This is a core version which contains all but deprecated classes.
Kinetic Simulation Algorithm Ontology (KiSAO)
The Kinetic Simulation Algorithm Ontology (KiSAO) classifies algorithms available for the simulation and analysis of models in biology, and their characteristics and the parameters required for their use.
has characteristic
'has characteristic' links algorithms to the characteristics they possess.
is hybrid of
The basic idea of hybrid simulation methods is to combine the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time. 'is hybrid of' is a relation between the complex simulation method and algorithms it applies to the different parts of the system.
Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Pahle J. Brief Bioinform. 2009 Jan;10(1):53-64. Epub 2009 Jan 16.
is parameter of
'is parameter of' is an inverse of 'has parameter' and links parameters to the algorithms which use them.
has parameter
'has parameter' links algorithms to the parameters they use.
is characteristic of
'is characteristic of' is an inverse of 'has characteristic' and links characteristics to the algorithms which possess them.
is similar to
A relationship indicating that two entities are similar to each other.
uses
A relation between algorithms indicating that one algorithm uses another one (for example, if the algorithm switches between several algorithms).
is generalization of
A relation between kinetic simulation algorithms, indicating that one is a generalization of another (e.g.if one extends another to suit systems with more general characteristics ).
is used by
A relation between algorithms indicating that one algorithm is used by another one (for example, if the algorithm switches between several algorithms). 'is used by' is inverse of 'uses'.
has type
Indicates the type of algorithm parameter value, such as, for example, xsd:double for 'absolute tolerance'.
has Runge-Kutta method order
Indicates the order of a 'Runge-Kutta based method' [urn.miriam.biomodels.kisao:KISAO_0000064]. A 'Runge-Kutta based method' has order p if for sufficiently smooth initial value problems (y'=f(x,y), y(x0)=y0), ||y(x0 + h)-y1||<=Kh^(p+h).
modelling and simulation algorithm
modeling and simulation algorithm
Algorithm used to instantiate a simulation from a mathematical model.
NLN
weighted stochastic simulation algorithm
weighted SSA
The weighted stochastic simulation algorithm manipulates the probabilities measure of biochemical systems by sampling, in order to increase the fraction of simulation runs exhibiting rare events.
EXACT
Kuwahara H, Mura I. (2008) An efficient and exact stochastic simulation method to analyse rare events in biochemical systems. J Chem Phys. 129(16):165101.
Cain
Gillespie first reaction algorithm
Gillespie's first reaction method
Stochastic simulation algorithm using the reaction probability density function (next-reaction density function), giving the probability that the next reaction will happen in a given time interval. To choose the next reaction to fire, the algorithm calculates a tentative reaction time for each reaction and then select the smallest.
Gillespie DT. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, Volume 2 , pages 403-434 (1976).
Gillespie DT. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, Vol. 81, No. 25. (1977), pp. 2340-2361.
StochSim
multi-state agent-based simulation method
Morton-Firth
The agent-based simulation method instantiates each molecule as an individual software object. The interactions between those objects are determined by interaction probabilities according to experimental data. The probability is depended on the state the molecule is in at that specific time (molecules have multiple-state). Additionally, ''pseudo-molecules'' are introduced to the system in order to simulate unimolecular reactions. For simulation, continuous time is broken down into discrete, independent ''slices''. During each time slice one molecule is selected randomly, a second molecule or pseudo-molecule is selected afterwards (leading to either a unimolecular or a bimolecular reaction). The reaction will only take place if a produced random number exceeds the reaction probability calculated beforehand. In that case, the system is updated after that reaction.
Morton-Firth, C.J, Bray, D. Predicting temporal fluctuations in an intracellular signalling pathway. Journal of Theoretical Biology, volume 192, pages 117-128 (1998).
SBML-SAT
SUNDIALS
libRoadRunner
CVODE
VODE
VODEPK
code value ordinary differential equation solver
The CVODE is a package written in C that solves ODE initial value problems, in real N-space, written as y'=f(t,y), y(t0)=y0. It is capable for stiff and non-stiff systems and uses two different linear multi-step methods, namely the Adam-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] method and the backward differentiation formula [http://identifiers.org/biomodels.kisao/KISAO_0000288].
BioNetGen
JSim
OpenCOR
PySCeS
tellurium
VCell
Cohen S, Hindmarsh C. Cvode, A Stiff/nonstiff Ode Solver In C. Computers in Physics, Vol. 10 (2), pages 138-143 (1996).
Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
RELATED
PVODE
parallel code value ordinary differential equation solver
PVODE is a general-purpose solver for ordinary differential equation (ODE) systems that implements methods for both stiff and nonstiff systems. [...] In the stiff case, PVODE uses a backward differentiation formula method [http://identifiers.org/biomodels.kisao/KISAO_0000288] combined with preconditioned GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000253] iteration. Parallelism is achieved by distributing the ODE solution vector into user-specified segments and parallelizing a set of vector kernels accordingly. For PDE-based ODE systems, we provide a module that generates a band block-diagonal preconditioner for use with the GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000253] iteration. PVODE is based on CVODE [http://identifiers.org/biomodels.kisao/KISAO_0000019].
Byrne GD, Hindmarsh AC. PVODE, an ODE Solver for Parallel Computers. International Journal of High Performance Computing Applications, Vol. 13 (4), pages 354-365 (1999).
Stochsim 1.2 and more recent versions
StochSim nearest-neighbour algorithm
The nearest-neighbour algorithm allows for the representation of spatial information, by adding a two-dimensional lattice in the form of a probabilistic cellular automata. That way, nearest neighbour interactions do additionally influence reactions taking place in the systems. Reactions between entities are calculated using the agent-based simulation algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000017].
Le Novere N, Shimizu TS. STOCHSIM: modelling of stochastic biomolecular processes. Bioinformatics, volume 17 (6), pages 575-576 (2001).
MesoRD
SmartCell
Elf and Ehrenberg method
Elf algorithm
NSM
next-subvolume method
Sub-volume stochastic reaction-diffusion method that is a combination of the Direct Method [http://identifiers.org/biomodels.kisao/KISAO_0000029] for sampling the time for a next reaction or diffusion event in each subvolume, with Gibson and Bruck's Next Reaction Method [http://identifiers.org/biomodels.kisao/KISAO_0000027], which is used to keep track of in which subvolume an event occurs next. The subvolumes are kept sorted in a queue, implemented as a binary tree, according to increasing time of the next event. When an event has occurred in the subvolume at the top of the queue, new event times need to be sampled only for one (the event is a chemical reaction) or two (the event is a diffusion jump) subvolume(s).
Elf J, Ehrenberg M. Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Systems Biology, IEE Proceedings, volume 1 (2), pages 230-236 (2004).
Gibson-Bruck next reaction algorithm
Gibson and Bruck algorithm
Gibson-Bruck's next reaction algorithm
Gillespie-Gibson stochastic simulation algorithm
SSA-GB
next reaction method
As with the first reaction method [http://identifiers.org/biomodels.kisao/KISAO_0000015], a putative reaction time is calculated for each reaction, and the reaction with the shortest reaction time will be realized. However, the unused calculated reaction times are not wasted. 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.
E-Cell
Gibson MA, Bruck J. Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. Journal of Physical Chemistry A, Vol. 104, pages 1876-1889 (2000).
slow-scale stochastic simulation algorithm
slow-scale stochastic SSA
ssSSA
Attempt to overcome the problem of stiff systems by developing an ''approximate theory that allows one to stochastically advance the system in time by simulating the firings of only the slow reaction events''.
Cao Y, Gillespie DT, Petzold LR. The slow-scale stochastic simulation algorithm. Journal of Chemical Physics, Vol. 122, No. 1. (1 January 2005).
BetaWB
ByoDyn
Gillespie direct algorithm
Doob-Gillespie method
Gillespie's algorithm
Gillespie's direct method
SSA
stochastic simulation algorithm
Stochastic simulation algorithm using the reaction probability density function (next-reaction density function), giving the probability that the next reaction will happen in a given time interval. To choose the next reaction to fire, the algorithm directly and separately calculates the identity of the reaction and the time it will fire.
iBioSim
Euler forward method
explicit Euler method
explicit Gaussian first order Runge-Kutta
The Euler method is an explicit one-step method for the numerical integration of ODES with a given initial value. The calculation of the next integration step at time t+1 is based on the state of the system at time point t.
