With the time course simulation, you can calculate the trajectory for the species in your model over a given time interval. There are different methods to calculate such trajectories and depending on your model, one or several of them may be appropriate to do a time course simulation of your model. COPASI supports three different methodologies to calculate a trajectory. The first method is to do a deterministic time course simulation of your model using the LSODA Petzold83 algorithm. For systems with small particle numbers, it is sometimes better to do a stochastic simulation rather than a deterministic one. COPASI supports a method for the stochastic calculation of time series, which is called stochastic and uses the next reaction method described by Gibson and Bruck. Since the deterministic simulation is inappropriate for some systems but on the other hand, the stochastic simulation is too time consuming, there are some methods that try to combine the advantages of both deterministic and stochastic simulation. Most of those methods are termed hybrid methods. COPASI also includes such a hybrid method which in some systems where deterministic simulation would lead to incorrect results will give the correct time series but is still computationally less demanding than a pure stochastic simulation. The default method in COPASI to calculate a time course is LSODA Petzold83. LSODA is part of the ODEPACK library Hindmarsh83. LSODA was written by Linda R. Petzold and Alan C. Hindmarsh. It solves systems dy/ dt = f with a dense or banded Jacobian when the problem is stiff, but it automatically selects between non-stiff (Adams) and stiff (BDF) methods. It uses the non-stiff method initially, and dynamically monitors data in order to decide which method to use. Integrate Reduced Model This parameter is a boolean value to determine whether the integration shall be performed using the mass conservation laws, i.e., reducing the number of system variables or to use the complete model. A value of '1' (the default) instructs COPASI to make use of the mass conservation laws, whereas a value of '0' instructs COPASI to determine all variables through ODEs. Relative Tolerance This parameter is a numeric value specifying the desired relative tolerance the user wants to achieve. A smaller value means that the trajectory is calculated more accurate. The default value is 1.0* 10-6. Please note that best achievable relative tolerance is approximately 2.22* 10-16. Absolute Tolerance This parameter is a positive numeric value specifying the desired absolute tolerance the user wants to achieve. Please note that for species the absolute tolerance is applied to the concentration value. The default value is 1.0* 10-12. Adams Max Order This parameter is a positive integer value specifying the maximal order the non-stiff Adams integration method shall attempt before switching to the stiff BDF method. The default and maximal order is '12'. BDF Max Order This parameter is a positive integer value specifying the maximal order the stiff BDF integration method shall attempt before switching to smaller internal step sizes. The default and maximal order is '5'. Max Internal Steps This parameter is a positive integer value specifying the maximal number of internal steps the integrator is allowed to take before the next desired reporting time. The default value is '10000'. This stochastic simulation method utilizes the algorithm developed by Gibson and Bruck (see Gibson00 for details). For each reaction a putative stochastic reaction time is calculated and the reaction with the shortest reaction time will be realized. The set of reactions is organized in a priority queue to allow for the efficient search for the fastest reaction. In addition, by using a so-called dependency graph only those reaction times are recalculated in each step, that are dependent on the reaction, which has been realized. This simulation method requires all the reactions to be irreversible. However, COPASI provides a tool, that converts all reversible reactions into irreversible ones. Because the algorithm internally works on discrete particle numbers rather than concentrations, the particle numbers in the system must not exceed a value of approximately 264. The current implementation of the Next Reaction Method is very inefficient when the model contains assignment rules, which leads to extended calculation times. There is a restriction regarding global quantities: If a differential equation is provided for a global quantity (the quantity is of type "ode") the model cannot be simulated stochastically with the current version of COPASI. If the global quantity is of type "assignment" stochastic simulation is possible but not as efficient as for models without assignments. No restrictions apply for "fixed" global quantities. Max Internal Steps This parameter is a positive integer value specifying the maximal number of internal steps the integrator is allowed to take before the next desired reporting time. The default value is '1000000'. Subtype This parameter is ignored in the current version of COPASI. Use Random Seed This flag can be '0' or '1' and determines if the user-defined random seed should be used for the calculation. The default is '0' meaning that the random seed is set to a random value before each run and consecutively calculated trajectories will be different. If the value of this flag is set to '1', the user-defined random seed will be used and each calculated trajectory will be the same for the same value of the given random seed. Random Seed This unsigned integer is used as random seed in the calculations, if the flag Use Random Seed is set to '1'. The default value is '1'. This hybrid simulation method developed by us combines a deterministic numerical integration of ODEs with a stochastic simulation algorithm. The whole biochemical network is partitioned into a deterministic and a stochastic subnet internally. The deterministic subnet contains all reactions, in which only species with high particle numbers take part. All reactions with at least one low-numbered species are in the stochastic subnet, because