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Mathematical Modelling of Genetic Regulatory Networks

Explore the dynamics of genetic regulatory networks through ODEs to understand induction, repression, AND, XOR logic operations. Should reversible reactions be considered? Find out in this detailed study from ETH Zürich iGEM team 2006.

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Mathematical Modelling of Genetic Regulatory Networks

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  1. Half Adder - Modelling ETH Zürich | iGEM team 2006

  2. ODE’s: Constitutive A A u·kA kA-Deg d[A]/dt = u·kA - [A]·kA-Deg simplified: d[A]/dt = 0  [A] = const. (high/low dep. on input u)

  3. ODE’s: Induction I A kI kA I + DNA I · DNA I · DNA + A k-I d[A]/dt = [I·DNA]·kA d[I]/dt = [I·DNA]·k-I – [I]·[DNA]·kI d[DNA]/dt = [I·DNA]·k-I – [I]·[DNA]·kI d[I·DNA]/dt = [I]·[DNA]·kI – [I·DNA]·k-I [I·DNA]+[DNA] = const.[I·DNA]+[I] = const.

  4. ODE’s: Repression kR R R + DNA R · DNA A k-R kA DNA DNA + A d[A]/dt = [DNA]·kA d[R]/dt = [R·DNA]·k-R – [R]·[DNA]·kR d[DNA]/dt = [R·DNA]·k-R – [R]·[DNA]·kR d[R·DNA]/dt = [R]·[DNA]·kR – [R·DNA]·k-R [R·DNA]+[DNA] = const.[R·DNA]+[R] = const.

  5. AND – 1 a PoPS E1-active A E1 PoPS e1 e2 E2 B PoPS b

  6. A B E2 E1 E2 E1 Out AND – 1 (ODE’s) uA kA k-E kE1-act E1-act kE1 kE kE2 kOut kB d[A]/dt = kAuA d[B]/dt = kBuB d[E1]/dt = kE1[A] - kE[E1][E2] + k-E[E1E2] d[E2]/dt = kE2[B] - kE[E1][E2] + k-E[E1E2] + kE1-act[E1E2] d[E1E2 ]/dt = kE[E1][E2] - k-E[E1E2] - kE1-act[E1E2] d[E1-act]/dt = kE1-act[E1 E2] d[Out]/dt = kout[E1-act] uB

  7. AND – 2 a PoPS A E1 PoPS i1 E1 E2 i2 E2 B PoPS b

  8. AND – 3

  9. m/([X]m-m) A B I2 I1 only qualitative! 1 Out [X]  AND – 3 (ODE’s) uA kI1,deg kI1 k’Out kOut,deg kI2 k’’Out kI2,deg uB d[A]/dt = kAuA d[B]/dt = kBuB d[I1]/dt = kI1 m1/([A]m1-m1) – kI1,deg[I1] d[I2]/dt = kI2 m1/([B]m1-m1) – kI2,deg[I1] d[Out]/dt = k’Outm3/([I1]m3-m3) k’’Outm4/([I2]m4 - m4) – kOut,deg[Out]

  10. AND – 4

  11. XOR – 1

  12. I’2 I2 I1 I’1 I1 I’1 I’2 I2 Out XOR – 1 (ODE’s) kI1,deg kI1 kA1 Question: Should kI1I‘1 and kI2I‘2 be modelled as reversible reactions? uA kI1I’1 kA2 kI’1,deg kOut,deg d[I1]/dt = kA1uA – kI1I’1[I1][I’1] – kI1,deg[I1] d[I’1]/dt = kA2uA – kI1I’1[I1][I’1] – kI’1,deg[I’1] d[I2]/dt = kB1uB – kI2I’2[I2][I’2] – kI2,deg[I2] d[I’2]/dt = kB2uB – kI2I’2[I2][I’2] – kI’2,deg[I’2] d[I1I’1]/dt = kI1I’1[I1][I’1] d[I2I’2]/dt = kI2I’2[I2][I’2] d[Out]/dt = kI1(1- m1/([I1]m1-m1)) + kI2(1- m2/([I2]m2 - m2)) – kOut,deg[Out] kI’2,deg kB2 uB kI2I’2 kI2 kB1 kI2,deg OR

  13. XOR – 2

  14. XOR – 2 (ODE’s)

  15. XOR – 3

  16. XOR – 3 (ODE’s)

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