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition (1992).
GSL
Euler backward method
implicit Euler method
implicit Gaussian first order Runge-Kutta
The Euler backward method is an implicit one-step method for the numerical integration of ODES with a given initial value. The next state of a system is calculated by solving an equation that considers both, the current state of the system and the later one.
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001).
explicit fourth-order Runge-Kutta method
ERK4
RK4
Runge-Kutta method
The Runge-Kutta method is a method for the numerical integration of ODES with a given initial value. The calculation of the next integration step at time t+1 is based on the state of the system at time point t, plus the product of the size of the interval and an estimated slope. The slope is a weighted average of 4 single slope points (beginning of interval-midpoint-midpoint-end of interval).
Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987).
Rosenbrock method
Kaps-Rentrop method
generalized fourth order Runge-Kutta method
Some general implicit processes are given for the solution of simultaneous first-order differential equations. These processes, which use successive substitution, are implicit analogues of the (explicit) Runge-Kutta processes. They require the solution in each time step of one or more set of simultaneous linear equations, usually of a special and simple form. Processes of any required order can be devised, and they can be made to have a wide margin of stability when applied to a linear problem.
Rosenbrock HH. Some general implicit processes for the numerical solution of differential equations. The Computer Journal, pages 329-330 (1963).
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition, pages 742-746 (1992).
sorting stochastic simulation algorithm
sorting SSA
sorting direct method
In order to overcome the problem of high complexity of the stochastic simulation algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000029] when simulating large systems, the sorting direct method maintains a loosely sorted order of the reactions as the simulation executes.
McCollum J M, Peterson G D, Cox C D, Simpson M L, Samatova N F. The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behaviour. Computational Biology and Chemistry, Volume 30 (1), pages 39-49 (2006).
tau-leaping method
tauL
Approximate acceleration procedure of the stochastic simulation algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000029] that divides the time into subintervals and ''leaps'' from one to another, firing all the reaction events in each subinterval.
Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V.
Poisson tau-leaping method
explicit tau-leaping
explicit tau-leaping method with basic pre-leap check
explicit tau-leaping method with basic preleap check
poisson tau-leaping
Explicit tau-leaping method with basic pre leap check.
implicit tau-leaping method
Contrary to the explicit tau-leaping [http://identifiers.org/biomodels.kisao/KISAO_0000039 and http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000239] , the implicit tau-leaping allows for much larger time-steps when simulating stiff systems.
Rathinam M, Petzold L R, Cao Y, Gillespie D T. Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. The Journal of Chemical Physics, Volume 119 (24), pages 12784-12794 (2003).
trapezoidal tau-leaping method
trapezoidal implicit tau-leaping method
Formula for accelerated discrete efficient stochastic simulation of chemically reacting system [which] has better accuracy and stiff stability properties than the explicit and implicit [http://identifiers.org/biomodels.kisao/KISAO_0000045] tau-leaping formulas for discrete stochastic systems, and it limits to the trapezoidal rule in the deterministic regime.
Cao Y, Petzold L. Trapezoidal tau-leaping formula for the stochastic simulation of biochemical systems. Foundations of Systems Biology in Engineering (FOSBE), pages 149-152 (2005).
adaptive explicit-implicit tau-leaping method
Modification of the original tau-selection strategy [http://identifiers.org/biomodels.kisao/KISAO_0000040], designed for explicit tau-leaping, is modified to apply to implicit tau-leaping, allowing for longer steps when the system is stiff. Further, an adaptive strategy is proposed that identifies stiffness and automatically chooses between the explicit and the (new) implicit tau-selection methods to achieve better efficiency.
Cao Y, Gillespie DT, Petzold LR. Adaptive explicit-implicit tau-leaping method with automatic tau selection. The Journal of Chemical Physics, Vol. 126 (22) (2007).
Bortz-Kalos-Lebowitz algorithm
BKL
DMC
KMC
dynamic Monte Carlo
dynamic Monte Carlo method
kinetic Monte Carlo
kinetic Monte Carlo method
n-fold way
The Bortz-Kalos-Lebowitz (or: kinetic Monte-Carlo-) method is a stochastic method for the simulation of time evolution of processes using (pseudo-)random numbers.
Bortz AB, Kalos MH, Lebowitz JL. A new algorithm for Monte Carlo simulation of Ising spin systems. Journal of Computational Physics, Vol. 17 (1), pages 10-18 (1975).
Smoluchowski equation based method
Method based on the Smoluchowski equation.
Smoluchowski M. Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem, Vol. 92, No. 129. (1917).
Smoldyn
Brownian diffusion Smoluchowski method
In the Brownian diffusion Smoluchowski method, ''each molecule is treated as a point-like particle that diffuses freely in three-dimensional space. When a pair of reactive molecules collide, such as an enzyme and its substrate, a reaction occurs and the simulated reactants are replaced by products. [..] Analytic solutions are presented for some simulation parameters while others are calculated using look-up tables''. Supported chemical processes include molecular diffusion, treatment of surfaces, zeroth-order-, unimolecular-, and bimolecular reactions.
Andrews SS, Bray D. Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol, volume 1 (3-4), pages 137-151 (December 2004).
Greens function reaction dynamics
GFRD
Green's function reaction dynamics
Method that simulates biochemical networks on particle level. It considers both changes in time and space by ''exploiting both the exact solution of the Smoluchowski Equation to set up an event-driven algorithm'' which allows for large jumps in time when the considered particles are far away from each other [in space] and thus cannot react. GFRD combines the propagation of particles in space with the reactions taking place between them in one simulation step.
Van Zon JS, Ten Wolde PR. Green's-function reaction dynamics: A particle-based approach for simulating biochemical networks in time and space. Journal of Chemical Physics, Volume 123 (23) (2005).
Runge-Kutta based method
modified Euler method
A method of numerically integrating ordinary differential equations, which uses a sampling of slopes through an interval and takes a weighted average to determine the right end point. This averaging gives a very accurate approximation.
deterministic cellular automata update algorithm
A cellular automaton is a discrete model of a regular grid of cells with a finite number of dimensions. Each cell has a finite number of defined states. The automaton changes its state in a discrete manner, meaning that the state of a cell at time t is determined by a function of the states of its neighbours at time t - 1. These neighbours are a selection of cells relative to the specified cell. Famous examples for deterministic cellular automata are Conway's game of life or Wolfram's elementary cellular automata.
Burks, Arthur W (Editor). Essays on Cellular Automata. University of Illinois Press (1970).
LSODE
Livermore solver for ordinary differential equations
LSODE solves stiff and nonstiff systems of the form dy/dt = f. In the stiff case, it treats the Jacobian matrix sf/dy as either a dense (full) or a banded matrix, and as either user-supplied or internally approximated by difference quotients. It uses Adams methods (predictor-corrector) [http://identifiers.org/biomodels.kisao/KISAO_0000364] in the nonstiff case, and Backward Differentiation Formula (BDF) methods (the Gear methods) [http://identifiers.org/biomodels.kisao/KISAO_0000288] in the stiff case.
Radhakrishnan K, Hindmarsh AC. Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations. Lawrence Livermore National Laboratory Report, Vol. UCRL-ID-113855 (1993).
Hindmarsh AC. LSODE and LSODI, two new initial value ordinary differential equation solvers. SIGNUM Newsletter, Volume 15 (4), pages 10-11 (1980).
binomial tau-leaping method
BtauL
binomial tau-leap spatial stochastic simulation algorithm
Coarse grained modified version of the next subvolume method [http://identifiers.org/biomodels.kisao/KISAO_0000022] that allows the user to consider both diffusion and reaction events in relatively long simulation time spans as compared with the original method and other commonly used fully stochastic computational methods.
Marquez-Lago T., Burrage K. Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics. The Journal of Chemical Physics, Vol. 127 (10) (2007).
Gillespie multi-particle method
GMP
Gillespie's multi-particle method
particle-based spatial stochastic method
Combination of the multiparticle method for diffusion [http://identifiers.org/biomodels.kisao/KISAO_0000334] and the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029].
