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Åbo Akademi University & TUCS, Turku, Finland Ion PETRE Andrzej MIZERA COPASI Complex Pathway Simulator

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2 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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3 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

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4 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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5 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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6 Lotka-Volterra system The differential equations for the Lotka-Volterra system:

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7 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. Lotka-Volterra system In order to obtain time course graphs we need to use numerical integration. “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)

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8 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 Lotka-Volterra system

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9 stationary point 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 Lotka-Volterra system

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10 How to implement a model in COPASI

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11 COPASI model implementation model time units: hours volume unit: m 3 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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12 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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13 Time course simulation Tasks Time Course duration: 20 h interval size: 0.01 define the plots by the ‘Output Assistant’

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14 one point! Steady state analysis Tasks Steady-State Remarks: 1. Obtaining a steady state might require relaxation of the ‘Resolution’ parameter! 2. 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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15 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. value:1) remarks 1.For numerical reasons it is important to set the lower bound to 1e-09 instead of 0. 2.In order to perform consecutive parameter estimations without loosing the so far estimated values it is necessary to check the ‘update model’ box. 3.In our example the data file was obtained by a numerical integration of the Lotka-Volterra system in Matlab.

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16 objective function value: Parameter estimation

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17 predator: 20, prey: 20, A: , B: , C: , D: ode23 ode45 ode113 Numerical difficulties

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18 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! Numerical difficulties

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19 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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20 Summary COPASI provides a set of standard tools utilized in systems biology, e.g. Metabolic Control Analysis, Sensitivities and other.

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21 Thank you for your attention :)

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