here stochastic effects are expected. Which particle numbers are considered low or high can be specified by the user with the Lower Limit and the Upper Limit parameters (Species with particle numbers between those limits do not change their status. This leads to a hysteresis-like behavior and avoids many unnecessary swaps, if the particle numbers fluctuate in the middle range). The partitioning of the biochemical network can change dynamically during the simulation. After a certain number of steps, which the user can define using the parameter Partitioning Interval, the partitioning is recalculated using the current particle numbers in the system. During one run the deterministic subnet and the stochastic subnet are simulated in parallel. A 4th-order Runge-Kutta method is used to numerically integrate the deterministic part of the system. For the stochastic part the simulation method by Gibson and Bruck (Gibson00) is utilized. The reaction probabilities of the stochastic subnet are approximated as constant during one stochastic step, even though in theory they can change due to the effects of the deterministic subnet. The current implementation of the Hybrid Runge Kutta Method is very inefficient when the model contains assignment rules, which leads to extended calculation times. Max Internal Steps This parameter is a positive integer value specifying the maximal number of internal steps the integrator is allowed to take before the next desired reporting time. The default value is '1000000'. Lower Limit This parameter is a double value specifying the lower limit for particle numbers. Species with a particle number below this value are considered as having a low particle number. The lower limit cannot be higher than the upper limit. The default value is '800'. Upper Limit This parameter is a double value specifying the upper limit for particle numbers. Species with a particle number above this value are considered as having a high particle number. The upper limit cannot be lower than the lower limit. The default value is '1000'. Runge Kutta Stepsize This positive double value is the step size of the Runge-Kutta solver for the integration of the deterministic part of the system. The default value is '0.001'. Partitioning Interval This positive integer value specifies after how many steps the internal partitioning of the system should be recalculated. The default is '1', i.e. after every step the partitioning of the system is checked. Use Random Seed This flag can be '0' or '1' and determines if the user-defined random seed should be used for the calculation. The default is '0' meaning that the random seed is set to a random value before each run and consecutively calculated trajectories will be different. If the value of this flag is set to '1', the user-defined random seed will be used and each calculated trajectory will be the same for the same value of the given random seed. Random Seed This unsigned integer is used as random seed in the calculations, if the flag Use Random Seed is set to '1'. The default value is '1'. The steady state is the state in which the state variables of the model, e.g. the species concentrations do not change in time. Mathematically this is expressed by setting the differential equations that describe the time evolution of the metabolic system to zero. This forms a system of algebraic non-linear equations. To solve them, COPASI can use a series of strategies using more than one numerical method. All calculations are done based on particle numbers and particle number rates rather than concentrations internally. The reduced model (see ) is used. The Jacobian (which is used in the Newton method and when eigenvalues of the Jacobian are requested) is calculated using finite differences. The eigenvalues of the Jacobian are calculated using CLAPACK. Use Newton This parameter is a boolean value to determine whether to use the damped Newton method on the non-linear algebraic equations defining the steady-state. The initial concentrations set by the user are taken as guesses for the solution. A value of '1' (the default) indicates that COPASI shall use the damped Newton method. The damped Newton method is a variant of the famous Newton method for the solution of systems of non-linear equations. The solution is obtained from an iterative procedure that refines an initial guess until the residual error is smaller than required. If a limit number of iterations is reached without an acceptable solution, the method halts without a solution. The iteration of the plain Newton method is: x i = x i - 1 - f ( x i - 1 ) f ' ( x i - 1 ) In the damped method if x i - 1 has a larger residual error than x i one looks at x i = x i - 1 - f ( x i - 1 ) f ' ( x i - 1 ) * 2 - n   where   n = 0, ... , 32 and accepts the first such value that has a smaller residual error than x i . If none is found, the procedure halts without a solution (because it is at a local minimum). Use Integration This parameter is a boolean value to determine whether to use the ODE solver () to follow the time course defined by the differential equations until a steady state is reached. If at 1010 units of time no steady state has been reached the method halts with no solution. If Use Newton is '1' an attempt to find the steady-state via the damped Newton method is made at each intermediate time point. A value of '1' (the default) indicates that COPASI shall use integration. Use Back Integration This parameter is a boolean value to determine whether to use the ODE solver () to reverse the time course (going backwards in time) defined by the differential equations until a steady state is reached. If at -1* 1010 units of time no steady state has been reached the method halts with no solution. If Use Newton is '1' an attempt to find the steady-state via the damped Newton method is made at each intermediate time point. A value of '1' (the default) indicates that COPASI shall use back integration. Accept Negative Concentrations This parameter is a boolean value to determine whether to accept a steady-state, which contains negative concentrations. A value of '1' indicates that negative