Rodriguez VJ, Kaandorp JA, Dobrzynski M, Blom JG. Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli. Bioinformatics, Vol. 22 (15), pages 1895-1901 (2006).
Stundzia and Lumsden method
RD SSA
reaction-diffusion stochastic simulation algorithm
Sub-volume stochastic reaction-diffusion method that using Green's function to link the bulk diffusion coefficient D in Fick's differential law to the corresponding transition rate probability for diffusion of a particle between finite volume elements. This generalized stochastic algorithm enables to numerically calculate the time evolution of a spatially inhomogeneous mixture of reaction-diffusion species in a finite volume. The time step is stochastic and is generated by a probability distribution determined by the intrinsic reaction kinetics and diffusion dynamics.
Stundzia AB, Lumsden CJ. Stochastic simulation of coupled reaction-diffusion processes. J Comput Phys, Vol. 127 (1), pages 196-207 (1996).
estimated midpoint tau-leaping method
explicit tau-leaping method with estimated-mid point technique
Estimated-Midpoint tau-Leap Method: For the selected leaping time tau which satisfies the Leap Condition, compute the expected state change lambda' = tau sumj( aj(x)vj ) during [t, t + tau). Then, with x' =x + [lambda'/2], generate for each j = 1,...,M a sample value kj of the Poisson random variable P(aj(x'), tau). Compute the actual state change, lambda = sumj( kjvj ), and effect the leap by replacing t by t + tau and x by x + lambda.
Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section VI.
k-alpha leaping method
Alternative to the tau-leaping [http://identifiers.org/biomodels.kisao/KISAO_0000039], where one leaps a fixed number of reaction-events.
Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section VIII.
nonnegative Poisson tau-leaping method
modified poisson tau-leaping
The explicit tau-leaping procedure attempts to speed up the stochastic simulation of a chemically reacting system by approximating the number of firings of each reaction channel during a chosen time increment Tau as a Poisson random variable. Since the Poisson random variable can have arbitrarily large sample values, there is always the possibility that this procedure will cause one or more reaction channels to fire so many times during Tau that the population of some reactant species will be driven negative. Two recent papers have shown how that unacceptable occurrence can be avoided by replacing the Poisson random variables with binomial random variables, whose values are naturally bounded. This paper describes a modified Poisson tau-leaping procedure that also avoids negative populations, but is easier to implement than the binomial procedure. The new Poisson procedure also introduces a second control parameter, whose value essentially dials the procedure from the original Poisson tau-leaping at one extreme to the exact stochastic simulation algorithm at the other; therefore, the modified Poisson procedure will generally be more accurate than the original Poisson procedure [http://identifiers.org/biomodels.kisao/KISAO_0000040].
Cao Y, Gillespie DT, Petzold LR. Avoiding negative populations in explicit Poisson tau-leaping. Journal of Chemical Physics, Vol. 123, 4104 (2005).
Fehlberg method
RKF45
Runge-Kutta-Fehlberg method
The method was developed by the German mathematician Erwin Fehlberg and is based on the class of Runge-Kutta methods. The Runge-Kutta-Fehlberg method uses an O(h4) method together with an O(h5) method that uses all of the points of the O(h4) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation that would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration of ordinary differential equations.
Erwin Fehlberg (1970). Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing (Arch. Elektron. Rechnen), vol. 6, pp. 61-71.
Takahashi K, Kaizu K, Hu B, Tomita M. A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics volume 20(4), pages 538-46 (2004).
ECell3
Matlab
Dormand-Prince method
DOPRI
Prince-Dormand method
Dormand-Prince is an explicit method for the numerical integration of ODES with a given initial value. It is an 'embedded Runge-Kutta method' [http://identifiers.org/biomodels.kisao/KISAO_0000302] RK5 (4) which (a) has a 'small' principal truncation term in the Fifth order and (b) has an extended region of absolute stability.
Dormand J, Prince P. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, Volume 6, pages 19-26 (1980).
LSODA
Livermore solver for ordinary differential equations with automatic method switching
LSODA solves systems dy/dt = f with a dense or banded Jacobian when the problem is stiff, but it automatically selects between non-stiff (Adams [http://identifiers.org/biomodels.kisao/KISAO_0000289]) and stiff (BDF [http://identifiers.org/biomodels.kisao/KISAO_0000288]) methods. It uses the non-stiff method initially, and dynamically monitors data in order to decide which method to use.
Petzold LR. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. SIAM Journal on Scientific and Statistical Computing, Vol. 4 (1), pages 136-148 (1983).
A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
Petzold LR, Hindmarsh, AC. LSODAR: Livermore solver of ordinary differential equations with automatic method switching and root finding, Computing and Mathematics Research Division, 1-316 Lawerence Livermore National Laboratory (1987).
LSODAR
Livermore solver for ordinary differential equations with automatic method switching and root finding
ordinary differential equation solver for stiff or non-stiff systems with root finding
LSODAR is a variant of LSODA [http://identifiers.org/biomodels.kisao/KISAO_0000088] with a root finding capability added. Thus it solves problems dy/dt = f with dense or banded Jacobian and automatic method selection, and at the same time, it finds the roots of any of a set of given functions of the form g(t,y). This is often useful for finding stop conditions, or for finding points at which a switch is to be made in the function f.
LSODI
Livermore solver for ordinary differential equations, implicit version
LSODI solves systems given in linearly implicit form, including differential-algebraic systems.
LSODIS
Livermore solver for ordinary differential equations, implicit sparse version
LSODIS is a set of general-purpose FORTRAN routines solver for the initial value problem for ordinary differential equation systems. It is suitable for both stiff and nonstiff systems. LSODIS treat systems in the linearly implicit form A(t,y) dy/dt = g(t,y), A = a square matrix, i.e. with the derivative dy/dt implicit, but linearly so.
Seager M, Balsdon S. LSODIS - A sparse implicit ODE solver. 10th World Congress on System Simulation and Scientific Computation, pages 437-439 (1983).
LSODPK
Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration
LSODPK is a set of FORTRAN subroutines for solving the initial value problem for stiff and nonstiff systems of ordinary differential equations. In solving stiff systems, LSODPK uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods [http://identifiers.org/biomodels.kisao/KISAO_0000354]. The user must select the desired Krylov method and supply a pair of routine to evaluate, preprocess, and solve the (left and/or right) preconditioner matrices. Aside from preconditioning, the implementation is matrix-free, meaning that explicit storage of the Jacobian (or related) matrix is not required. The method is experimental because the scope of problems for which it is effective is not well-known, and users are forewarned that LSODPK may or may not be competitive with traditional methods on a given problem. LSODPK also includes an option for a user-supplied linear system solver to be used without Krylov iteration.
Livermore solver
Method to solve ordinary differential equations developed at the Lawrence Livermore National Laboratory.
sub-volume stochastic reaction-diffusion algorithm
Stochastic method using a combination of discretisation of compartment volumes into voxels and Gillespie-like algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000241] to simulate the evolution of the system.
modelling and simulation algorithm characteristic
modeling and simulation algorithm characteristic
Simulation algorithm property, which can, for example, describe the model, such as the type of variables (discrete or continuous), and information on the treatment of spatial descriptions, or can be a numerical characteristic, such as the system's behaviour (deterministic or stochastic) as well as the progression mechanism (fixed or adaptive time steps).
type of variable
Type of variables used for the simulation.
type of system behaviour
A characteristic describing the rules the algorithm uses to simulate the temporal evolution of a system, specifically whether or not the final state is uniquely determined from a precise initial state.
type of progression time step
Type of time steps used by the algorithm.
spatial description
Algorithm, possessing this characteristic, takes into account the location of the reacting components.
deterministic system behaviour
Algorithm, possessing this characteristic, simulates the temporal evolution of a system deterministically, so that from a precise initial state the algorithm will always end up in the same final state.
stochastic system behaviour
Algorithm, possessing this characteristic, simulates the temporal evolution of a system using probabilistic rules, so that between two simulations, the same precise initial state may result in a different final state.
discrete variable
Algorithm, possessing this characteristic, allows values of system's variables to change by discrete (integral) amounts.
