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Reaction Kinetics, Metabolic Networks, Petri nets CS 6280 Lecture 3 P.S. Thiagarajan 31.08.07 CS6280 Lecture3 1 The Role of Chemical Reactions Bio-Chemical reactions A network of Bio-Chemical reactions Interacting networks of Bio-Chemical reactions Cell functions Rate Laws • Rate law: – An equation that relates the concentrations of the reactants to the rate. • Mass action law: – The reaction rate is proportional to the probability of collision of the reactants – Proportional to the concentration of the reactants to the power of their molecularities. 31.08.07 CS6280 Lecture3 3 Mass action law V1 S1 + S2 V2 2P V = (V1) - (V2) = k1. [S1] [S2] – k2 [P]2 [S1] ([S2]} is the concentration (Moles/litre) of S1 (S2) k1 and k2 are the rate constants V1, the rate of the forward reaction V2, the rate of the backward reaction V, the net rate Molecularity is 1 for each substrate (reactant) of the forward reaction and 2 for the backward reaction 31.08.07 CS6280 Lecture3 4 Michaelis-Menton Kinetics • Describes the rate of enzyme-mediated reactions in an amalgamated fashion: – Based on mass action law. – Much slower (seconds to minutes) k1 k2 E+S k -1 ES E+P 31.08.07 CS6280 Lecture3 5 Michaelis-Menton Kinetics k1 E+S k -1 ES k2 E+P Use mass action law to model each reaction. dS/dt = -k1 ([E].[S]) + k-1 ([ES]) dES/dt = k1 ([E].[S]) – (k-1 + k2) [ES] dE/dt = -k1[E][S] + (k-1 + k2) [ES] dP/dt = k2 [ES] 31.08.07 CS6280 Lecture3 6 Michaelis-Menton Kinetics This is the Michaelis-Menten equation! 31.08.07 CS6280 Lecture3 7 Michaelis-Menton Kinetics Consider the case v = Vmax / 2 The KM of an enzyme is therefore the substrate concentration at which the reaction occurs at half of the maximum rate. 31.08.07 CS6280 Lecture3 8 31.08.07 CS6280 Lecture3 9 Michaelis-Menton Kinetics • At KM, 50% of active sites have substrate bound. • At higher [S] a point is reached (at least theoretically) where all of the enzyme has substrate bound and is working flat out. • Adding more substrate will not increase the rate of the reaction, hence the levelling out observed in the graph. 31.08.07 CS6280 Lecture3 10 Parameter Estimation • Change of variables used to linearize the law. • Don’t have to do non-linear regression. • Instead, can do linear regression. 31.08.07 CS6280 Lecture3 11 31.08.07 CS6280 Lecture3 12 13 31.08.07 CS6280 Lecture3 14 Variations • Reversible form of Michaelis-Menten. k1 E+S k -1 ES k2 E+P k -2 More complicated equation but similar form. See the book. 31.08.07 CS6280 Lecture3 15 Variations • Enzymes don’t merely accelerate reactions. • They also play regulatory roles. – Their production and degradation adapted to current requirements. • Enzyme’s effectiveness targeted by inhibitors and activators (effectors). 31.08.07 CS6280 Lecture3 16 Variations • Regulatory interactions between an enzyme and an effector characterized by: – How the enzyme binds the effector EI, ESI or both – Which complexes can release the product ES alone or ESI or both ES and ESI 31.08.07 CS6280 Lecture3 17 General Inhibitory Scheme 31.08.07 CS6280 Lecture3 18 Competitive Inhibition 31.08.07 CS6280 Lecture3 19 Competitive Inhibition S and I compete for the binding place High S may out-compete I 31.08.07 CS6280 Lecture3 20 Uncompetitive Inhibition Inhibitor binds only to the ES complex. Does not compete but inhibits by binding elsewhere and inhibiting . S can’t out-compete I. 31.08.07 CS6280 Lecture3 21 Other forms Inhibitions • Non-competitive inhibition • Mixed inhibition • Partial inhibition 31.08.07 CS6280 Lecture3 22 23 Hill Coefficients • Suppose a dimeric (two identical sub-units linked together) protein has two identical binding sites. • The binding to the first ligand (at the first site) can facilitate binding to the second ligand. – Cooperative binding. • The degree of cooperation is indicated by the Hill coefficient. 