concentrations are acceptable whereas a value of '0' (the default) indicates that such states are discarded. Iteration Limit This parameter is a positive integer to determine the maximum number of iterations the damped Newton method shall perform before it fails. The default is '50'. Derivation Factor This is a numeric value to determine the step size used to calculate f ' ( x i - 1 ) . The default is '0.001'. Resolution This is a positive numeric value to determine the resolution used to decide whether the current state is acceptable as a steady-state. If the absolute change of each state variable is smaller than the resolutions the state is accepted. The default is 10-9. Note that this value is interpreted as a concentration value, even though the calculation internally uses particle numbers. The reason for that is purely heuristic: In many cases the modeler will choose the units in a way that concentration values are neither extremely large nor extremely small numerically so that the default value for this parameter leads to useful results. However generally it is not save to just keep the default value without checking. Metabolic control analysis (MCA) is a sensitivity analysis of metabolic systems. In MCA one studies the relative control exerted by each step on the system's variables (e.g. fluxes and species concentrations). This control is measured by applying a perturbation to the step being studied and then measuring the effect on the variable of interest after the system has settled to a new steady state. A control coefficient is a relative measure of how much a perturbation on a parameter affects a system variable (e.g. fluxes or concentrations). It is defined Kacser73 Heinrich74 Burns85 as: C ν i A = ∂ A ∂ ν i ν i A where A is the variable, i the step (enzyme) and v the steady-state rate of the perturbed step. The most common control coefficients are those for fluxes and species concentrations, but any variable of the system can be analyzed with MCA and have control coefficients defined by equations analogous to equation 1. In fact, there is no need even for the system to be in a steady state. Gepasi only calculates directly the steady-state concentration- and flux-control coefficients, those for other variables can still be estimated by simulating small perturbations. A very important property of steady-state metabolic systems was uncovered with the MCA formalism. This concerns the summation of all the flux control coefficients of a pathway. By various procedures Kacser73 Heinrich75 Giersch88 Reder88 it can be demonstrated that for a given reference flux the sum of all flux-control coefficients (of all steps) is equal to unity: ∑ i C ν i J = 1 For a given reference species concentration the sum of all concentration-control coefficients is zero: ∑ i C ν i [M] = 0 where the summations are over all the steps of the system. According to the first summation theorem, increases in some of the flux-control coefficients imply decreases in the others so that the total remains unity. As a consequence of the summation theorems, one concludes that the control coefficients are global properties and that in metabolic systems, control is a systemic property, dependent on all of the system's elements (steps). In enzyme kinetics the behavior of isolated enzymes is studied through the dependence of the initial rates of reaction with the concentration of the substrate(s). Enzyme kinetic studies are centered on derivation of rate equations and the determination of their kinetic constants such as Michaelis constants or limiting-rates or even on the elementary rate constants of a specific reaction mechanism. In metabolic control analysis the properties of each (isolated) enzyme are measured in a way very similar to the flux-control properties: using a sensitivity, known as the elasticity coefficient Kacser73 Heinrich74 Burns85. In this case, one has to consider the effect of perturbations of a reaction parameter on the local reaction rate. By local one means that this sensitivity refers to the isolated reaction which has the same characteristics (effector and enzyme concentrations, temperature, and so on) as in the whole system at the operating point (steady state) of interest. The elasticity coefficients are defined as the ratio of relative change in local rate to the relative change in one parameter (normally the concentration of an effector). Infinitesimally, this is written as: ε p ν i = ∂ ν i ∂ p p ν i where v is the rate of the enzyme in question and p is the parameter of the perturbation. Each enzyme has as many elasticity coefficients as the number of parameters that affect it. One can immediately recognize the concentration of the reaction substrates, products and modifiers as parameters of the reaction. Unlike control coefficients, elasticity coefficients are not systemic properties but rather measure how isolated enzymes are sensitive to changes in their parameters. The elasticity coefficients can be obtained from the kinetic functions by partial derivation. Again like the control coefficients, the elasticity coefficients are not constants, they are dependent on the value of the relevant parameter and so are different for each steady-state. A particularly useful and important feature of MCA is that it can relate the kinetic properties of the individual reactions (local properties) with (global) properties of the whole intact pathway. This is done through the connectivity theorems Kacser73 that relate the control coefficients and the elasticity coefficients of steps with common intermediate species. The connectivity theorem for flux-control coefficients Kacser73 states that, for a common species S, the sum of the products of the flux-control coefficient of all (i) steps affected by S and its elasticity coefficients towards S, is zero: ∑ i C ν i J ε [S] ν i = 0 For the concentration-control coefficients, the following two equations apply Westerhoff84: ∑ i C ν i [A] ε [S] ν i = 0 , where  A ≠ S ∑ i C ν i [A] ε [S] ν i = -1 The first equation applies to the case in which the reference species (A) is different from the perturbed species (S). Whereas the second applies