continuous variable
Algorithm, possessing this characteristic, allows the values of a system's variables to change by continuous (non-integral) amounts.
progression with adaptive time step
Algorithm, possessing this characteristic, does not use fixed time steps to update the state of a system during the whole simulation, but on the contrary adapts the length of the time steps to the local situation.
progression with fixed time step
Algorithm, possessing this characteristic, uses time steps of constant length to update the state of a system during the whole simulation.
modelling and simulation algorithm parameter
modeling and simulation algorithm parameter
Parameter that can be used in the simulation experiment settings.
particle number lower limit
This parameter of 'Pahle hybrid method' [http://identifiers.org/biomodels.kisao/KISAO_0000231] 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 'particle number lower limit' cannot be higher than the 'particle number upper limit' [http://identifiers.org/biomodels.kisao/KISAO_0000204].
particle number upper limit
This parameter of 'Pahle hybrid method' [http://identifiers.org/biomodels.kisao/KISAO_0000231] 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 'particle number upper limit' cannot be lower than the 'particle number lower limit' [http://identifiers.org/biomodels.kisao/KISAO_0000203].
partitioning interval
This positive integer value specifies after how many steps the internal partitioning of the system should be recalculated.
relative tolerance
RTOL
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 accurately.
absolute tolerance
ATOL
This parameter is a positive numeric value specifying the desired absolute tolerance the user wants to achieve.
use reduced model
integrate reduced model
Boolean value which indicates whether the simulation/analysis should be performed using the complete model or an equivalent reduced model. Reduced models can be determined in many ways such as using mass conservation laws.
Jacek Puchałka, Andrzej M Kierzek. Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophys J. 2004, 86(3):1357-1372.
maximum Adams order
Adams max order
maximum non-stiff order
This parameter is a positive integer value specifying the maximal order the non-stiff Adams integration method [http://identifiers.org/biomodels.kisao/KISAO_0000289] shall attempt before switching to the stiff BDF method [http://identifiers.org/biomodels.kisao/KISAO_0000288].
maximum BDF order
BDF max order
maximum stiff order
This parameter is a positive integer value specifying the maximal order the stiff BDF integration method [http://identifiers.org/biomodels.kisao/KISAO_0000288] shall attempt before switching to smaller internal step sizes.
number of history bins
The 'number of history bins' is only enabled for models that contain delayed or multistep reactions for specifying the granularity with which the delayed reaction solver should retain the history of species values, for species that participate in delayed reactions.
tau-leaping epsilon
epsilon
tolerance
The leap condition is chosen such that the expected change in the propensity function aj(x) is bounded by Epsilon * a0 where Epsilon is an error control parameter between 0 and 1. This parameter is the basic error control mechanism for the Tau-Leaping algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000039]. As Epsilon decreases the leaps become shorter and the simulation is more accurate.
minimum reactions per leap
threshold
'minimum reactions per leap' parameter is used in hybrid methods, which adaptively switch between the tau-leaping algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000039] to the SSA Direct Method [http://identifiers.org/biomodels.kisao/KISAO_0000029] when the number of reactions in a single tau-leaping leap step is less than the threshold.
Pahle hybrid method
The hybrid method combines the stochastic 'Gibson-Bruck's next reaction method' [http://identifiers.org/biomodels.kisao/KISAO_0000027] with different algorithms for the numerical integration of ODEs [http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000374]. The biochemical network is dynamically partitioned into a deterministic and a stochastic subnet depending on the current particle numbers in the system. The user can define limits for when a particle number should be considered low or high. The stochastic subnet contains reactions involving low numbered species as substrate or product. All the other reactions form the deterministic subnet. The two subnets are then simulated in parallel using the stochastic and deterministic solver, respectively. The reaction probabilities in the stochastic subnet are approximated as constant between two stochastic reaction events.
COPASI
COPASI--a COmplex PAthway SImulator. Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, Kummer U. Bioinformatics. 2006 Dec 15;22(24):3067-74. Epub 2006 Oct 10.
LSOIBT
Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment
LSOIBT solves linearly implicit systems in which the matrices involved are all assumed to be block-tridiagonal. Linear systems are solved by the LU method.
LSODES
Livermore solver for ordinary differential equations with general sparse Jacobian matrix
LSODES solves systems dy/dt = f and in the stiff case treats the Jacobian matrix in general sparse form. It determines the sparsity structure on its own, or optionally accepts this information from the user. It then uses parts of the Yale Sparse Matrix Package (YSMP) to solve the linear systems that arise, by a sparse (direct) LU factorization/backsolve method.
LSODKR
Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding.
LSODKR is an initial value ODE solver for stiff and nonstiff systems. It is a variant of the LSODPK [http://identifiers.org/biomodels.kisao/KISAO_0000093] and LSODE [http://identifiers.org/biomodels.kisao/KISAO_0000071] solvers, intended mainly for large stiff systems. The main differences between LSODKR and LSODE [http://identifiers.org/biomodels.kisao/KISAO_0000071] are the following: a) for stiff systems, LSODKR uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods. The user must supply routines for the preconditioning operations, b) within the corrector iteration, LSODKR does automatic switching between functional (fixpoint) iteration and modified Newton iteration, c) LSODKR includes the ability to find roots of given functions of the solution during the integration.
type of solution
A characteristic describing the type of the solution produced by the method, specifically whether it is exact or approximate.
exact solution
Algorithm, possessing this characteristic, provides an exact solution to the initial problem.
approximate solution
Approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial time exact algorithms solving NP-hard problems, one settles for polynomial time sub-optimal solutions. Unlike heuristics, which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run time bounds. Ideally, the approximation is optimal up to a small constant factor (for instance within 5% of the optimal solution). Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size.
type of method
A characteristic describing the way the method finds a solution, specifically whether it solves an equation involving only the current state of the system (explicit) or both the current and the later one (implicit).
explicit method type
Explicit methods calculate the state of a system at a later time from the state of the system at the current time. Mathematically, if Y(t) is the current system state and Y((t+delta t) is the state at the later time (delta t is a small time step), then, for an explicit method Y(t+delta t) = F(Y(t)), to find Y(t+delta t).
implicit method type
Implicit methods find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if Y(t) is the current system state and Y((t+delta t) is the state at the later time (delta t is a small time step), then, for an implicit method one solves an equation G(Y(t), Y(t+delta t))=0, to find Y(t+delta t).
Gillespie-like method
Stochastic simulation algorithm using an approach alike the one described in Gillespie's papers of 1976 and 1977.
error control parameter
Parameter controlling method accuracy.
method switching control parameter
Parameters describing threshold conditions for algorithms that switch between different methods.
granularity control parameter
Parameter controlling granularity.
tau-leaping delta
Tau-leaping delta specifies how close two symmetric transition rates must be before we classify them as in partial-equilibrium. Only applies to the implicit tau routine [http://identifiers.org/biomodels.kisao/KISAO_0000045].
critical firing threshold
nonnegative tau-leaping second control parameter
The 'nonnegative Poisson tau-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000084] is based on the fact that negative populations typically arise from multiple firings of reactions that are only a few firings away from consuming all the molecules of one of their reactants. To focus on those reaction channels, the modified tau-leaping algorithm introduces a second control parameter nc, a positive integer that is usually set somewhere between 5 and 20. Any reaction channel with a positive propensity function that is currently within nc firings of exhausting one of its reactants is then classified as a critical reaction. The modified algorithm chooses tau in such a way that no more than one firing of all the critical reactions can occur during the leap.
Parameter describing partitioning of the system.
partitioning control parameter
coarse-graining factor
The time in each Monte-Carlo iteration of 'binomial tau-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000074] is updated with the time increments tau=f/(a1+a2+...+aM). Here 1/(a1+a2+...+aM) is the averaged microscopic increment of the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] and f is a coarse-graining factor, controlling the speed-up.
Chatterjee A, Vlachos DG, Katsoulakis MA. Binomial distribution based tau-leap accelerated stochastic simulation. J Chem Phys. 2005;122(2):024112.
