Bioinformatics Modelling dynamic behaviour: Modularisation

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1 Bioinformatics Modelling dynamic behaviour: David Gilbert Bioinformatics Research Centre Department of Computing Science, University of Glasgow

2 Lecture outline Modelling Enzymatic Reactions Signal Transduction Cascades Modelling: Building blocks & composition/decomposition Positive & negative feedback Modelling the effect of drug inhibitors Analysing the behaviour of models 2

3 Motivation Quantitative models of biochemical networks are a central component of modern systems biology. Building & managing these complex models is a major challenge that can benefit from the application of formal methods adopted from theoretical computing science. Hence - a general introduction to the field of formal modelling emphasizes the intuitive biochemical basis of the modelling process, accessible for an audience with a background in computing science and/or model engineering. 3

4 Some (Bio)Chemical Conventions Concentration of Molecule A = [A], usually in units mol/litre (molar) Rate constant = k, with indices indicating constants for various reactions (k 1, k 2...) Therefore: A B d[ A] dt =! d[ B] dt =! k 1[ A ] 4

5 Reversible, Single-Molecule Reaction A B, or A B B A, or Differential equations: d[ A] dt d[ B] dt =! k = k 1 forward reverse 1 [ A] + [ A]! k k 2 2 [ B] [ B] Main principle: Partial reactions are independent! 5

6 Irreversible, two-molecule reaction The last piece of the puzzle A+B C Differential equations: d[ A] dt d[ A] dt d[ B] d[ C] = =! dt dt =! k[ A][ B] Non-linear! Underlying idea: Reaction probability = Combined probability that both [A] and [B] are in a reactive mood : * * p ( AB) = p( A) p( B) = k [ A] k [ B] = k[ A][ ] 1 2 B 6

7 Biological description bigraph ODEs 7

8 Biological description bigraph ODEs 8

9 E Mass action, MA1 model E A B A E A B A: substrate, B: product E: enzyme E A substrate-enzyme complex E +A k 1 $ E A k 3$ E + B " k 2 9

10 Mass action equations 1. E + A -(k1) E A 2. E A -(k2) E+A 3. E A -(k3) E+B OR 1. E + A =(k1/k2)= E A 2. E A -(k3) E+B E +A k1 $ " k2 E A k 3 $ E + B?Can you code the differential equations? 10

11 Differential equations 11

12 MA2, MA3 models MA2 A+E k1 $ " k2 A E k 3 $ B E k4 $ " k 5 B + E MA3 A+E k1 $ " k 2 A E k3 $ " k 6 B E k4 $ " k 5 B + E 12

13 Multiple substrates 13

14 Michaelis-Menten V = V max " [A] (K M +[A]) k cat = V max [E T ] d[a] dt = d[b] dt = k cat "[E T ]" [A] (K M +[A]) V : Reaction velocity K M : Michaelis constant - concentration of substrate at which reaction rate is half max value [E T ] : total enzyme concentration 14

15 Assumptions It is critical to note that the Michaelis-Menten equation only holds at the initial stage of a reaction before the concentration of the product is appreciable, and makes the following assumptions: 1. No product reverts to initial substrate 2. MM Equation holds at initial stage of reaction before concentration of product is appreciable 3. [E] << [A] 15

16 Metabolic pathways vs Signalling Pathways (can you give the mass-action equations?) Metabolic (initial substrate) S E1 Input Signal X Signalling cascade S S1 P1 E2 S S2 P2 E3 S (final product) Classical enzyme-product pathway S3 P3 Output Product become enzyme at next stage 16

17 A special case: enzyme reactions Underlying assumptions of the Michaelis-Menten approximation: Free diffusion, random collisions Irreversible reactions Quasi steady state In cell signaling pathways, all three assumptions will be frequently violated: Reactions happen at membranes and on scaffold structures Reactions happen close to equilibrium and both reactions have non-zero fluxes Enzymes are themselves substrates for other enzymes, concentrations change rapidly, d[es]/dt d[p]/dt 17

18 Cell signaling pathways 18

19 Cell signaling pathways 19

20 Cell signaling pathways Common components: Receptors binding to ligands R(inactive) + L RL(active) Proteins forming complexes P1 + P2 P1P2-complex Proteins acting as enzymes on other proteins (e.g., phosphorylation by kinases) P1 + K P1* + K 20

