# Reaction Kinetics- Metabolic Networks

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Reaction Kinetics, Metabolic Networks, Petri nets
CS 6280 Lecture 3 P.S. Thiagarajan

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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.
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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
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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

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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]

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Michaelis-Menton Kinetics

This is the Michaelis-Menten equation!

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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.

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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.
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Parameter Estimation
• Change of variables used to linearize the law. • Don’t have to do non-linear regression. • Instead, can do linear regression.

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13

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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.

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Variations
• Enzymes don’t merely accelerate reactions. • They also play regulatory roles.

• Enzyme’s effectiveness targeted by inhibitors and activators (effectors).

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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

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General Inhibitory Scheme

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Competitive Inhibition

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Competitive Inhibition
S and I compete for the binding place

High S may out-compete I

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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.

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Other forms Inhibitions
• Non-competitive inhibition • Mixed inhibition • Partial inhibition

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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.
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Hill Coefficients
• A Hill coefficient of 1 indicates completely independent binding.

• A coefficient > 1 indicates cooperative binding.
– Oxygen binding to hemoglobin:
 Hill coefficient of 2.8 – 3.0
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Hill Equation
General form of Michaelis-Menten

General form of the Hill equation

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Sigmoidal Plots

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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

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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.
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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.

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The Glycolysis Metabolic Pathway

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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.

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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.

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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.

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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.
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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
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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!)
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Metabolic Networks
• dSi/dt =  nij . Vj • Stoichiometric matrix
–N – N(i, j) = nij

• dS/dt = N V

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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

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Initial values chosen “randomly”

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Stoichiometric Analysis
• Use the structure of the network and the stoichiometric coefficients.
 And other information

– Sensitivity to different changes in the steady state.
 Metabolic control analysis.

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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

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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

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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

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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

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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.
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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!
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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.
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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.
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• 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

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• dS/dt = N V • dS/dt = N V = 0

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In steady state, the reaction v8 will go to 0 !

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Elementary Fluxes
• Elementary flux: a minimal set of nonzero-rate reactions
– producing a steady state. – Respect the irreversibility (if any) of the reactions

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2
1 1 1
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1

1 0

=

1 1 1

+

0 1
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v2 is irreversible

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The Kernel space of NT
• The Kernel space of NT yields conservation (invariant) principles.
– involving the metabolite concentrations.

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The 3rd and 4th rows are not linearly independent

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Approximations
• Quasi steady state approximations • Quasi equilibrium approximations • Replace differential equations by algebraic equations. • Other constraints • Metabolic pathways can be very large and complex!
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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)…
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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).
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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 –…
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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…
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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:
– – – – –
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Petri nets (many types) Statecharts. Live Sequence Charts Process algebras Hybrid automata (many types)
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