31.08.07 CS6280 Lecture3 24 Hill Coefficients • A Hill coefficient of 1 indicates completely independent binding. – Independent of whether or not additional ligands are already bound. • A coefficient > 1 indicates cooperative binding. – Oxygen binding to hemoglobin: Hill coefficient of 2.8 – 3.0 31.08.07 CS6280 Lecture3 25 Hill Equation General form of Michaelis-Menten General form of the Hill equation 31.08.07 CS6280 Lecture3 26 Sigmoidal Plots 31.08.07 CS6280 Lecture3 27 The Role of Chemical Reactions Bio-Chemical reactions A network of Bio-Chemical reactions Interacting networks of Bio-Chemical reactions Cell functions The Role of Chemical Reactions Bio-Chemical reactions A network of Bio-Chemical reactions Bio-pathways Interacting networks of Bio-Chemical reactions Cell functions Biopathways 31.08.07 CS6280 Lecture3 30 Metabolic Pathways • Cells require energy and material: – To grow and reproduce – Many other processes • Metabolism: – Acquire energy and use it to grow and build new cells • Highly organized process • Involves thousands of reactions catalyzed by enzymes. 31.08.07 CS6280 Lecture3 31 Metabolic Pathways • Two types of reactions: – Catabolic: break down complex molecules to acquire energy and produce building blocks. breakdown of food in cellular respiration – Anabolic: construct complex compounds from simpler building blocks by expending energy. 31.08.07 CS6280 Lecture3 32 The Glycolysis Metabolic Pathway 31.08.07 CS6280 Lecture3 33 Glycolysis • The universal cellular metabolic process. – Takes place in the cytoplasm • (6-carbon) glucose is split into two (3carbon) pyruvate molecules + ATP + NADH • 9 reactions, each catalyzed by an enzyme. 31.08.07 CS6280 Lecture3 34 Glycolisis • In steps 1 and 3, ATP is converted to ADP supplying energy into the reaction. • In steps 6 and 9 ADPis converted to ATP. 31.08.07 CS6280 Lecture3 35 The Glycolysis Metabolic Pathway • The individual nodes are the molecule types. • Arrows depict chemical reaction. They are labeled with the enzymes that catalyze them. • For some of the reactions, ADP is consumed and ATP is produced. 31.08.07 CS6280 Lecture3 36 Metabolic Networks • Basic constituents: – The substances with their concentrations – The (chain of) reactions and transport processes. that change these concentrations – Reactions are usually catalyzed by enzymes – Transport carried out by transport proteins or pores. 31.08.07 CS6280 Lecture3 37 Metabolic Networks • Stoichiometric Coefficients: – Reflect the proportion of substrate and product molecules in a reaction V1 S1 + S2 V2 dS1/dt = -v = dS2/dt dP/dt = 2v 2P (-1, -1, 2) Can also be (-1/2, -1/2, 1) Can even be (1, 1, -2) if the reverse reaction is being considered 31.08.07 CS6280 Lecture3 38 Metabolic Networks • System equations • m substances and r reactions. • dSi/dt = nij . Vj – i = 1, 2, ….,m - metabolites – j = 1, 2, …,r - reactions – nij = The stoichiometric coefficient of substrate (metabolite) i in the reaction j. – Vj the rates (functions of time!) 31.08.07 CS6280 Lecture3 39 Metabolic Networks • dSi/dt = nij . Vj • Stoichiometric matrix –N – N(i, j) = nij • dS/dt = N V 31.08.07 CS6280 Lecture3 40 Metabolic Networks • dS/dt = N V • S(t) are the functions we would like to know. – Need to solve simultaneous systems of differential equations. – Rate constants are often unknown! – Initial values not always known 31.08.07 CS6280 Lecture3 41 Initial values chosen “randomly” 31.08.07 CS6280 Lecture3 42 Stoichiometric Analysis • Use the structure of the network and the stoichiometric coefficients. – Deduce steady state flows. And other information – Sensitivity to different changes in the steady state. Metabolic control analysis. 31.08.07 CS6280 Lecture3 43 An example By convention, V1 V2 V3 V1 (V2) is positive from left to right V3 is positive from top to bottom S1 2S2 V4 S3 31.08.07 CS6280 Lecture3 44 An example By convention, V2 V3 V1 (V2) is positive from left to right V3 is positive from top to bottom V1 S1 1 V2 -1 V3 0 V4 -1 V1 S1 2S2 V4 S3 S2 S3 31.08.07 CS6280 Lecture3 45 An example By convention, V2 V3 V1 (V2) is positive from left to right V3 is positive from top to bottom V1 S1 1 0 V2 -1 2 V3 0 -1 V4 -1 0 V1 S1 2S2 V4 S3 S2 S3 31.08.07 CS6280 Lecture3 46 An example By convention, V2 V3 V1 (V2) is positive from left to right V3 is positive from top to bottom V1 S1 1 0 V2 -1 2 V3 0 -1 V4 -1 0 V1 S1 2S2 V4 S3 S2 S3 0 0 0 1 31.08.07 CS6280 Lecture3 47 Stoichiometric Matrix • Contains structural information about the pathway. • Can compute what are the admissible fluxes possible in steady state. – Flux: The total amount of a reactant passing through (the pathway; through an enzyme;..) in unit time. – But we are ignoring a good deal of the dynamics. 