21 Cell signaling pathways Fig. courtesy of W. Kolch 21

22 Cell signaling pathways Fig. courtesy of W. Kolch 22

23 Cell signaling pathways Fig. courtesy of W. Kolch 23

24 MAPK Pathway Responds to wide range of stimuli: cytokines, growth factors, neurotransmitters, cellular stress and cell adherence, STIMULUS Pivotal role in many key cellular processes: growth control in all its variations, cell differentiation and survival cellular adaptation to chemical and physical stress. Deregulated in various diseases: cancer; immunological, inflammatory and degenerative syndromes, Represents an important drug target. 24

25 Mass action for enzymatic reaction - phosphorylation S 1 R R: substrate, : product (phosphorylated R) S 1 : enzyme (kinase) R S 1 substrate-enzyme complex k 1 R+S $ R S k 3 $ R + S 1 " 1 p 1 k 2 25

26 Phosphorylation - dephosphorylation step Mass action MA1 R: unphosphorylated form : phosphorylated form S: kinase P: phosphotase R S unphosphorylated+kinase complex R P unphosphorylated+phosphotase complex R S R+S R+P k1 $ " k 2 R S " kr 3 R P p k 3 $ R + S p kr 1 " + P $ kr2 P 26

27 Phosphorylation - dephosphorylation step Mass action MA1 dydt = [-k1*s*r + k2*rs + k3*rs -k1*s*r + k2*rs + k3*rpp +k1*s*r - k2*rs - k3*rs -kr1*p*rp + kr2*rpp + k3*rs -kr1*p*rp + kr2*rpp + kr3*rpp +k1*rp*p - kr2*rpp - kr3*rpp ]; % S % R % RS % Rp % P % RpP R+S " k1 R S R + P p " kr1 R P p R S R S " k2 R + S R P p " k 3 R + S R P p p " kr2 R + P p " kr 3 R + P 27

28 Phosphorylation - dephosphorylation loop Mass action MA2 R: unphosphorylated form : phosphorylated form S 1 : kinase S 2 : phosphotase R S 1 unphosphorylated+kinase complex S 1 phosphorylated+kinase complex R S 2 unphosphorylated+phosphotase complex S 2 phosphorylated+phosphotase complex R S 1 k 1 R+S $ k 1 R S " 1 3 $ S $ 1 R " p + S 1 k 2 k 5 kr4 R+S 2 kr 5 " $ kr R S 3 2 " k 4 kr1 S 2 kr 2 " $ + S 2 S 2 28

29 Phosphorylation - dephosphorylation step Mass action MA3 R: unphosphorylated form : phosphorylated form S: kinase P: phosphotase R S unphosphorylated+kinase complex R P unphosphorylated+phosphotase complex R S R+S k1 $ " k2 R S k 3 $ " Rp S k 6 k4 $ " k 5 + S P R+P " kr4 $ kr5 R P " kr3 $ kr 6 P " kr1 $ kr2 + P 29

30 V = k 3 "[S]" Michaelis-Menten equation for phosphorylation-dephosphorylation [R] (K m1 +[R]) k '" [ ] 3 (K m2 +[ ]) R S Assumptions: 1. No product reverts to initial substrate 2. MM Equation holds at initial stage of reaction before concentration of product is appreciable 3. [Enzyme] << [Substrate] K m is [Substrate] at which the reaction rate is half its maximum value d /dt == reaction rate V k 3 x S == V max for the forward reaction k 3 == V max for the reverse reaction (Phosphotase is ignored) K m1 == (k 2 +k 3 )/k 1 (k s from mass-action 1) P 30

31 Questions Is Michaelis-Menten adequate for phosphorylation pathways? Is Mass Action sufficient/correct for these pathways? What is the effect of negative feedback? Can we confirm the negative feedback amplifer behaviour in both MM and MA models Can oscillators be built? Overall, what are the rules for component-based construction? 31

32 Composition vertical & horizontal S 1 S R R p P 1 2-stage cascade RR R P 1-stage cascade double phosphorylation step P 2 32