to the case in which the reference species is the same as the perturbed species. The connectivity theorems allow MCA to describe how perturbations on species of a pathway propagate through the chain of enzymes. The local (kinetic) properties of each enzyme effectively propagate the perturbation to and from its immediate neighbors. COPASI calculates (non-normalized) elasticity coefficients by numerical derivation with finite differences. To calculate control coefficients from steady-state data, COPASI applies the method described in Reder88. This method works with the reduced system where some variables are eliminated using conservation relations (see ). All coefficients are obtained unscaled by this method and are scaled with the appropriate steady state concentrations and fluxes (the same with the elasticities). Both scaled and unscaled coefficients and elasticities are displayed and available for output. Modulation Factor This parameter is ignored in the current version of COPASI. The rest of the options is described in the sections for and . The optimization methods described in this chapter attempt to minimize a given objective function. There are several ways to do this and COPASI supports many different methods for the minimization of an objective function. Evolutionary programming (EP) Fogel92Baeck93Baeck97 is a computational technique that mimics evolution and is based on reproduction and selection. An EP algorithm is composed of individuals that reproduce and compete, each one is a potential solution to the (optimization) problem and is represented by a "genome" where each gene corresponds to one adjustable parameter. At each generation of the EP, each individual reproduces asexually, i.e. divides into two individuals. One of these contains exactly the same "genome" as the parent while the other suffers some mutations (the parameter values of each gene change slightly). At the end of the generation, the algorithm has double the number of individuals. Then each of the individuals is confronted with a number of others to count how many does it outperform (the number of wins is the number of these competitors that represent worse solutions than itself). All the individuals are ranked by their number of wins, and the population is again reduced to the original number of individuals by eliminating those which have worse fitness (solutions). Number of Generations The parameter is a positive integer value to determine the number of generations the algorithm shall evolve the population. The default is '200'. Population Size The parameter is a positive integer value to determine the size of the population, i.e., the number of individuals that survive after each generation. The default is '20'. Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. Evolutionary Strategies with Stochastic Ranking (SRES) Runarsson00 is similar to . However, a parent has multiple offsprings during each generation. Each offspring will contain a recombination of genes with another parent and additional mutations. The algorithm assures that each parameter value will be within its boundaries. But constraints to the solutions may be violated. Whenever this happens the square of the size of the violation is summed up, i.e., we calculate φ = ∑ c j < l c j ( l c j - c j ) 2 + ∑ c j > u c j ( c j - u c j ) 2 where the constraints are given by c j ∈ ( l c i ,   u c i ) . The value φ is used within the selection, which is performed as described in Number of Generations The parameter is a positive integer value to determine the number of generations the algorithm shall evolve the population. The default is '200.' Population Size The parameter is a positive integer value to determine the size of the population, i.e., the number of individuals that survive after each generation. The default is '20'. Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. Pf This parameter is a numerical value in the interval (0, 1) determining the chance that individuals either outside the parameter boundaries or violating the constraints are compared during the selection. The default is '4.75'. The genetic algorithm (GA) Baeck97Baeck93Michalewicz94Mitchell95 is a computational technique that mimics evolution and is based on reproduction and selection. A GA is composed of individuals that reproduce and compete, each one is a potential solution to the (optimization) problem and is represented by a "genome" where each gene corresponds to one adjustable parameter. At each generation of the GA, each individual is paired with one other at random for reproduction. Two offspring are produced by combining their genomes and allowing for "cross-over", i.e., the two new individuals have genomes that are formed from a combination of the genomes of their parents. Also each new gene might have mutated, i.e. the parameter value might have changed slightly. At the end of the generation, the algorithm has double the number of individuals. Then each of the individuals is confronted with a number of others to count how many does it outperform (the number of wins is the number of these competitors that represent worse solutions than itself). All the individuals are ranked by their number of wins, and the population is again reduced to the original number of individuals by eliminating those which have worse fitness (solutions). Many features of a GA may be varied. The details of this particular implementation of the GA for optimization of biochemical kinetics are: Parameters are encoded in genes using floating-point representation, rather than the more usual binary representation. Mutation is carried out by adding to the gene a random number drawn from a normal distribution with zero mean and a standard deviation of 10% of the parameter value. Whenever this makes the parameter (gene) exceed one boundary, it is set to that boundary value. Cross-over is always performed at gene boundaries so that no gene is ever disrupted. The number of cross-over points is a random number between zero and half the number of adjustable parameters (uniform distribution). Selection is done by a tournament where each individual competes with a number