Brownian diffusion accuracy
Accuracy code of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057], which sets which neighbouring boxes are checked for potential bi-molecular reactions. Consider the reaction A + B -> C and suppose that A and B are within a binding radius of each other. This reaction will always be performed if A and B are in the same virtual box. If accuracy is set to at least 3, then it will also occur if A and B are in nearest-neighbour virtual boxes. If it is at least 7, then the reaction will happen if they are in nearest-neighbour boxes that are separated by periodic boundary conditions. And if it is 9 or 10, then all edge and corner boxes are checked for reactions, which means that no potential reactions are overlooked.
molecules per virtual box
Target molecules per virtual box is a parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057], which sets the box sizes so that the average number of molecules per box, at simulation initiation, is close to the requested number.
virtual box side length
The 'virtual box side length' is a parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. It requests the length of one side of a box.
surface-bound epsilon
A parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. Molecules that are bound to a surface are given locations that are extremely close to that surface. However, this position does not need to be exactly at the surface, and in fact it usually cannot be exactly at the surface due to round-off error. The tolerance for how far a surface-bound molecule is allowed to be away from the surface can be set with the epsilon statement.
neighbour distance
A parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. When a surface-bound molecule diffuses off of one surface panel, it can sometimes diffuse onto the neighbouring surface tile. It does so only if the neighbouring panel is declared to be a neighbour and also the neighbour is within a distance that is set with the neighbour distance statement.
virtual box size
Target size of virtual boxes for 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057].
Euler method
The Euler method, named after Leonhard Euler, is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
NFSim agent-based simulation method
A generalization a rule-based version of 'Gillespie's direct method' (SSA) [http://identifiers.org/biomodels.kisao/KISAO_0000029]. The method is guaranteed to produce the same results as the exact SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] by cycling over three primary steps. First, NFsim calculates the probability or propensity for each rule to take effect given the current molecular states. Second, it samples the time to the next reaction event and selects the corresponding reaction rule. Finally, NFsim executes the selected reaction by applying the rule and updating the molecular agents accordingly.
NFSim
Sneddon MW, Faeder JR and Emonet T. Efficient modelling, simulation and coarse-graining of biological complexity with NFsim. Nature Methods (2011) 8(2):177-83.
cellular automata update method
cellular automata
cellular spaces
cellular structures
homogeneous structures
iterative arrays
tessellation automata
tessellation structures
Cellular automata are mathematical idealizations of physical systems in which space and time are discrete, and physical quantities take on a finite set of discrete values. A cellular automaton consists of a regular uniform lattice (or ''array''), usually infinite in extent, with a discrete variable at each site (''cell''). A cellular automaton evolves in discrete time steps, with the value of the variable at one site being affected by the values of variables at sites in its ''neighbourhood'' on the previous time step. The neighbourhood of a site is typically taken to be the site itself and all immediately adjacent sites. The variables at each site are updated simultaneously (''synchronously''), based on the values of the variables in their neighbourhood at the preceding time step, and according to a definite set of ''local rules''.
Wolfram, Stephen (1983) Statistical mechanics of cellular automata. Reviews of Modern Physics 55 (3): 601-644.
hard-particle molecular dynamics
A collision-driven molecular dynamics algorithm for a system of non-spherical particles.
Aleksandar Donev, Salvatore Torquato, Frank H. Stillinger, Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details, Journal of Computational Physics, v.202 n.2, p.737-764, 20 January 2005
first-passage Monte Carlo algorithm
AED DKMC
AED diffusion kinetic Monte Carlo method
asynchronous event-driven diffusion Monte Carlo
We present a novel Monte Carlo algorithm for N diffusing finite particles that react on collisions. Using the theory of first-passage processes and time dependent Green's functions, we break the difficult N-body problem into independent single- and two-body propagations circumventing numerous diffusion hops used in standard Monte Carlo simulations. The new algorithm is exact, extremely efficient, and applicable to many important physical situations in arbitrary integer dimensions.
T. Oppelstrup, V. V. Bulatov, G. H. Gilmer, M. H. Kalos, and B. Sadigh. First-passage Monte- Carlo algorithm: Diffusion without all the hops. Physical Review Letters, 97(23):230602, 2006.
Gill method
Gill's method
Runge-Kutta-Gill method
Gill's fourth order method is a Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y); y(x0) = y0 which evaluates the integrand,f(x,y), four times per step. This method is a fourth order procedure for which Richardson extrapolation can be used.
Gill, S. 1951. A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Cambridge Philos. Soc., 47, pp 96-108.
CompuCell3D
Metropolis Monte Carlo algorithm
Metropolis algorithm
Metropolis-Hastings algorithm
A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration [http://identifiers.org/biomodels.kisao/KISAO_0000051] over configuration space.
Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). Equations of State Calculations by Fast Computing Machines. Journal of Chemical Physics 21 (6): 1087-1092.
Adams-Bashforth method
explicit Adams method
Given an initial value problem: y' = f(x,y), y(x0) = y0 together with additional starting values y1 = y(x0 + h), . . . , yk-1 = y(x0 + (k-1) h) the k-step Adams-Bashforth method is an explicit linear multistep method that approximates the solution, y(x) at x = x0+kh, of the initial value problem by yk = yk - 1 + h * ( a0 f(xk - 1,yk - 1) + a1 f(xk - 2,yk - 2) + . . . + ak - 1 f(x0,y0) ), where a0, a1, . . . , ak - 1 are constants.
Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
Adams-Moulton method
implicit Adams method
The (k-1)-step Adams-Moulton method is an implicit linear multistep method that iteratively approximates the solution, y(x) at x = x0+kh, of the initial value problem by yk = yk - 1 + h * ( b0 f(xk,yk) + b1 f(xk - 1,yk - 1) + . . . + bk - 1 f(x1,y1) ), where b1, . . . , bk - 1 are constants.
multistep method
multi-value method
A numerical method for differential equations which is based on several values of the solution.
KINSOL
FKINSOL
NKSOL
Newton-Krylov solver for nonlinear algebraic systems
KINSOL solves algebraic systems in real N-space, written as F(u)=0, F:RN->RN, given an initial guess u0. The basic method is either a modified or an inexact Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408]. The linear systems that arise are solved with either a direct (dense or banded) solver (serial version only), or one of the Krylov iterative solvers [http://identifiers.org/biomodels.kisao/KISAO_0000354]. In the Krylov case, the user can (optionally) supply a right preconditioner.
BROWN, P. N. AND SAAD, Y. 1990. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11, 450-481.
Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
IDA
implicit differential-algebraic solver
solver for differential-algebraic equation systems
IDA solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0. At each step, a Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408] leads to linear systems Jx=b, which are solved by one of five methods - two direct (dense or band; serial version only) and three Krylov [http://identifiers.org/biomodels.kisao/KISAO_0000354] (GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000353], BiCGStab [http://identifiers.org/biomodels.kisao/KISAO_0000392], or TFQMR [http://identifiers.org/biomodels.kisao/KISAO_0000396]). IDA is written in C, but derived from the package DASPK [http://identifiers.org/biomodels.kisao/KISAO_0000355] which is written in Fortran.
BROWN, P. N., HINDMARSH, A. C., AND PETZOLD, L. R. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 1467-1488.
finite volume method
FVM
The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations, which attempts to emulate continuous conservation laws of physics.
Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA.
Euler-Maruyama method
stochastic Euler scheme
The Euler-Maruyama method is a method for the approximate numerical solution of a stochastic differential equation, which truncates the Ito and Stratonovich Taylor series of the exact solution after the first order stochastic terms. This converges to the Ito solution with strong global order accuracy 1/2 or weak global order accuracy 1. It is a simple generalization of the Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261] for ordinary differential equations to stochastic differential equations.
Burrage, K., Burrage, P.M. and Tian, T. (2004) Numerical methods for strong solutions of stochastic differential equations: an overview. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 460 (2041). pp. 373-402.
Milstein method
The Milstein method is a technique for the approximate numerical solution of a stochastic differential equation.
Milstein M. G. Numerical Integration of Stochastic Differential Equations (Mathematics and Its Applications) Springer, 2004.
backward differentiation formula
BDF
Gear method
Gear's method
The backward differentiation formulas (BDF) are implicit multistep methods based on the numerical differentiation of a given function and are wildly used for integration of stiff differential equations.