33 Phosphorylation cascade: 2-stage, Mass Action MA1 k 1 R+S $ k 1 R S " 1 3 $ + S 1 k 2 S 1 k 1 k R+S 3 2 " R S 2 k 2 " $ + S 2 R kk 1 $ RR + " kk 2 kk RR R 3 p $ R + kk1 kk RR+ SS 3 2 " RR SS 2 kk 2 " $ R + SS 2 RR R 33

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35 Phosphorylation cascade: 2-stage, Michaelis-Menten d dt = k 3 " S 1 " R K m1 + R k 3 '" K m 2 + S 1 dr dt = kk 3 " " RR KK m1 + RR kk 3 '"R KK m2 + R R RR R 35

36 Phosphorylation cascade: 3-stage, Mass-Action MA1 R+S 1 R+S 2 k 1 $ " k R S 3 1 $ + S 1 k2 k 1 ' k 3 ' " S 2 k2 ' " $ + S 2 S 1 RR + kk 1 kk RR R 3 p $ R + $ " kk2 R RR+ SS 2 kk1 ' kk 3 ' " R SS 2 kk2 ' kkk1 RRR + R kkk 2 $ " " $ R + SS 2 kkk RRR RR 3 p $ RR + R RR R RRR+ SSS 2 kkk1 ' kkk 3 ' " RR SSS 2 kkk 2 ' " $ RR + SSS 2 RRR RR 36

37 Phosphorylation cascade: 3-stage, Michaelis-Menten d dt = k 3 " S 1 " R K m1 + R k 3'" K m 2 + R S 1 dr dt = kk 3 " " RR KK m1 + RR kk 3 '"R KK m2 + R RR R drr dt = kkk 3 " R " RRR KKK m1 + RRR kkk 3 '"RR KKK m2 + RR RRR RR 37

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41 Example: Levchenko Levchenko et al. Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold properties. Proc Natl Acad Sci USA, 97(11): ,

42 Schoeberl et al 2002 model Schoeberl et al. (2002), Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors, Nature Biotechnology 20,

43 S 1 Phosphorylation cascade + feedback $ R +S 1" R S 1 S 1 $ R +S 1" R S 1 $ R + R S 1" R R S 1 $ R S 1 R R P 1 P 1 RR R RR R P 2 P 2 S 1 S 1 R R P 1 P 1 RR R RR R P 2 P 2 $ R +P 1" R P 1 $ + R P 1" R P 1 $ R P 1 $ R +P 1" R P 1 43

44 Phosphorylation cascade + negative feedback: 2-stage, Mass Action MA1 R +S 1 ki $ " ki ' R S 1 R+S 1 R+S 2 k1 $ " k2 k R S 3 1 $ + S 1 k 1 ' " $ k 3 ' " S 2 k 2 ' + S 2 R S 1 RR + RR+ SS 2 kk 1 $ kk RR R 3 p $ R + " kk2 kk 1 ' " $ kk 3 ' " R SS 2 kk 2 ' R + SS 2 RR R 44

45 Phosphorylation cascade + negative feedback: 2-stage, Michaelis-Menten d dt = k 3 " S 1 " R K m1 " 1+ RR & p % ( + R $ ' K i ) k 3 " K m2 + R S 1 dr dt = kk 3 " " RR KK m1 + RR ) kk 3 "R KK m2 + R RR R Using Competitive Inhibition Ki is the dissociation constant for the SI complex [S] V = V max " [S]+ K m " 1+ [I ] & % ( $ [K i ]' 45

46 Phosphorylation cascade + negative feedback: 3- stage, Mass Action, MA1 $ " ki RR +S 1 ki R+S 1 k 1 $ " k2 RR S 1 k R S 3 1 $ + S 1 S 1 k1 " $ k R+S 3 2 " S 2 k2 RR + kk1 $ " kk2 + S 2 kk RR R 3 p $ R + R kk 1 " $ kk RR+ SS 3 2 " R SS 2 kk2 RRR + R kkk1 $ " kkk2 R + SS 2 kkk RRR RR 3 p $ RR + R RR R kkk 1 " $ kkk RRR+ SSS 3 2 " RR SSS 2 kkk2 RR + SSS 2 RRR RR 46