of others equal to 20% the population size. The competitors are chosen at random. The initial population contains one individual whose genes are the initial parameter values, the genes of all other individuals are initialized to a random value between their boundaries. If the boundaries span two orders of magnitude or more, the random distribution is exponential, otherwise normal. Whenever the fittest individual has not changed for the last 10 generations, the 10% less fit individuals are replaced by individuals with random genes. When the fittest individual has not changed for 30 generations, the worse 30% are substituted by individuals with random genes. When the fittest individual has not changed for 50 generations, the worse 50% are substituted by individuals with random genes. This procedure helps the algorithm escape local minima and is somewhat equivalent to increasing the mutation rate when the population has become uniform. Number of Generations The parameter is a positive integer value to determine the number of generations the algorithm shall evolve the population. The default is '200'. Population Size The parameter is a positive integer value to determine the size of the population, i.e., the number of individuals that survive after each generation. The default is '20'. Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. The genetic algorithm with stochastic ranking is very similar to the before described with tournament selection. With two exception which are the mutations are not forced to be within the boundaries and the selection is done through a bubble sort with a random factor as described in Runarsson00. Parameters are encoded in genes using floating-point representation, rather than the more usual binary representation. Mutation is carried out by adding to the gene a random number drawn from a normal distribution with zero mean and a standard deviation of 10% of the parameter value. Parameters may exceed boundaries. Whenever this happens or a constraint to the solution is violated the square of the size of the violation is summed up, i.e., we calculate φ = ∑ p i < l p i ( l p i - p i ) 2 + ∑ p i > u p i ( p i - u p i ) 2 + ∑ c j < l c j ( l c j - c j ) 2 + ∑ c j > u c j ( c j - u c j ) 2 where the parameters are given by p i ∈ ( l p i ,   u p i ) and the constraints by c j ∈ ( l c i ,   u c i ) . The value φ is used within the selection. Cross-over is always performed at gene boundaries so that no gene is ever disrupted. The number of cross-over points is a random number between zero and half the number of adjustable parameters (uniform distribution). Selection is done by the bubble sort described in Runarsson00. This sort incorporates a probability to compare objective values for individuals with a φ ≠ 0 . The pseudo code for the sort is: // Here sweepNum is optimal number of sweeps from paper, i.e., TotalPopulation for (i = 0; i < sweepNum; i++) { wasSwapped = false; for (j = 0; j < TotalPopulation - 1; j++) { // within bounds or random chance if ((phi(j) == 0 and phi(j + 1) == 0) or UniformRandom(0, 1) < Pf) { // compare objective function values if (Value(j) > Value(j + 1)) { swap(j, j + 1); wasSwapped = true; } } else // phi != 0 { // individual j further outside then j + 1 if (phi(j) > phi(j + 1)) { swap(j, j + 1); wasSwapped = true; } } } // if no swap then break if (wasSwapped == false) break; } The initial population contains one individual whose genes are the initial parameter values, the genes of all other individuals are initialized to a random value between their boundaries. If the boundaries span two orders of magnitude or more, the random distribution is exponential, otherwise normal. Whenever the fittest individual has not changed for the last 10 generations, the 10% less fit individuals are replaced by individuals with random genes. When the fittest individual has not changed for 30 generations, the worse 30% are substituted by individuals with random genes. When the fittest individual has not changed for 50 generations, the worse 50% are substituted by individuals with random genes. This procedure helps the algorithm escape local minima and is somewhat equivalent to increasing the mutation rate when the population has become uniform. Number of Generations The parameter is a positive integer value to determine the number of generations the algorithm shall evolve the population. The default is '200'. Population Size The parameter is a positive integer value to determine the size of the population, i.e., the number of individuals that survive after each generation. The default is '20'. Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. Pf This parameter is a numerical value in the interval (0, 1) determining the chance that individuals either outside the parameter boundaries or violating the constraints are compared during the selection. The default is '4.75'. The method of Hooke and Jeeves Bell66Hooke61Kaupe63Swann72 is a direct search algorithm that searches for the minimum of a nonlinear function without requiring (or attempting to calculate) derivatives of the function. Instead it is based on a heuristic that suggests a descent direction using the values of the function calculated in a number of previous iterations. Iteration Limit This parameter is positive integer determining the maximum number of iterations the algorithm shall perform. The default is '50'. Tolerance This parameter is a positive value determining the tolerance with which the solution shall be determined. If the improvement between two steps is less than the tolerance the algorithm stops. The default is ' 10 -5 '. Rho This parameter is a value in (0, 1) determining the factor with which the steps size is reduced between iterations. The default is '0.2'. Levenberg-Marquardt Levenberg44Marquardt63 is a gradient descent method. It is a hybrid between the steepest descent and the Newton methods. The Newton optimization method searches for the minimum of a nonlinear function by following descent directions determined from the function's first and second partial derivatives. The steepest descent method searches for a minimum based only on the first