C. William Gear. 1971. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall PTR, Upper Saddle River, NJ, USA.
Adams method
Adams' methods are multi-step methods used for the numerical integration of initial value problems in Ordinary Differential Equations (ODE's). Adams' algorithm consists of two parts: firstly, a starting procedure which provides y1, ... , yk-1 ( approximations to the exact solution at the points x0 + h, ... , x0 + (k - 1)h ) and, secondly, a multistep formula to obtain an approximation to the exact solution y(x0 + kh). This is then applied recursively, based on the numerical approximation of k successive steps, to compute y(x0 + (k + 1)h).
Merson method
Kutta-Merson method
Merson's method
Runge-Kutta-Merson method
A five-stage Runge-Kutta method with fourth-order accuracy.
R.H. Merson, An operational method for the study of integration processes, in: Proceedings of the Symposium on Data Processing, Weapons Research Establishment, Salisbury, Australia, 1957, pp. 110-1-110-25.
Hammer-Hollingsworth method
The numerical integration of ordinary differential equations by the use of Gaussian quadrature methods.
P. C. Hammer & J. W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, MTAC, v. 9, 1955, pp. 92-96. MR 17, 302.
Lobatto method
implicit Runge-Kutta method based on Lobatto quadrature
There are three families of Lobatto methods, called IIIA, IIIB and IIIC. These are named after Rehuel Lobatto. All are implicit Runge-Kutta methods, have order 2s − 2 and they all have c1 = 0 and cs = 1.
J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp., v. 18, 1964, pp. 233-244.
Butcher-Kuntzmann method
Gauss method
From a theoretical point of view, the Butcher-Kuntzmann Runge-Kutta methods belong to the best step-by-step methods for nonstiff problems. These methods integrate first-order initial-value problems by means of formulas based on Gauss-Legendre quadrature, and combine excellent stability features with the property of superconvergence at the step points.
F. Ceschin, J. Kuntzmann, Problèmes différentiels de conditions initiales. Paris 1963. Dunod.
Butcher, J. C. (1964) Implicit Runge-Kutta processes. Math. Comput. 18, 50-64.
Heun method
Heun's method
The method is named after Karl L. W. M. Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It can be seen as extension of the Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261] into two-stage second-order Runge-Kutta method.
embedded Runge-Kutta method
embedded RK
An embedded Runge-Kutta method is a method in which two Runge-Kutta estimates are obtained using the same auxiliary functions ki but with a different linear combination of these functions so that one estimate has an order one greater than the other.
Zonneveld method
An embedded Runge-Kutta method [http://identifiers.org/biomodels.kisao/KISAO_0000302] of order 4(3), proposed by J.A. Zonneveld in 1964.
Zonneveld, J. A (1970). Automatic numerical integration Mathematisch Centrum, Amsterdam
Radau method
implicit Runge-Kutta method based on Radau quadrature
Implicit Runge-Kutta methods based on Radau quadrature.
J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comput., 18(1964), 233-244.
Verner method
Verner's method
The first high order (6(5)) embedded Runge-Kutta formulas that avoid the drawback of giving identically zero error estimates for quadrature problems y' = f(x) were constructed by Verner in 1978.
J.H. Verner, Some Runge-Kutta formula pairs, SIAM J. Numer. Anal 28 (1991) 496-511
Lagrangian sliding fluid element algorithm
BTEX
LSFEA
blood-tissue exchange method
Because the analytic solutions to the partial differential equations require convolution integration, solutions are obtained relatively efficiently by a fast numerical method. Our approach centers on the use of a sliding fluid element algorithm for capillary convection, with the time step set equal to the length step divided by the fluid velocity. Radial fluxes by permeation between plasma, interstitial fluid, and cells and axial diffusion exchanges within each time step are calculated analytically. The method enforces mass conservation unless there is regional consumption.
Bassingthwaighte JB, Chan IS, and Wang CY. Computationally efficient algorithms for capillary convection-permeation-diffusion models for blood-tissue exchange. Ann Biomed Eng 20: 687-725, 1992.
finite difference method
FDM
The finite difference method is based on local approximations of the partial derivatives in a Partial Differential Equation, which are derived by low order Taylor series expansions.
MacCormack method
In computational fluid dynamics, the MacCormack method is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite difference method [http://identifiers.org/biomodels.kisao/KISAO_0000307] is introduced by R. W. MacCormack in 1969.
MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).
Crank-Nicolson method
In numerical analysis, the Crank-Nicolson method is a finite difference method [http://identifiers.org/biomodels.kisao/KISAO_0000307] used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time, implicit in time, and is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century.
Crank, J.; Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Phil. Soc. 43 (1): 50-67.
method of lines
MOL
NMOL
NUMOL
The method of lines is a general technique for solving partial differential equations (PDEs) by typically using finite difference relationships for the spatial derivatives and ordinary differential equations for the time derivative.
Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press.
type of domain geometry handling
S-System power-law canonical differential equations solver
ESSYNS GMA
Ordinary differential equations can be recast into a nonlinear canonical form called an S-system. Evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions, recognized as among the most general, are special cases of S-systems. Identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations. By capitalizing on the regular structure of S-systems, efficient formulas for a variable-order, variable-step Taylor-series method are developed.
Irvine, D.H. and Savageau, M.A., Efficient solution of nonlinear ordinary differential equations expressed in S-System canonical form, Siam. J. Numer. Anal., 27:704-735, 1990.
lattice gas automata
LGA
LGCA
lattice gas cellular automata
Lattice gas automata methods are a series of cellular automata methods used to simulate fluid flows. From the LGCA, it is possible to derive the macroscopic Navier-Stokes equations.
Dieter A. Wolf-Gladrow (2000). Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer.
enhanced Greens function reaction dynamics
eGFRD
GFRD [http://identifiers.org/biomodels.kisao/KISAO_0000058] decomposes the multi­body reaction diffusion problem to a set of single and two body problems. Analytical solutions for two body reaction diffusion are available via Smoluchowski equation. eGFRD allows to solve each sub­problem asynchronously by introducing the concept of first passage processes.
K. Takahashi, An exact Brownian dynamics method for cell simulation, in Computational Methods in Systems Biology, vol. 5307/2008.
E-Cell multi-algorithm simulation method
A modular meta-algorithm with a discrete event scheduler that can incorporate any type of time-driven simulation algorithm. It was shown that this meta-algorithm can efficiently drive simulation models with different simulation algorithms with little intrusive modification to the algorithms themselves. Only a few additional methods to handle communications between computational modules are required.
Takahashi, K., Kaizu, K., Hu, B., and Tomita, M., A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics, in press.
Gauss-Legendre Runge-Kutta method
Open Formula
So called 'Open Formula', two points formula, three points formula, four points formula, five points formula and six points formula of the Runge-Kutta method to solve the initial value problem of the ordinary differential equation. These formulas use the points and weights from the Gauss-Legendre Quadrature formulas for finding the value of the definite integral.
Maitree Podisuk, Sirirat Khuntidilokwongsa, and Witchaya Rattanametawee. 2006. Gauss-Legendre Quadrature formula in Runge-Kutta method with modified model of Newton cooling law. In Proceedings of the 8th WSEAS international conference on Mathematical methods and computational techniques in electrical engineering (MMACTEE'06). World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, 312-317.
Monte Carlo method
Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results.
Metropolis, N.; Ulam, S. (1949). "The Monte Carlo Method". Journal of the American Statistical Association (American Statistical Association) 44 (247): 335-341.
BioRica
BioRica hybrid method
The simulation schema for a given BioRica node is given by a hybrid algorithm that deals with continuous time and allows for discrete events that roll back the time according to these discrete interruptions.
Cvijovic M, Soueidan H, Sherman DJ, Klipp E, Nikolski M. Exploratory simulation of cell ageing using hierarchical models. Genome Inform. 2008;21:114-25.
Cash-Karp method
An family of explicit Runge-Kutta formulas, which are very efficient for problems with smooth solution as well as problems having rapidly varying solutions. Each member of this family consists of a fifty-order formula that contains embedded formulas of all orders 1 through 4. By computing solutions at several different orders, it is possible to detect sharp fronts or discontinuities before all the function evaluations defining the full Runge-Kutta step have been computed.