47 Phosphorylation cascade + negative feedback: 3-stage, Michaelis-Menten d dt = k 3 " S 1 " R K m1 " 1+ RRR & p % ( + R $ ' K i ) k 3 " K m2 + R S 1 dr dt = kk 3 " " RR KK m1 + RR ) kk 3 "R KK m2 + R RR R drr dt = kkk 3 " R " RRR KKK m1 + RRR ) kkk 3 "RR KKK m2 + RR RRR RR Using Competitive Inhibition Ki is the dissociation constant for the SI complex [S] V = V max " [S]+ K m " 1+ [I ] & % ( $ [K i ]' 47

48 Oscillations! Phosphorylation cascade + negative feedback: 3-stage, Mass Action Conditions S1=3 Inhibitor=0.5 48

49 Kholodenko Model 49

50 Kholodenko Simulation Time = 500 Seconds Time = 5000 Seconds Simulation from Richard Orton 50

51 Combination of positive & negative feedback Mathematical Model Input [Ras] v1 v2 [Ras*] (m1) [RKIP] v9 v10 [Raf] [RKIPp] (m5) v3 v4 [MEK] [Raf*] (m2) v5 [MEK-pp](m3) v6 Negative feedback Positive feedback [ERK] v7 v8 [ERK-pp](m4) Modeling and Analysis of Two Feedback Loop Dynamics in Ras/Raf-1/MEK/ERK Signaling Pathway Kwang-Hyun Cho, Sung-Young Shin, Walter Kolch, Olaf Wolkenhauer. ICSB

52 Combination of positive & negative feedback: Simulation No Feedback Positive Feedback Negative Feedback Positive & Negative Feedback 52

53 Combination of positive & negative feedback: Simulation vs. Experimental Data h 2 h 3 h 4 h 6 h TPA ERK-pp (activated ERK) Western blots COS1 cell lysates total ERK ] s s e l t i n u [. l a m r o N Comparison of experimentad data and simulation result m5(erk-pp) raw-erk Experiment Simulation Time [hour] 53

54 Adding a drug: 3-stage, Inhibitor on 2nd stage, Mass Action RR +S 1 ki $ " ki ' RR S 1 R+S 1 k 1 $ " k R S 3 1 $ + S 1 k2 k1 ' " $ k R+S 3 ' 2 " S 2 k 2 ' + S 2 S 1 kk 1 $ RR + " kk 2 kk RR R 3 p $ R + kk 1 ' " $ kk RR+ SS 3 ' 2 " R SS 2 kk2 ' R + SS 2 R U + RR U + R ku 1 $ " ku2 ku 1 $ " ku2 U RR U R kk 1 $ U RR + R kk p U RR R " p 3 $ U R + kk 2 kk U RR+ SS 3 ' 2 " U R SS 2 kk 2 ' kkk 1 RRR + R kkk 2 $ " kk 1 ' " $ U R + SS 2 kkk RRR RR 3 p $ RR + R kkk 1 ' kkk RRR+ SSS 3 ' 2 " RR SSS 2 kkk 2 ' " $ RR + SSS 2 RR U RR RRR R U R RR 54

55 Adding a drug: 3-stage, Inhibitor on 2nd stage, Mass Action RR +S 1 ki $ " ki ' RR S 1 R+S 1 k 1 $ " k R S 3 1 $ + S 1 k2 k1 ' " $ k R+S 3 ' 2 " S 2 k 2 ' + S 2 S 1 kk 1 $ RR + " kk 2 kk RR R 3 p $ R + kk 1 ' " $ kk RR+ SS 3 ' 2 " R SS 2 kk2 ' R + SS 2 R U + RR U + R ku 1 $ " ku2 ku 1 $ " ku2 U RR U R kk 1 $ U RR + R kk p U RR R " p 3 $ U R + kk 2 kk U RR+ SS 3 ' 2 " U R SS 2 kk 2 ' kkk 1 RRR + R kkk 2 $ " kk 1 ' " $ U R + SS 2 kkk RRR RR 3 p $ RR + R kkk 1 ' kkk RRR+ SSS 3 ' 2 " RR SSS 2 kkk 2 ' " $ RR + SSS 2 RR U RR S 2 SS 2 R U R RRR SSS 2 RR 55

56 Real cascade & feedback Ras Raf U0126 MEK ERK 56

57 Is the ERK pathway a negative feedback amplifier? Sauro HM, Kholodenko BN. Quantitative analysis of signaling networks. Prog Biophys Mol Biol Sep;86(1):

