A structured approach for the engineering of biochemical network - - PowerPoint PPT Presentation

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A structured approach for the engineering of biochemical network - - PowerPoint PPT Presentation

Sep 08, 2023 22 likes •423 views

A structured approach for the engineering of biochemical network models, illustrated for signalling pathways Rainer Breitling 3 , David Gilbert 1 , Monika Heiner 2 , Robin Donaldson 1 (1) Bioinformatics Research Centre University of Glasgow,

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A structured approach for the engineering of biochemical network models, illustrated for signalling pathways

Rainer Breitling3, David Gilbert1, Monika Heiner2, Robin Donaldson1

(1) Bioinformatics Research Centre University of Glasgow, Glasgow, UK (2) Computer Science Department, Brandenburg University of Technology, Cottbus, Germany (3) Groningen Bioinformatics Centre, University of Groningen, Groningen, Netherlands. www.brc.dcs.gla.ac.uk/~drg/workshops/ismb08

Biology

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Tutorial outline

  • I. Biological introduction

Rainer Breitling

  • II. Petri net introduction

Monika Heiner

  • III. Biological applications

David Gilbert

  • IV. Model checking

Robin Donaldson (each 50 min + 10 min break/discussion)

R.Breitling@rug.nl

2 Biology

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A structured approach … Part I Biology

Rainer Breitling

Groningen Bioinformatics Centre, University of Groningen, Groningen, Netherlands.

Biology

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Outline

  • Part 1: Why modelling?
  • Part 2: The statistical physics of modelling:

A  B (where do differential equations come from?)

  • Part 3: Translating biology to mathematics

(finding the right differential equations)

Biology

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Biology = Concentrations

Biology

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Humans think small-scale...

(the “7 items” rule)

...but biological systems contain (at least) dozens of relevant interacting components!

  • phone number length

(memory constraint)

  • ptimal team size

(manipulation constraint)

  • maximum complexity for

rational decision making

Biology

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Humans think linear...

...but biological systems contain:

  • non-linear interaction between components
  • positive and negative feedback loops
  • complex cross-talk phenomena

Biology

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Biochemical Pathway Simulation

Wet lab experiments

Prediction

Computational Simulation

Validation

What is the best formalism?

How to deal with lack of information?

Predictions on what?

How to collect quantitative measurements in vivo?

How to manipulate regulatory mechanisms?

Biology

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The simplest chemical reaction

A  B

  • irreversible, one-molecule reaction
  • examples: all sorts of decay processes, e.g. radioactive, fluorescence,

activated receptor returning to inactive state

  • any metabolic pathway can be described by a combination of

processes of this type (including reversible reactions and, in some respects, multi-molecule reactions)

Biology

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The simplest chemical reaction

A  B

various levels of description:

  • homogeneous system, large numbers of molecules =
  • rdinary differential equations, kinetics
  • small numbers of molecules = probabilistic equations,

stochastics

  • spatial heterogeneity = partial differential equations,

diffusion

  • small number of heterogeneously distributed molecules =

single-molecule tracking (e.g. cytoskeleton modelling)

Biology

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Kinetics Description

  • Imagine a box containing N molecules.

How many will decay during time t? k*N

  • Imagine two boxes containing N/2 molecules each.

How many decay? k*N

  • Imagine two boxes containing N molecules each.

How many decay? 2k*N

  • In general:

Main idea: Molecules don’t talk differential equation (ordinary, linear, first-order) exact solution (in more complex cases replaced by a numerical approximation)

Biology

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Kinetics Description

If you know the concentration at one time, you can calculate it for any other time! (and this really works)

Biology

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Probabilistic Description

Probability of decay of a single molecule in some small time interval: Probability of survival in Δt: Probability of survival for some time t: Transition to large number of molecules:

  • r

Main idea: Molecules are isolated entities without memory

Biology

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Probabilistic Description – 2

Probability of survival of a single molecule for some time t: Probability that exactly x molecules survive for some time t: Most likely number to survive to time t:

Biology

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Probabilistic Description – 3

Markov Model (pure death!)

Decay rate: Probability of decay: Probability distribution of n surviving molecules at time t: Description: Time: t -> wait dt -> t+dt Molecules:

n -> no decay -> n n+1 -> one decay -> n

Final Result (after some calculating): The same as in the previous probabilistic description

Biology

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Spatial heterogeneity

  • concentrations are different in different places, n

= f(t,x,y,z)

  • diffusion superimposed on chemical reactions:
  • partial differential equation

Biology

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Spatial heterogeneity

  • one-dimensional case

(diffusion only, and conservation of mass)

∆x

inflow

  • utflow
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Spatial heterogeneity – 2

Biology

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Summary of Physical Chemistry

  • Simple reactions are easy to model accurately
  • Kinetic, probabilistic, Markovian approaches lead to the same

basic description

  • Diffusion leads only to slightly more complexity
  • Next step: Everything is decay...

Biology

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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 (k1, k2...) Therefore: AB

Biology

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Reversible, Single-Molecule Reaction

A  B, or A  B || B  A, or Differential equations:

forward reverse

Main principle: Partial reactions are independent!

Biology

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Reversible, single-molecule reaction – 2

Differential Equation: Equilibrium (=steady- state):

Biology

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Irreversible, two-molecule reaction

A+BC Differential equations:

Underlying idea: Reaction probability = Combined probability that both [A] and [B] are in a “reactive mood”: The last piece of the puzzle Non-linear!

Biology

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A simple metabolic pathway

ABC+D Differential equations:

d/dt decay forward reverse [A]=

  • k1[A]

[B]=

+k1[A]

  • k2[B]

+k3[C][D]

[C]=

+k2[B]

  • k3[C][D]

[D]=

+k2[B]

  • k3[C][D]

Biology

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Metabolic Networks as Bigraphs

ABC+D

d/dt decay forward reverse [A]

  • k1[A]

[B]

+k1[A]

  • k2[B]

+k3[C][D]

[C]

+k2[B]

  • k3[C][D]

[D]

+k2[B]

  • k3[C][D]

k1 k2 k3 A

  • 1

B

1

  • 1

1

C

1

  • 1

D

1

  • 1

Biology

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Biological description  bigraph  differential equations

KEGG

Biology

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Biological description  bigraph  ODEs

EC 1.1.1.2 substance A substance B

A B

k1

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Biological description  bigraph  ODEs

EC 1.1.1.2 substance A substance B

A B

k EA EB k*

k1 k2

E

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A special case: enzyme reactions

In a quasi steady state, we can assume that [ES] is constant. Then: If we now define a new constant Km (Michaelis constant), we get:

Biology

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A special case: enzyme reactions

Substituting [E] (free enzyme) by the total enzyme concentration we get: Hence, the reaction rate is:

Biology

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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

Biology

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Metabolic pathways vs Signalling Pathways

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

Biology

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Cell signaling pathways

Biology

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Cell signaling pathways

Biology

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Cell signaling pathways

Biology

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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

Biology

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Cell signaling pathways

Biology

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Cell signaling pathways

  • Fig. courtesy of W. Kolch

Biology

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Cell signaling pathways

  • Fig. courtesy of W. Kolch

Biology

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Cell signaling pathways

  • Fig. courtesy of W. Kolch

Biology