J. R. Cash, A. H. Karp. "A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides", ACM Transactions on Mathematical Software 16: 201-222, 1990.
hybridity
The basic idea of hybrid simulation methods is to combine the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time.
equation-free probabilistic steady-state approximation
We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. <...> The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.
Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106.
nested stochastic simulation algorithm
nested SSA
This multiscale method is a small modification of the Gillespie's direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029], in the form of a nested SSA, with inner loops for the fast reactions, and outer loop for the slow reactions. The number of groups can be more than two, and the grouping into fast and slow variables can be done dynamically in an adaptive version of the scheme.
E W., Liu D. and Vanden-Eijnden E. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys. 2005 Nov 15;123(19):194107.
minimum fast/discrete reaction occurrences number
Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which describes the minimum number of fast/discrete reaction occurrences before their effects cause convergence to a quasi-steady-state distribution.
number of samples
Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which determines the number of samples taken from the distribution.
maximum discrete number
Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which controls the maximum number of molecules of some reactant species in order for the reaction to be considered discrete.
minimum fast rate
Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which controls the minimum rate of the reaction in order for it to be considered fast.
constant-time kinetic Monte Carlo algorithm
SSA-CR
The computational cost of the original SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] scaled linearly with the number of reactions in the network. Gibson and Bruck developed a logarithmic scaling version of the SSA which uses a priority queue or binary tree for more efficient reaction selection [http://identifiers.org/biomodels.kisao/KISAO_0000027]. More generally, this problem is one of dynamic discrete random variate generation which finds many uses in kinetic Monte Carlo and discrete event simulation. We present here a constant-time algorithm, whose cost is independent of the number of reactions, enabled by a slightly more complex underlying data structure.
Slepoy A, Thompson AP, Plimpton SJ. A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. J Chem Phys. 2008 May 28;128(20):205101.
R-leaping algorithm
R-leap method
A novel algorithm is proposed for the acceleration of the exact stochastic simulation algorithm by a predefined number of reaction firings (R-leaping) that may occur across several reaction channels. In the present approach, the numbers of reaction firings are correlated binomial distributions and the sampling procedure is independent of any permutation of the reaction channels. This enables the algorithm to efficiently handle large systems with disparate rates, providing substantial computational savings in certain cases.
Auger A, Chatelain P, Koumoutsakos P. R-leaping: accelerating the stochastic simulation algorithm by reaction leaps. J Chem Phys. 2006 Aug 28;125(8):084103.
exact R-leaping algorithm
ER-leap method
exact R-leap method
exact accelerated stochastic simulation algorithm
We present a SSA which, similar to R-leap [http://identifiers.org/biomodels.kisao/KISAO_0000330], accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] by executing multiple reactions per algorithmic step, but which samples the reactant trajectories from the same probability distribution as the SSA. This 'exact R-leap' or 'ER-leap' algorithm is a modification of the R-leap algorithm which is both exact and capable of substantial speed-up over SSA.
Eric Mjolsness, David Orendorff, Philippe Chatelain, and Petros Koumoutsakos. An exact accelerated stochastic simulation algorithm. J Chem Phys. 2009 April 14; 130(14): 144110.
ER-leap initial leap
L (initial step) is a parameter of 'exact R-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000331]. ''We will assume that the reaction event to be bounded occurs within a run of L events in the SSA algorithm[http://identifiers.org/biomodels.kisao/KISAO_0000029], in order to execute L reactions at once in the manner of the R-leap algorithm[http://identifiers.org/biomodels.kisao/KISAO_0000230]''.
accelerated stochastic simulation algorithm
accelerated SSA
An algorithm, which accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] either at the expense of its accuracy or exact.
multiparticle lattice gas automata
multiparticle lattice gas cellular automata
An algorithm which allows for an arbitrary number of particles, while keeping the benefits of the cellular automata approach [http://identifiers.org/biomodels.kisao/KISAO_0000315].
Chopard B., et al.Multiparticle lattice gas automata for reaction diffusion systems. Int. J. Mod. Phys. C 1994;5:47-63.
generalized stochastic simulation algorithm
Gillespie direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029] follows unit-by-unit changes in the total numbers of each reactant species, it is especially well suited to the study of systems in which reactant densities are low and the application of methods based on continuum approximations, such as the traditional ordinary differential equations of chemical kinetics, is questionable. The 'generalized stochastic simulation algorithm' branch presents methods, which extend Gillespie direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029] to suit to systems with other characteristics.
D-leaping method
We propose a novel, accelerated algorithm for the approximate stochastic simulation of biochemical systems with delays. The present work extends existing accelerated algorithms by distributing, in a time adaptive fashion, the delayed reactions so as to minimize the computational effort while preserving their accuracy.
Bayati B, Chatelain P, Koumoutsakos. D-leaping: Accelerating stochastic simulation algorithms for reactions with delays. Journal of Computational Physics. 2009 Sept;228(16):5909-5918.
finite element method
FEA
FEM
finite element analysis
A numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261], Runge-Kutta [http://identifiers.org/biomodels.kisao/KISAO_0000064], etc.
Pelosi G. The finite-element method, Part I: R. L. Courant: Historical Corner. IEEE Antennas and Propagation Magazine, Vol. 49, No. 2. (April 2007), pp. 180-182.
h-version of the finite element method
h-FEM
h-method
Classical form of the 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337], in which polynomials of fixed degree p are used and the mesh is refined to increase accuracy. Can be considered as a special case of the h-p version [http://identifiers.org/biomodels.kisao/KISAO_0000340].
Babuska, I., T. Strouboulis, A. Mathur, C. S. Upadhyay, Pollution error in the h-version of the finite element method and the local quality of a-posteriori error estimators, Finite Elements in Analysis and Design, 17: 273-321, 1994.
p-version of the finite element method
p-FEM
p-method
The p version of 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] uses a fixed mesh but increases the polynomial degree p to increase accuracy. Can be considered as a special case of the h-p version [http://identifiers.org/biomodels.kisao/KISAO_0000340].
Babuska, I., B.A. Szabo and I.N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal., 18: 515-545, 1981.
h-p version of the finite element method
hp-FEM
hp-method
In h-p version of 'finite difference method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] the two approaches of mesh refinement and degree enchacement are combined.
Babuska, I., and B. Guo, The h-p version of the finite element method, Comput. Mech., 1: 203-220, 1986.
mixed finite element method
A 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] in which both stress and displacement fields are approximated as primary variables.
D. N. Arnold. 1990. Mixed finite element methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 82, 1-3 (September 1990), 281-300.
level set method
LSM
level-set method
An algorithm for moving surfaces under their curvature. This algorithm rely on numerically solving Hamilton-Jacobi equations with viscous terms, using approximation techniques from hyperbolic conservation laws.
Osher, S.; Sethian, J. A. (1988), "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations", J. Comput. Phys. 79: 12-49.
generalized finite element method
GFEM
PUM
partition of unity method
The GFEM is a generalization of the classical 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] — in its h [http://identifiers.org/biomodels.kisao/KISAO_0000338], p [http://identifiers.org/biomodels.kisao/KISAO_0000339], and h-p versions [http://identifiers.org/biomodels.kisao/KISAO_0000340]— as well as of the various forms of meshless methods used in engineering.
Babuska, Ivo; Uday Banerjee, John E. Osborn (June 2004). Generalized Finite Element Methods: Main Ideas, Results, and Perspective. International Journal of Computational Methods 1 (1): 67-103.
h-p cloud method
h-p clouds
method of clouds
A meshless method, which uses a partition of unity to construct the family of h-p cloud functions.
Duarte, C. A. and Oden, J. T. (1996), H-p clouds—an h-p meshless method. Numerical Methods for Partial Differential Equations, 12: 673-705.
mesh-based geometry handling
In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc.
meshless geometry handling
Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis.
extended finite element method
X-FEM
XFEM
A numerical method to model arbitrary discontinuities in continuous bodies that does not require the mesh to conform to the discontinuities nor significant mesh refinement near singularities. In X-FEM the standard finite element approximation [http://identifiers.org/biomodels.kisao/KISAO_0000337] is enriched and the approximation space is extended by an additional family of functions.
