 ][ � E ] − ( k 2 + k 3 )[ � S 0 E ] − k 7 [ � S 0 E ] d τ ε d [ � S 1 E ] = k 4 [ S 1 ][ � E ] − ( k 5 + k 6 )[ � S 1 E ]+ k 7 [ � S 0 E ] d τ ε d [ � S 2 F ] = l 1 [ S 2 ][ � F ] − ( l 2 + l 3 )[ � S 2 F ] − l 7 [ � S 2 F ] d τ ε d [ � S 1 F ] = l 4 [ S 1 ][ � F ] − ( l 5 + l 6 )[ � S 1 F ]+ l 7 [ � S 2 F ] d τ Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 11 / 22
Michaelis-Menten Approximation of ODEs = ( a 1 [ S 1 ] + a 2 [ S 2 ])[ � a 3 [ S 0 ][ � d [ S 0 ] F T ] E T ] 1 + c 2 [ S 1 ] + d 2 [ S 2 ] − d τ 1 + b 1 [ S 0 ] + c 1 [ S 1 ] = ( a 4 [ S 1 ] + a 5 [ S 0 ])[ � a 6 [ S 2 ][ � d [ S 2 ] E T ] F T ] 1 + b 1 [ S 0 ] + c 1 [ S 1 ] − d τ 1 + c 2 [ S 1 ] + d 2 [ S 2 ] [ S 1 ] = S T − [ S 0 ] − [ S 2 ] k 1 ( k 5 + k 6 + k 7 ) l 1 ( l 5 + l 6 + l 7 ) b 1 = d 1 = ( k 2 + k 3 + k 7 )( k 5 + k 6 ) ( l 2 + l 3 + l 7 )( l 5 + l 6 ) k 4 l 4 c 1 = c 2 = k 5 + k 6 l 5 + l 6 l 4 l 6 k 4 k 6 a 1 = a 4 = l 5 + l 6 k 5 + k 6 l 1 l 6 l 7 k 1 k 6 k 7 a 2 = a 5 = ( l 2 + l 3 + l 7 )( l 5 + l 6 ) ( k 2 + k 3 + k 7 )( k 5 + k 6 ) k 1 ( k 3 + k 7 ) l 1 ( l 3 + l 7 ) a 3 = a 6 = ( k 2 + k 3 + k 7 ) ( l 2 + l 3 + l 7 ) Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 12 / 22
Dulac’s Criterion = ( a 1 [ S 1 ] + a 2 [ S 2 ])[ � a 3 [ S 0 ][ � f ([ S 0 ] , [ S 2 ]) = d [ S 0 ] F T ] E T ] 1 + c 2 [ S 1 ] + d 2 [ S 2 ] − d τ 1 + b 1 [ S 0 ] + c 1 [ S 1 ] = ( a 4 [ S 1 ] + a 5 [ S 0 ])[ � a 6 [ S 2 ][ � g ([ S 0 ] , [ S 2 ]) = d [ S 2 ] E T ] F T ] 1 + b 1 [ S 0 ] + c 1 [ S 1 ] − d τ 1 + c 2 [ S 1 ] + d 2 [ S 2 ] Theorem (Dulac) If the sign of df dg d [ S 0 ] + d [ S 2 ] does not change across a simply connected domain in R 2 , the system does not exhibit oscillations in the domain. Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, 2017 13 / 22
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