58 Negative Feedback Amplifier A negative feedback amplifier stems from the field of electronics and consists of an amplifier with a negative feedback loop from the output of the amplifier to its input. The negative feedback loop results in a system that is much more robust to disturbances in the amplifier. The negative feedback amplifier was invented in 1927 by Harold Black of Western Electric and was originally used for reducing distortion in long distance telephone lines. The negative feedback amplifier is now a key electrical component used in a wide variety of applications Input Amplifier Output Feedback 58

59 Negative Feedback Amplifier Input After Feedback Amplifier Input u - e A Ae y Output F Negative Feedback Loop Steady State Equation y Au = 1+ AF y = Ae e = u Fy y = A (u Fy) y = Au AFy y + AFy = Au y (1 + AF) = Au 59

60 The negative feedback imparts signalling robustness Standard Amplifier Negative Feedback Amplifier Output (y) Output (y) u Amplifier (A) gain + + A y Amplifier (A) gain u + + A - y y=a*u F y=a*u/(1+a*f) A large change in amplifier gain leads to a small change in output (y) 60

61 Feedback Output Increasing -> Increasing Feedback Feedback Increasing -> <- Amplifier Decreasing 61

62 The negative feedback imparts signalling robustness Negative Feedback Amplifier u Standard Amplifier + + A y u + + A - y y=a*u F y=a*u/(1+a*f) Sudden drop in Amplifier (A) gain Sudden drop in Amplifier (A) gain Output (y) Δy Output Output (y) Δy Output Time Time 62

63 Application to Biology The ERK cascade is a well known biological amplifier and contains numerous negative feedback loops. At first sight, it has the correct structure to be a negative feedback amplifier. Sauro & Kholodenko (2004) If the ERK cascade is a negative feedback amplifier it should be robust to disturbances within the cascade. From a biological point of view, these disturbances could be caused by drugs, such as U0126, aimed at decreasing the activity of the ERK cascade. This suggests that these drugs will be relatively ineffective. In fact, current drugs aimed at decreasing the activity of the MAPK pathway have proved less efficient in in vivo applications than anticipated from in vitro inhibition assays. 63

64 Raf/MEK/ERK amplifies the signal Rec. GST-BXB NIH COS Rec. MEK1-His NIH COS Rec. GST-ERK NIH COS WB: Raf-1 WB: MEK WB: ERK Cell line Raf-1 MEK ERK Concentration per cell COS femtomol ratio NIH 3T femtomol ratio 64

65 How to test if the ERK pathway is a Generate input: Stimulate with GF NFA? Ras-GTP Raf-1 Disturb the Amplifier : Use a MEK inhibitior, such as U0126 U0126 MEK1/2 Negative Feedback Remove negative feedback ERK1/2 Measure signal output: i.e. ERK phosphorylation 65

66 Hypothesis: Braking the feedback should sensitise the ERK pathway to MEK-inhibitor Feedback intact Feedback removed Ras-GTP Ras-GTP U0126 Raf-1 MEK1/2 Negative Feedback phospho-erk MEK inhibitor U0126 Raf-1 MEK1/2 ERK1/2 ERK1/2 66

67 How to test if the ERK pathway is a NFA? Strategy In vivo system that allows us to compare feedback broken to feedback intact model. Computational Model of ERK pathway with/without feedback 67

68 Computational Modeling 1: Build the model Non-linear ordinary differential equations (ODE s). ODE s were solved using Math Lab and Gepasi. Models are based on the Schoeberl et al. (2002) model Mass Action Kinetics instead of Michaelis Menten Kinetic parameters are from literature, previous models and guesstimates Schoeberl et al. (2002), Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors, Nature Biotechnology 20,

69 Amplifier / negative feedback Model amplifier strength by Adding inhibitor to 2nd stage Modifying kk3, kkk3 [I.e. modifying rate of production of RRp, RRRp] Add/remove cascade elements Then plot amp strength versus output, e.g. [U] vs [RRRp]?Model feedback strength by Leaving out feedback loop varying ki, and plot ki vs [RRRp] Notes: avoid saturation; use signal in linear range;?model degradation in S1 signal? 69