derivatives of the function. While the Newton method converges quadratically towards the minimum in its vicinity, it may not converge at all if it is far away from it. On the other hand the steepest descent method only converges linearly but is guaranteed to converge. Levenberg first suggested an improvement to the Newton method in order to make it more robust, i.e. to overcome the problem of non-convergence. His suggestion was to add a factor to the diagonal elements of the Hessian matrix of second derivatives when not close to the minimum (this can be judged by how positive definite the matrix is). The effect when this factor is large compared to the elements of Hessian is that the method then becomes the steepest descent method. Later Marquardt suggested that the factor should be multiplicative rather than additive and also defined a heuristic to make this factor increase or decrease. The method known as Levenberg-Marquardt is thus an adaptive method that effectively changes between the steepest descent to the Newton method. The original suggestions of Levenberg and Marquardt were effective to enhance the Gauss-Newton method, a variant of the Newton method specifically for minimizing least-squares functions. In this case the advantage is also that the second derivatives do not need to be calculated as they are estimated from the gradient of the residuals. Subsequently Goldfeld et al. Goldfeld66 extended the method to the case of general non-linear functions. Iteration Limit This parameter is positive integer determining the maximum number of iterations the algorithm shall perform. The default is '200'. Tolerance This parameter is a positive value determining the tolerance with which the solution shall be determined. If the improvement between two steps is less than the tolerance the algorithm stops. The default is ' 10 -5 '. This method also known as the simplex method is due to Nelder and Mead Nelder65. A simplex is a polytope of N + 1 vertices in N dimensions. The objective function is evaluated at each vertex. Dependent on these calculated values a new simplex is constructed. The simplest step is to replace the worst point with a point reflected through the centroid of the remaining N points. If this point is better than the best current point, then we can try stretching exponentially out along this line. On the other hand, if this new point isn't much better than the previous value then we are stepping across a valley, so we shrink the simplex towards the best point. Iteration Limit This parameter is a positive integer to determine the maximum number of iterations the method is to perform. The default value is '200'. Tolerance This parameter is a positive number and provides an alternative termination criteria. If the variance of the values of the objective function at the vertices of the current simplex is smaller than the tolerance the algorithm stops. The default is ' 1.0 * 10 -5 '. Scale This parameter is a positive number and determines the size of the initial simplex. The edges of the polytope are inverse proportional to the scale, i.e., a larger value makes the initial simplex smaller. The default is '10'. The particle swarm optimization method suggested by Kennedy and Eberhart Kennedy95 is inspired by a flock of birds or a school of fish searching for food. Each particle has a position X i and a velocity V i in the parameter space. Additionally, it remembers its best achieved objective value O and position M i . Dependent on its own information and the position of its best neighbor (a random subset of particles of the swarm) a new velocity is calculated. With this information the position is updated. The pseudo code for the algorithm for each particle is: N = best neighbor's position for i = 0 to number of parameters do R1 = uniform random number in [0, 1] R2 = uniform random number in [0, 1] V[i]= w * V[i] + C * R1 * (M[i]- X[i]) + C * R2 * (N[i]- X[i]) X[i] = X[i] + V[i] enddo current_O = evaluate objective function if current_O < O then do O = current_O M = X enddo Iteration Limit This parameter is a positive integer to determine the maximum number of iterations the method is to perform. The default value is '2000'. Swarm Size This parameter is a positive integer specifying the number of particles in the swarm. The default value is '50'. Std. Deviation This parameter is a positive number and provides an alternative termination criteria. If the standard deviation of the values of the objective function of each particle and the standard deviation of the best positions is smaller than the provided value the algorithm stops. The default value is ' 1.0 * 10 -6 '. Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. Praxis is a direct search method (for a review see Swann72) that searches for the minimum of a nonlinear function without requiring (or attempting to calculate) derivatives of that function. Praxis was developed by Brent Brent72 after the method proposed by Powell Powel64. The inspiration for Praxis was the well-known method of minimising each adjustable parameter (direction) at a time - the principal axes method. In Praxis directions are chosen that do not coincide with the principal axes, in fact if the objective function is quadratic then these will be conjugate directions, assuring a fast convergence rate. This implementation of the Praxis method was originally written in FORTRAN at the Stanford Linear Accelerator Center (dated 3/1/73). The original FORTRAN code is available from http://www.netlib.org/opt/praxis. The original code was translated automatically by the F2C program from AT&T Bell Labs (available at http://www.netlib.org/f2c/). The C code was then enhanced to suite COPASI's reentrant optimization method interface. Tolerance Convergence stopping criterion: the method stops when two consecutive estimates differ by less than Tolerance. This parameter is a positive integer to determine the number of parameter sets to be drawn before the algorithm stops. The default value is ' 1.0 * 10 -5 '. Random search is an optimization method that attempts to find the optimum by testing the objective function's value on a series of combinations of random values of the adjustable parameters. The random values are generated complying with any boundaries selected by the user, furthermore, any combinations of parameter values that do not fulfill constraints on the variables are excluded. This means that the method is capable of handling bounds on the adjustable parameters and fulfilling constraints. For infinite number of iterations this method is guaranteed to find the global optimum of the objective function. In general one is interested in processing a very large number of iterations. Number of Iterations This parameter is a positive integer to determine the number of parameter sets to be drawn before the algorithm stops. The default value is '100000'. Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. Simulated annealing is an optimization algorithm first proposed by Kirkpatrick et al. Kirkpatrick83 and was inspired by statistical mechanics and the way in which perfect crystals are formed. Perfect crystals are formed by first melting the substance of interest, and then cooling it very slowly. At large temperatures the particles vibrate with wide amplitude and this allows a search for global optimum. As the temperature decreases so do the vibrations until the system settles to the global optimum (the perfect crystal). The simulated annealing optimization algorithm uses a similar concept: the objective function is considered a measure of the energy of the system and this is maintained constant for a certain number of iterations (a temperature cycle). In each iteration, the parameters are changed to a nearby location in parameter space and the new objective function value calculated; if it decreased, then the new state is accepted, if it increased then the new state is accepted with a probability that follows a Boltzmann distribution (higher temperature means higher probability of accepting the new state). After a fixed number of iterations, the stopping criterion is checked; if it is not time to stop, then the system's temperature is reduced and the algorithm continues. Simulated annealing is a stochastic algorithm that is guaranteed to converge if ran for an infinite number of iterations. It is one of the most robust global optimization algorithms, although it is also one of the slowest. (Be warned that simulated annealing can run for hours or even days!). This implementation of simulated annealing is based on the code of Corana et al. Corana87. This implementation has tuned some of the parameters of the original implementation as follows: Each temperature cycle takes 10 * max ( 5 * p , 100 ) * p random steps, with p being the number of optimization parameters. At each step, a new candidate solution is accepted if: a) it reduced the objective function value, or b) with a probability equal to e - Δ f T , where Δf is the increase in objective function value, and T is the current temperature. The stopping criterion is applied to the last two temperature cycles (i.e. the change in objective function between this temperature and the previous must have been smaller than the Tolerance in the last two cycles). Start Temperature Initial temperature of the system. The higher the temperature, the larger the probability that a global optimum is found. Note that the temperature should be very high in the beginning of the method (the system should be above the "melting" temperature). This value has the same units as the objective function, so what represents "high" is different from problem to problem. The default is '1'. Cooling Factor Rate by which the temperature is reduced from one cycle to the next, given by the formula: T new = T old * "Cooling Factor" . The simulated annealing algorithm works best if the temperature is reduced at a slow rate, so this value should be close to 1. (but values closer to 1 will also cause the algorithm to run longer). The default is '0.85'. Tolerance Convergence stopping criteria: the method stops when the change in the objective function has been smaller than this value in the last two temperature steps. The default is ' 10 -6 '. (This is an absolute tolerance) Random Number Generator The parameter is an enumeration value to determine which random number generator this method shall use. COPASI provides two random number generators R250 Maier91 (selected through the value 0) and the Mersenne Twister Matsumoto98 (selected through the value 1 (default)). Seed The parameter is a positive integer value to determine the seed for the random number generator. A value of zero instructs COPASI to select a "random" value. Steepest descent Fogel92 is an optimization method that follows the direction of steepest descent on the hyper-surface of the objective function to find a local minimum. The direction of steepest descent is defined by the negative of the gradient of the objective function. Iteration Limit This parameter is positive integer determining the maximum number of iterations the algorithm shall perform. The default is '100'. Tolerance This parameter is a positive value determining the tolerance with which the solution shall be determined. If the improvement between two steps is less than the tolerance the algorithm stops. The default is ' 10 -6 '. The Truncated Newton method is a sophisticated variant of the Newton optimisation method. The Newton optimisation method searches for the minimum of a nonlinear function by following descent directions determined from the function’s first and second partial derivatives. The Truncated Newton method does an incomplete (truncated) solution of a system of linear equations to calculate the Newton direction. This means that the actual direction chosen for the descent is between the steepest descent direction and the true Newton direction. A more complete description of the method can be found in Gill81 and Nash84. The particular implementation of the Truncated Newton method used here takes into account simple bounds on the optimisation parameters. Like the Newton method itself, the Truncated Newton variant converges quadratically when in the vicinity of the minimum. On average this method is one of the best gradient descent