A structured approach … Part III Biological applications David Gilbert Bioinformatics Research Centre University of Glasgow, Glasgow, UK drg@brc.dcs.gla.ac.uk Biological Applications 1
MAPK Pathway • Responds to wide range of stimuli: cytokines, growth factors, neurotransmitters, cellular stress STIMULUS and cell adherence,… • 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. drg@brc.dcs.gla.ac.uk Biological Applications 2
MA1: Mass action for enzymatic reaction - phosphorylation k 1 → E | A k 3 E + A E + B → ← k 2 • A: substrate E • B: product (phosphorylated A) • E: enzyme (kinase) • E|A substrate-enzyme complex A B drg@brc.dcs.gla.ac.uk Biological Applications 3
Differential equations Enzymatic reaction k 1 → A | E k 3 A + E B + E → ← k 2 d [ A ] dt = − k 1 × [ A ] × [ E ] + k 2 × [ A | E ] d [ A | E ] = + k 1 × [ A ] × [ E ] − k 2 × [ A | E ] − k 3 × [ A | E ] dt d [ B ] dt = + k 3 × [ A | E ] d [ E ] dt = − k 1 × [ A ] × [ E ] + k 2 × [ A | E ] + k 3 × [ A | E ] drg@brc.dcs.gla.ac.uk Biological Applications 4
MA2 model drg@brc.dcs.gla.ac.uk Biological Applications 5
MA3 model drg@brc.dcs.gla.ac.uk Biological Applications 6
Multiple substrates drg@brc.dcs.gla.ac.uk Biological Applications 7
Metabolic pathways vs Signalling Pathways (Petri Nets) drg@brc.dcs.gla.ac.uk Biological Applications 8
Mass action for enzymatic reaction - phosphorylation S 1 R R p • R: substrate, • R p : product (phosphorylated R) • S 1 : enzyme (kinase) • R|S 1 substrate-enzyme complex drg@brc.dcs.gla.ac.uk Biological Applications 9
Phosphorylation - dephosphorylation step Mass action model 1 • R: unphosphorylated form S • R p : phosphorylated form • S: kinase • P: phosphotase R • R|S unphosphorylated+kinase complex R p • R|P unphosphorylated+phosphotase complex P drg@brc.dcs.gla.ac.uk Biological Applications 10
Phosphorylation - dephosphorylation loop Mass action model 2 • R: unphosphorylated form S 1 • R p : phosphorylated form • S 1 : kinase • S 2 : phosphotase • R|S 1 unphosphorylated+kinase complex • R p| S 1 phosphorylated+kinase complex R R p • R|S 2 unphosphorylated+phosphotase complex • R p |S 2 phosphorylated+phosphotase complex S 2 drg@brc.dcs.gla.ac.uk Biological Applications 11
Phosphorylation - dephosphorylation step Mass action (all singing/dancing) • R: unphosphorylated form S • R p : phosphorylated form • S: kinase • P: phosphotase R • R|S unphosphorylated+kinase complex R p • R|P unphosphorylated+phosphotase complex P drg@brc.dcs.gla.ac.uk Biological Applications 12
Michaelis-Menten equation for phosphorylation-dephosphorylation S R R p • Assumptions: P 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 • dR p /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) drg@brc.dcs.gla.ac.uk Biological Applications 13
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? drg@brc.dcs.gla.ac.uk Biological Applications 14
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Composition Vertical & horizontal S 1 S R R p R R p R pp P 1 P 2-stage cascade RR p RR 1-stage cascade double phosphorylation step P 2 drg@brc.dcs.gla.ac.uk Biological Applications 16
Composition Vertical & horizontal drg@brc.dcs.gla.ac.uk Biological Applications 17
Two stage, double phosphorylation drg@brc.dcs.gla.ac.uk Biological Applications 18
Phosphorylation cascade: 2-stage, Mass Action model 1 k 1 → R | S 1 k 3 S 1 R + S 1 R p + S 1 → ← k 2 k 1 ' k 3 ' ← R p + P R + P R | P ← 1 1 1 → k 2 ' R R p kk 1 → RR | R p kk 3 RR + R p RR p + R p → ← kk 2 RR p RR kk 1 ' kk 3 ' ← RR p + P RR + P RR | P ← 2 2 2 → kk 2 ' Engineering Biochemical Network models BioSysBio08 19