70 Adding a drug: 3-stage, Inhibitor on 2nd stage, Mass Action RR +S 1 ki $ " ki ' RR S 1 R+S 1 k 1 $ " k R S 3 1 $ + S 1 k2 k1 ' " $ k R+S 3 ' 2 " S 2 k 2 ' + S 2 S 1 kk 1 $ RR + " kk 2 kk RR R 3 p $ R + kk 1 ' " $ kk RR+ SS 3 ' 2 " R SS 2 kk2 ' R + SS 2 R U + RR U + R ku 1 $ " ku2 ku 1 $ " ku2 U RR U R kk 1 $ U RR + R kk p U RR R " p 3 $ U R + kk 2 kk U RR+ SS 3 ' 2 " U R SS 2 kk 2 ' kkk 1 RRR + R kkk 2 $ " kk 1 ' " $ U R + SS 2 kkk RRR RR 3 p $ RR + R kkk 1 ' kkk RRR+ SSS 3 ' 2 " RR SSS 2 kkk 2 ' " $ RR + SSS 2 RR U RR RRR R U R RR 70

71 Phosphorylation cascade + negative feedback: 3- stage, Inhibitor on 2nd stage, Mass Action RR + ki $ " ki ' RR k 1 $ k R+S 1 R S " 1 3 $ + S 1 k 2 k 1 ' " $ k R+S 3 ' 2 " S 2 k 2 ' + S 2 S 1 kk 1 $ RR + " kk 2 kk RR R 3 p $ R + kk 1 ' " $ kk RR+ SS 3 ' 2 " R SS 2 kk 2 ' R + SS 2 R U + RR U + R ku 1 $ " ku 2 ku 1 $ " ku2 U RR U R kk 1 $ U RR + R kk p U RR R " p 3 $ U R + kk 2 kk U RR+ SS 3 ' 2 " U R SS 2 kk 2 ' kkk 1 kk 1 ' " $ U R + SS 2 RRR + RR $ kkk p RRR RR " p 3 $ RR + R kkk 2 kkk 1 ' kkk RRR+ SSS 3 ' 2 " RR SSS 2 kkk 2 ' " $ RR + SSS 2 RR U RR RRR R U R RR 71

72 Computational Modeling 2: Feedback broken Results Feedback intact Prediction: Braking the feedback modulates drug response 72

73 Computational Modeling 2: Results Sensitivity of kinetic parameters is decreased due to Negative Feedback 73

74 The experimental systems Negative feedback loops intact EGFR Sos 4557W EGFR inhibitor One feedback loop eliminated by constitutively active RasV12 mutant Both feedback loops eliminated by BXB-ER (4-OHT regulatable Raf-1 mutant) Ras RasV12 4-OHT Raf Raf BXB-ER MEK ERK MEK MEK U0126 U0126 U0126 MEK inhibitor ERK ERK 74

75 Breaking the ERK feedback with BXBER Raf-1 Regulatory Domain CR1 CR2 Kinase Domain CR3 S 29 P S 43 P S 289 P S 296 P S 301 P S 642 P ERK feedback phosphorylation sites BXB-ER CR3 ER hormone binding HA S 642 P pperk levels min BXB-ER stimulated with 4-OHT (4-Hydroxy Tamoxifen, a synthetic estrogen) Raf-1 stimulated with EGF 75

76 Ablation of feedback by BXBER decreases robustness to MEK-inhibitor U0126 Computer Simulation 76

77 Ablation of feedback by BXBER decreases robustness to MEK-inhibitor U0126 Experiment 77

78 Signal recovery after MEK inhibition Simulation Experiment U0126 added perk1/2, +EGF perk1/2, + BXBER/4HT min stimulation 78

79 Implications for drug targeting The aim of a drug is to cause a disruption to the network in such a way that it restores the network to its healthy wild-type state. Targets must be susceptible to disruption for the drug to have any effect. The analysis of feedback suggests that targets inside the feedback loop will prove difficult drug targets because any attempt to disturb these targets will be resisted by the feedback loop. 79

80 Summary Modelling Enzymatic Reactions Signal Transduction Cascades Modelling: Building blocks & composition/decomposition Positive & negative feedback, & drug inhibitors Analysing the behaviour of models 80