Study of Tricyclic Cascade Networks using Dynamic Optimization

Transcription

1 Study of Tricyclic Cascade Networks using Dynamic Optimization

2 Systems Biology interdisciplinary field that focuses on the systematic study of complex interactions in biological systems Signal transduction pathways o enable cells to integrate external and internal signals and to respond to them, o are present in major developmental changes in organism (embryo development, but also in cancer, asthma, diabetes ) H. Kestler, C. Wawra, B. Kracher and M. Kuhl: Network modeling of signal transduction: establishing the global view. Bioessays. 30:

3 Basic module Protein Inactive - X Active X* Enzymes S* and P X phosphorylation/dephosphorylation in MAPK cascades methylation/demethylation in bacterial chemotaxis

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6 Direct Approach: Inverse Approach: Both direct and inverse approaches work synergistically! Dynamic optimization very well suited for studying biochemical networks it allows dealing with large-scale, nonlinear dynamic models can handle a great variety of objective functions and constraints.

7 balance equations, conservation equations kinetic expressions

8 balance equations, conservation equations kinetic expressions Kinetic parameters: a X,, a X *, d X, d X *, k X

9 Concentration ratios Saturation parameters Activity coefficient

10 Inverse Approach:

11 max. activity Concentration kinase activity Input stimulus 0 Time max. activity max. activity 90% Concentration upstream kinase kinase activity Concentration kinase activity upstream kinase 10% Time Time

12 Steady-state kinase activity % Typical graded response Ultrasensitive response 10% α 0.1 X α 0.9 X Input stimulus Go from a typical graded response to a switch-like response

13 Optimal ultrasensitivity for fixed values of ratios between total concentrations of enzymes and substrate: Ultrasensitivity [S]/[X] = 0.01

14 Optimal ultrasensitivity for fixed values of ratios between total concentrations of enzymes and substrate: Ultrasensitivity Ultrasensitivity Ultrasensitivity [S]/[X] = 0.01 [S]/[X] = 0.1 [S]/[X] = 1

15 Ultrasensitivity Saturation parameter K X Saturation parameter K* X Concentration ratio [P X ]/[X] Concentration ratio [P X ]/[X] Concentration ratio [P X ]/[X] Concentration ratio [S]/[X] Concentration ratio [S]/[X] Concentration ratio [S]/[X] Steady-state kinase activity 1 Ultrasensitivity can be achieved for small values of concentration ratios The closer to saturation, the more ultrasensitive the monocyclic cascade. 0 Input stimulus

16 Kinetic parameters: a X,, a * X, d X, d * X, k X, a Y, a * Y, d Y, d * Y, k Y, k * Y

17 Concentration ratios ρ S / X,ρ P X / X,ρ X /Y,ρ P Y /Y Saturation parameters K X = K Y = d X + a X d Y + a Y k X k Y K * X = K * Y = d * X +1 * a X d * Y + k * Y * a Y Activity coefficients α X = k X ρ S / X ρ P X / X k Y * k Y α Y = ρ X /Y ρ P Y /Y

18 Maximal values of ultrasensitivity objective measure, with corresponding values of saturation parameters for activation reaction of each level in a signaling cascade model: Concentration ratio [S]/[X] Ultrasensitivity Saturation parameter K X Saturation parameter K Y Concentration ratio [S]/[X] Concentration ratio [S]/[X] Concentration ratio [X]/[Y] Concentration ratio [X]/[Y] Concentration ratio [X]/[Y]

19 saturated unsaturated Maximum ultrasensitivity when the first kinase is saturated, but not the second kinase An optimal multicycle cascade does not correspond to a series of optimal monocyclic cascades

20 Kinetic parameters: a X,, a * X, d X, d * X, k X, a Y, a * Y, d Y, d * Y, k Y, k * Y, a Z, a * Z, d Z, d * Z, k Z, k * Z

21 Concentration ratios ρ S / X,ρ P X / X,ρ X /Y,ρ P Y /Y,ρ Y / Z,ρ P Z / Z Activity coefficients α X = k X ρ S / X ρ P X / X k Y * k Y α Y = ρ X /Y ρ P Y /Y Saturation parameters K X = K Y = K Z = k Z * k Z α Z = ρ Y / Z ρ P Y / Z d X + a X d Y + k X k Y a Y d Z + a Z k Z K * X = K * Y = K * Z = d * X +1 * a X d * Y + k * Y * a Y d * Z + * a Z k Z *

22 Optimal ultrasensitivity for various combinations of saturation parameters: K X = 0.1 K Y =10 K Z =10 K X = 0.1 K Y =10 K Z =1 K X = 0.1 K Y =1 K Z =10 K X =1 K Y =10 K Z =10 K X =1 K Y =10 K Z =1 K X =1 K Y =1 K Z =10

23 Concentration ratio [X]/[Y] Ultrasensitivity Optimal ultrasensitivity for fixed values of ratios between total concentrations of enzymes and substrate Concentration ratio [Y]/[Z]

24 Concentration ratio [X]/[Y] Ultrasensitivity Optimal ultrasensitivity for fixed values of ratios between total concentrations of enzymes and substrate Concentration ratio [Y]/[Z] Saturation parameter K X Saturation parameter K Y Saturation parameter K Z Concentration ratio [X]/[Y] Concentration ratio [Y]/[Z] Concentration ratio [Y]/[Z] Concentration ratio [Y]/[Z]

25 saturated unsaturated unsaturated Optimal ultrasensitivity is achieved if the first kinase is saturated by its target kinase, but not the subsequent two kinases.

26 Ultrasensitivity K X K Y K Z Monocyclic ~ 3 ~ 0.1 Bicyclic ~ 5.5 ~ 0.1 ~ 200 Tricyclic ~9.8 ~ 0.1 ~ 350 ~ 150

27 max. activity max. activity max. activity Concentration kinase activity Input stimulus 90% Concentration kinase activity upstream kinase Concentration upstream kinase kinase activity 10% 0 Time Time Time Amplification in signaling cycles - a measure of response strength - measure of how fast a signal is transduced through a cycle and time needed to reach 90% of the maximum substrate activation time needed for the substrate activity to decrease to within 10% of the ground state

28 d X k X Γ Γ a X Γ d * X * a X Γ Γ

29 Optimal multicycle cascades may not correspond to multiple levels of an optimal single level cascade The larger the number of levels in the cascade, the more robust that cascade - variation in concentration regimes Fast signal propagation can be achieved with different sets of kinetic parameters

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