methods. This implementation of the Truncated Newton method was originally written in FORTRAN by Stephen Nash (Operations Research and Applied Statistics Dept., George Mason University, Fairfax, VA 22030, USA). The original code is available from http://www.netlib.org/opt/tn). The original code was translated automatically to C by the F2C program from AT&T Bell Labs (available at http://www.netlib.org/f2c/ ). The C code was then enhanced to suite COPASI's reentrant optimization method interface. None The algorithm is not tunable. COPASI allows the calculation of Lyapunov exponents of a trajectory as well as the average divergence of the system. The exponents are calculated for the reduced system (see ), so the maximum number of exponents that can be calculated is the number of independent variables. If less than this number of exponents is requested, the largest exponents are calculated. COPASI uses the well known algorithm proposed by Wolf et al. (Wolf85). This algorithm integrates one reference trajectory and simultaneously N difference trajectories (if N is the number of exponents requested) in a system linearized around the reference trajectory. This integration is carried out for a short time interval (the "Orthonormalization interval", see below) and then the difference vectors are reorthonormalized. The exponents for this time interval are calculated from how much each of the difference trajectories converged or diverged from the reference trajectory during the interval. This calculation is repeated and the "local" exponents are averaged over the the whole trajectory. The divergence is calculated (if requested) as the average of the trace of the Jacobian. Although it is not numerically necessary the divergence is also calculated for the same short intervals that are used for the Lyapunov exponents. This allows comparing the local values of the divergence with the local exponents. If you are only interested in the end result of the Lyapunov exponents and the average divergence you can just use the default report that is provided by COPASI (or just look at the result in the GUI). If you want to have access to the "local" results for the single orthonormalization intervals however, you will have to define a plot or report manually. In this version of COPASI you will need to used the "expert" feature of the object selection dialog to access the exponents. They are located in the "Lyapunov Exponents" branch under the "Task List" entry. Output takes place after each reorthonormalization interval. The output can contain each of the ten largest exponents, both the local value from the last interval and the and the average value of all intervals calculated so far. Correspondingly the divergence can be output both as an average over the last interval or as an average over the whole trajectory. The Jacobian that is used for both Lyapunov exponents and divergence calculation is calculated using finite differences. The integration of the reference and difference trajectories is done using LSODA Hindmarsh83. Orthonormalization interval This is the time interval after which an orthonormalization of the difference trajectories takes place. This parameter is critical for the accuracy of the Lyapunov exponents. Smaller values generally lead to more accurate results, but take longer time (since the numerical integration needs to be restarted for many short calculations). One way to judge the adequacy of this parameter is to compare the sum of exponents with the divergence of the system. Those two values should be the same (only if you request the calculation of all exponents), and since the calculation of the divergence is very robust, a mismatch would typically mean that the orthonormalization interval needs to be smaller. Note that this parameter mostly affects the accuracy of the exponents with the largest absolute values. Since large positive exponents are unusual, this means that the largest negative exponents suffer from accuracy problems related to this parameter. If you don't need the exact values of the strongly negative exponents you can chose a larger value for this parameter and enjoy a much faster calculation. The default value is 1.0. Overall time This parameter specifies the overall time of the calculation. The integration will be repeated in small steps given by the "Orthonormalization interval" parameter until the overall time is reached. This value is also critical for the accuracy of the exponents. Since COPASI cannot guess how fast the exponents converge, no save default value can be given for this parameter. One indication would be that if the system does not run into a steady state one of the exponents should be zero. If this is not the case in the result, probably the overall time was to short to allow the exponents to converge to their average value. The default value is 1000. The rest of the options apply for the LSODA numerical integrator that is used for the calculation. They are described in the section for COPASI allows the calculation of generalized sensitivities of a model. This is done by numerical differentiation using finite differences. The user specifies a list of functions to be differentiated and one or two lists of variables. The differentiation is performed with respect to these variables. Since every function in the list is differentiated with respect to every variable in the list the result will generally be a two-dimensional matrix. If also a list of variables for second derivatives is given then all the first derivatives are again differentiated with respect to all the variables in the second list. The result will be a three-dimensional array of second derivatives. Delta factor This is used to determine the delta value for the finite difference numerical differentiation. The delta is calculated as the product of the delta factor and the current absolute value of the variable. If the resulting value is smaller than the "Delta minimum" parameter, it is discarded and the "Delta minimum" value is used instead. Default value is 1.0*10-6 . Delta minimum The minimal delta for numerical differentiation. Default value is 1.0*10-12 .