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Åbo Akademi University TUCS, Turku, Finland
Ion PETREAndrzej MIZERA

• COPASI
• Complex Pathway Simulator

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Outline

• General information about COPASI
• The Lotka-Volterra model of predator-prey
  interactions
• Implementation of predator-prey model in COPASI
• Time-course simulation, steady state analysis,
  parameter estimation in COPASI
• Comments on numerical difficulties
• Summary

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General information on COPASI

• COPASI - Complex Pathway Simulator
• Software application for simulation and analysis
  of biochemical networks
• Free for non-commercial use
• Features
• stochastic and deterministic time course
  simulation
• steady state analysis
• metabolic control analysis/sensitivity analysis
• parameter estimation using data from time course
  and/or steady state experiments
• imports and exports SBML
• http//www.copasi.org

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Lotka-Volterra system

• The Lotka-Volterra system of chemical reactions
  describes an ecological predator-prey
  (fox-rabbit) model.
• chemical reactions
• growth of prey population
• prey ? 2 prey
• consumption of preys
• predator prey ? predator
• death of predators
• predator ? ?
• increase of predator population
• predator prey ? 2 predator prey

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Lotka-Volterra system

• The differential equations for the Lotka-Volterra
  system are obtained by applying the mass-action
  law.
• The Lotka-Volterra system assumes that
• the prey population x grows at a rate
  proportional to the current population (A x dt),
• but when predators y are present, the prey
  population decreases at a rate proportional to
  the number of predator/prey encounters
• (B x y dt)
• the predator population declines at a rate
  proportional to the current population (C y dt),
• but increases at a rate proportional to the
  predator/prey meetings (D x y dt),
• where A, B, C, and D are positive constants.

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Lotka-Volterra system

• The differential equations for the Lotka-Volterra
  system

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Lotka-Volterra system

• The system consists of two ordinary, non-linear,
  first order differential equations.
• The equations have periodic solutions which do
  not have a simple expression in terms of the
  usual trigonometric functions.

No matter what the population of prey and
predator are, neither species will die out, nor
will its population grow indefinitely.
Differential Equations, Dynamical Systems, and
an Introduction to Chaos, M. Hirsch, R. L.
Devaney, S. Smale (2004)

• In order to obtain time course graphs we need to
  use numerical integration.

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Lotka-Volterra system

• An equilibrium solution is a set of
  concentrations/particle numbers which will not
  change over time, and hence can be found by
  solving the following set of equations
• Solving it for x(t) and y(t) gives 2 solutions
• and

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Lotka-Volterra system

• An equilibrium solution is a set of
  concentrations/particle numbers which will not
  change over time, and hence can be found by
  solving the following set of equations
• Solving it for x(t) and y(t) gives 2 solutions
• and

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How to implement a model in COPASI
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COPASI model implementation

• model
• time units hours
• volume unit m3
• quantity unit
• compartments forest
• metabolites predator, prey
• chemical reactions
• growth of prey population (A)
• prey ? 2 prey
• consumption of preys (B)
• predator prey ? predator
• death of predators (C)
• predator ? ?
• increase of predator population (due to prey
  consumption) (D)
• predator prey ? 2 predator prey

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COPASI model implementation

• global quantities rate constants of the
  reactions
• A initial value 1
• B initial value 0.01
• C initial value 1
• D initial value 0.02
• initial concentrations of the metabolites
• predator 20
• prey 20

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Time course simulation

• Tasks ? Time Course
• duration 20 h
• interval size 0.01
• define the plots by the Output Assistant

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Steady state analysis

• Tasks ? Steady-State
• Remarks
• Obtaining a steady state might require relaxation
  of the Resolution parameter!
• COPASI does not find all possible steady states.
  In our example, after the (0,0) steady state is
  obtained, although (50,100) is also a solution to
  the steady state problem.

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Parameter estimation

• Multiple Task ? Parameter Estimation
• define the experimental data file in
  Experimental Data
• set the Experiment Type to Time Course
• associate the columns in the data file with
  appropriate model objects
• estimate
• predator initial particle number (init. value
  2)
• prey initial particle number (init. value 40)
• 2 reaction rate constants B (init. value 1) and
  D (init. value1)
• remarks
• For numerical reasons it is important to set the
  lower bound to 1e-09 instead of 0.
• In order to perform consecutive parameter
  estimations without loosing the so far estimated
  values it is necessary to check the update
  model box.
• In our example the data file was obtained by a
  numerical integration of the Lotka-Volterra
  system in Matlab.

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Parameter estimation
objective function value 0.194162
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Numerical difficulties
ode23
ode45
ode113
predator 20, prey 20, A 8.68876, B
0.0872658, C 8.3535, D 0.168216
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Numerical difficulties

• COPASI does not solve analytically the
  differential equations, but integrates them with
  a certain numerical method.
• Each numerical method has its own precision which
  in general strongly depends on the equations
  being solved. Usually we do not know the
  numerical properties of the system, hence do not
  know if a particular method is suitable for it.
• Parameter Estimation task heavily utilizes the
  numerical integration to evaluate the fit. Thus,
  the obtained parameters depend not only on the
  experimental data but also on the chosen
  numerical method!

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Summary

• COPASI enables the user to implement a molecular
  model in a straightforward way by defining the
  chemical reactions in a simple format.
• COPASI allows to perform both deterministic and
  stochastic simulations in an easy and fast way.
• COPASI does not require any sophisticated
  knowledge of mathematics. It automatically
  generates mathematical model (differential
  equations) from the molecular model (chemical
  reactions).
• COPASI implements a rich variety of parameter
  estimation methods. They are based on different
  heuristics and can be applied consecutively in
  any order the next method starts with
  parameters estimated by the previous one.

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Summary

• COPASI provides a set of standard tools utilized
  in systems biology, e.g. Metabolic Control
  Analysis, Sensitivities and other.

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Thank you for your attention )