Phosphorylation cascade: 2-stage, Michaelis-Menten dR p K m 1 + R − k 3 ' × R p dt = k 3 × S 1 × R S 1 K m 2 + R p R R p dRR p = kk 3 × R p × RR kk 3 ' × RR p − dt KK m 1 + RR KK m 2 + RR p RR p RR Engineering Biochemical Network models BioSysBio08 20
3-stage Phosphorylation cascade (Mass Action) S 1 k 1 → R | S 1 k 3 R + S 1 R p + S 1 → ← k 2 k 1 ' k 3 ' ← R p + P R + P R p | P ← 1 1 1 → k 2 ' R R p kk 1 → RR | R p kk 3 RR + R p RR p + R p → ← P 1 kk 2 kk 1 ' kk 3 ' ← RR p + P RR + P RR p | P ← 2 2 2 → RR RR p kk 2 ' kkk 1 RRR | RR p → kkk 3 RRR + RR p RRR p + RR p → P 2 ← kkk 2 kkk 1 ' kkk 3 ' ← RRR p + P RRR + P RRR p | P ← 3 3 3 → RRR RRR p kkk 2 ' P 3 Engineering Biochemical Network models BioSysBio08 21
Phosphorylation cascade: 3-stage, Michaelis-Menten S 1 dR p K m 1 + R − k 3 ' × R p dt = k 3 × S 1 × R K m 2 + R p R R p dRR p = kk 3 × R p × RR kk 3 ' × RR p − dt KK m 1 + RR KK m 2 + RR p RR RR p kkk 3 × RR p × RRR dRRR p kkk 3 ' × RRR p = − dt KKK m 1 + RRR KKK m 2 + RRR p RRR RRR p Engineering Biochemical Network models BioSysBio08 22
3-stage drg@brc.dcs.gla.ac.uk Biological Applications 23
Phosphorylation cascade + feedback S 1 S 1 R p R R p R P 1 P 1 RR p RR RR p RR P 2 P 2 S 1 S 1 R p R R p R P 1 P 1 RR p RR RR p RR P 2 P 2 drg@brc.dcs.gla.ac.uk Biological Applications 24
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Phosphorylation cascade + negative feedback: 2-stage, Mass Action model 1 ki → RR p | S 1 RR p + S 1 ← ki ' S 1 k 1 → R | S 1 k 3 R + S 1 R p + S 1 → ← R R p k 2 k 1 ' k 3 ' ← R p + P R + P R | P ← 1 1 1 → k 2 ' RR p RR kk 1 → RR | R p kk 3 RR + R p RR p + R p → ← kk 2 kk 1 ' kk 3 ' ← RR p + P RR + P RR | P ← 2 2 2 → kk 2 ' Engineering Biochemical Network models BioSysBio08 27
Phosphorylation cascade + negative feedback: 2-stage, Michaelis-Menten dR p − k 3 ' × R p k 3 × S 1 × R S 1 dt = K m 1 × 1 + RR p K m 2 + R p + R K i R R p dRR p = kk 3 × R p × RR kk 3 ' × RR p − dt KK m 1 + RR KK m 2 + RR p RR p RR [ S ] V = V max × • Using Competitive Inhibition [ S ] + K m × 1 + [ I ] [ K i ] • Ki is the dissociation constant for the SI complex Engineering Biochemical Network models BioSysBio08 28
Phosphorylation cascade + negative feedback: 3-stage, Mass Action, model 1 ki → RRR p | S 1 RRR p + S 1 ← S 1 ki ' k 1 → R | S 1 k 3 R + S 1 R p + S 1 → ← k 2 R R p k 1 ' k 3 ' ← R p + P R + P ← R p | P 1 1 1 → k 2 ' kk 1 → RR | R p kk 3 RR + R p RR p + R p → ← kk 2 RR p RR kk 1 ' kk 3 ' ← RR p + P RR + P ← RR p | P 2 2 2 → kk 2 ' kkk 1 RRR | RR p → kkk 3 RRR + RR p RRR p + RR p → ← kkk 2 RRR RRR p kkk 1 ' kkk 3 ' ← RRR p + P RRR + P ← RRR p | P 3 3 3 → kkk 2 ' Engineering Biochemical Network models BioSysBio08 29
Phosphorylation cascade + negative feedback: 3-stage, Michaelis-Menten S 1 dR p − k 3 ' × R p k 3 × S 1 × R dt = K m 1 × 1 + RRR p K m 2 + R p R R p + R K i dRR p = kk 3 × R p × RR kk 3 ' × RR p RR RR p − dt KK m 1 + RR KK m 2 + RR p kkk 3 × RR p × RRR dRRR p kkk 3 ' × RRR p RRR RRR p = − dt KKK m 1 + RRR KKK m 2 + RRR p [ S ] V = V max × • Using Competitive Inhibition [ S ] + K m × 1 + [ I ] [ K i ] • Ki is the dissociation constant for the SI complex Engineering Biochemical Network models BioSysBio08 30
3-stage, negative feedback drg@brc.dcs.gla.ac.uk Biological Applications 31
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