31.08.07 CS6280 Lecture3 48 31.08.07 CS6280 Lecture3 49 So What can it do for us? • Apply linear algebra to compute: – The rank of N. – The basis for the kernel space – deduce steady state behavior of the rates. – Deduct invariants and conservation principles. • If you have forgotten basic linear algebra…. – Look it up! 31.08.07 CS6280 Lecture3 50 Kernel Space • C = m × n matrix – m rows n columns – Entries: rational (real) numbers polynomials …. • C is a linear transform – C: V n V m • Kernal = {v | C.v = 0} – is itself a vector space. 31.08.07 CS6280 Lecture3 51 Kernel Space • C = m × n matrix – m rows n columns • C is a linear transform – C: V n V m • Kernal = {v | C.v = 0} . • Dimension of Kernal = n – Rank (C) • Rank (C) – The maximal number of pair-wise linearly independent column (row) vectors of C. 31.08.07 CS6280 Lecture3 52 Steady State • dS/dt = N V • dS/dt = N V = 0 – The knowledge about the rates V at steady state is contained in the kernel space of N. – If we have the basis vectors for kernel space then we know: “all” the rates which can hold at steady state. fluxes 31.08.07 CS6280 Lecture3 53 Steady State • dS/dt = N V • dS/dt = N V = 0 31.08.07 CS6280 Lecture3 54 In steady state, the reaction v8 will go to 0 ! 31.08.07 CS6280 Lecture3 55 31.08.07 CS6280 Lecture3 56 31.08.07 CS6280 Lecture3 57 Elementary Fluxes • Elementary flux: a minimal set of nonzero-rate reactions – producing a steady state. – Respect the irreversibility (if any) of the reactions 31.08.07 CS6280 Lecture3 58 2 1 1 1 31.08.07 1 1 0 = 1 1 1 + 0 1 59 CS6280 Lecture3 31.08.07 CS6280 Lecture3 60 v2 is irreversible 31.08.07 CS6280 Lecture3 61 31.08.07 CS6280 Lecture3 62 The Kernel space of NT • The Kernel space of NT yields conservation (invariant) principles. – involving the metabolite concentrations. 31.08.07 CS6280 Lecture3 63 31.08.07 CS6280 Lecture3 64 The 3rd and 4th rows are not linearly independent 31.08.07 CS6280 Lecture3 65 31.08.07 CS6280 Lecture3 66 Approximations • Quasi steady state approximations • Quasi equilibrium approximations • Replace differential equations by algebraic equations. • Other constraints • Metabolic pathways can be very large and complex! 31.08.07 CS6280 Lecture3 67 Control analysis • Infer global behavior from local behaviors. • How do local perturbations affect global (steady) states? • Which effectors have the most effect on reaction rates (downstream)? • Applications in bio-technology, medicine (metabolic disorders)… 31.08.07 CS6280 Lecture3 68 Petri nets • A natural representation. – A well established model of distributed discrete event systems. • The connection matrix of a Petri net contains the same information as the stoichiometric matrix. • Kernel spaces of the connection matrix correspond to T-invariants (fluxes) and the dual kernel spaces correspond to S-invariants (conservation principles). 31.08.07 CS6280 Lecture3 69 The role of Petri nets • Petri nets have additional structural notions such as: siphons, traps,… • Petri nets also have dynamics associated with them. This can be used sometimes to discover information about – feasible initial states – Reachability properties –… 31.08.07 CS6280 Lecture3 70 System Biology Applications. • Many ! • Ordinary Petri nets Metabolic pathway analysis • Hybrid Functional Petri nets. signaling pathways Supported by the tool Cell Illustrator • Stochastic Petri nets, Colored Petri nets, Hybrid Petri nets… 31.08.07 CS6280 Lecture3 71 Models of Computation • Biological systems will consist of many – Components – Processes – Agents • Using a global description (say a finite state machine) will result in a very large incomprehensible object. • Need models that can explicitly model independent activities. • Many candidates: – – – – – 31.08.07 Petri nets (many types) Statecharts. Live Sequence Charts Process algebras Hybrid automata (many types) CS6280 Lecture3 72

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posted: | 4/18/2008 |

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