Stazi, F. L.; Budyn, E.; Chessa, J.; Belytschko, T. An extended finite element method with higher-order elements for curved cracks. Computational Mechanics, Volume 31, Issue 1-2, pp. 38-48 (2003).
method of finite spheres
MFS
Method of finite spheres is truly meshless in the sense that the nodes are placed and the numerical integration is performed without a mesh. Some of the novel features of the method of finite spheres are the numerical integration scheme and the way in which the Dirichlet boundary conditions are incorporated.
S. De and K. J. Bathe. The method of finite spheres. Computational Mechanics, Volume 25, Number 4, 329-345.
probability-weighted dynamic Monte Carlo method
PW-DMC
probability-weighted DMC
We have developed a probability-weighted DMC method by incorporating the weighted sampling algorithm of equilibrium molecular simulations. This new algorithm samples the slow reactions very efficiently and makes it possible to simulate in a computationally efficient manner the reaction kinetics of physical systems in which the rates of reactions vary by several orders of magnitude.
Resat H, Wiley HS, Dixon DA. Probability-weighted dynamic Monte Carlo method for reaction kinetics simulations. J Phys Chem B. 2001;105:11026-11034.
multinomial tau-leaping method
MtauL
The multinomial tau-leaping method is an extension of the binomial tau-leaping method [http://identifiers.org/biomodels.kisao/KISAO_0000074] to networks with arbitrary multiple-channel reactant dependencies. Improvements were achieved by a combination of three factors: First, tau-leaping steps are determined simply and efficiently using a-priori information and Poisson distribution based estimates of expectation values for reaction numbers. Second, networks are partitioned into closed groups of reactions and corresponding reactants in which no group reactant set is found in any other group. Third, product formation is factored into upper bound estimation of the number of times a particular reaction occurs.
Pettigrew MF, Resat H. Multinomial tau-leaping method for stochastic kinetic simulations. J Chem Phys. 2007 Feb 28;126(8):084101.
hybrid method
A simulation methods which combines the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time.
generalized minimal residual algorithm
GMRES
An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The generalized minimal residual method extends the minimal residual method (MINRES) [http://identifiers.org/biomodels.kisao/KISAO_0000388], which is only applicable to symmetric systems, to non-symmetric systems.
Youcef Saad and Martin H. Schultz, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput. 7, 856 (1986).
Krylov subspace projection method
Krylov subspace method
Krylov subspace method is an iterative linear equation method, which builds up Krylov subspaces and look for good approximations to eigenvectors and invariant subspaces within the Krylov spaces.
DASPK
DDASPK
SDASPK
differential algebraic system solver with Krylov preconditioning
In DASPK, we have combined the time-stepping methods of DASSL [http://identifiers.org/biomodels.kisao/KISAO_0000255] with preconditioned iterative method GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000386], for solving large-scale systems of DAEs of the form F(t, y, y') = 0, where F, y, y' are N-dimensional vectors, and a consistent set of initial conditions y(t0) = y0, y'(t0) = y'0 is given. DASPK is written in Fortran.
Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 6 (November 1994), 1467-1488.
NARROW
DASSL
DDASSL
SDASSL
differential algebraic system solver
DASSL is designed for the numerical solution of implicit systems of differential/algebraic equations written in the form F(t,y,y')=0, where F, y, and y' are vectors, and initial values for y and y' are given.
Linda R. Petzold: A Description of DASSL: A Differential/Algebraic System Solver, SAND82-8637 (September 1982).
conjugate gradient method
Conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too large to be handled by direct methods. Such systems often arise when numerically solving partial differential equations.
M. R. Hestenes and E. L. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49 (1952), p. 409.
biconjugate gradient method
BCG
Bi-CG
BiCG
The biconjugate gradient method provides a generalization of conjugate gradient method [http://identifiers.org/biomodels.kisao/KISAO_0000357] to non-symmetric matrices.
Fletcher, R. (1976). "Conjugate gradient methods for indefinite systems". Numerical Analysis. Lecture Notes in Mathematics (Springer Berlin / Heidelberg) 506: 73-89.
implicit-state Doob-Gillespie algorithm
The algorithm uses a representation of the system together with a super-approximation of its ‘event horizon’ (all events that may happen next), and a specific correction scheme to obtain exact timings. Being completely local and not based on any kind of enumeration, this algorithm has a per event time cost which is independent of (i) the size of the set of generable species (which can even be infinite), and (ii) independent of the size of the system (ie, the number of agent instances). The algorithm can be refined, using concepts derived from the classical notion of causality, so that in addition to the above one also has that the even cost is depending (iii) only logarithmically on the size of the model (ie, the number of rules).
V. Danos, J. Feret, W. Fontana, and J. Krivine. 2007. Scalable simulation of cellular signalling networks. In Proceedings of the 5th Asian conference on Programming languages and systems (APLAS'07), Zhong Shao (Ed.). Springer-Verlag, Berlin, Heidelberg, 139-157.
rule-based simulation method
Rule-based models provide a powerful alternative to approaches that require explicit enumeration of all possible molecular species of a system. Such models consist of formal rules governing interactive behaviour. Rule-based simulation methods simulate such models.
Adams predictor-corrector method
The combination of evaluating a single explicit integration method ('Adams-Bashforth method' [http://identifiers.org/biomodels.kisao/KISAO_0000279]) (the predictor step) in order to provide a good initial guess for the successive evaluation of an implicit method ('Adams-Moulton method' [http://identifiers.org/biomodels.kisao/KISAO_0000280]) (the corrector step) using iteration.
Moulton, Forest R. (1926), New methods in exterior ballistics, University of Chicago Press.
Mathematica
NDSolve method
The Mathematica computation system function NDSolve is a general numerical differential equation solver. It can handle a wide range of ordinary differential equations as well as some partial differential equations. NDSolve can also solve some differential-algebraic equations, which are typically a mix of differential and algebraic equations.
Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica Tutorial Collection, 2008.
symplecticness
Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness (phase volume preservation).
J. M. Sanz-Serna (1992). Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica, 1, pp 243-286
partitioned Runge-Kutta method
PRK
SPRK
symplectic partitioned Runge-Kutta method
If a Hamiltonian system possesses a natural partitioning, it is possible to integrate its certain components using one Runge-Kutta method and other components using a different Runge-Kutta method. The overall s-stage scheme is called a partitioned Runge-Kutta method.
P. Rentrop (1985): Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math., Vol. 47, p.545-564.
partial differential equation discretization method
A method which solves partial differential equations by discretizing them, i.e. approximating them by equations that involve a finite number of unknowns.
type of problem
A characteristic describing the type of the problems which can be solved by the algorithm.
stochastic differential equation problem
SDE problem
partial differential equation problem
PDE problem
differential-algebraic equation problem
DAE
ordinary differential equation problem
ODE problem
delay differential equation problem
DDE problem
linearity of equation
Linear differential equations are of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function of the same nature as y.
one-step method
A numerical method for differential equations which uses one starting value at each step.
implicit midpoint rule
implicit Gaussian second order Runge-Kutta method
The implicit midpoint rule is a second-order case of the more general implicit s-stage Runge-Kutta methods [http://identifiers.org/biomodels.kisao/KISAO_0000064 and (http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000240)].
Bulirsch-Stoer algorithm
GBS
Gragg-Bulirsch-Stoer algorithm
The Bulirsch-Stoer method is an adaptive method which uses Gragg's modified midpoint method [http://identifiers.org/biomodels.kisao/KISAO_0000382] to estimate the solution of an initial value problem for various step sizes. The estimates are fit to a "diagonal" rational function or a polynomial as a function of the step size and the limit as the step size tends to zero is taken as the final estimate.
Bulirsch R. and Stoer J. (1966) Numerical treatment of ordinary differential equations by extrapolation methods, Numerische Mathematik, Volume 8, Number 1, 1-13.
Richardson extrapolation based method
A method based on ideas of Richardson extrapolation, which is a process for obtaining increased accuracy in a discretized approximation by extrapolating results from coarse discretizations to an arbitrarily fine one.
Richardson, L. F. (1911). The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London, Series A 210 (459-470): 307-357.
midpoint method
The midpoint method is an explicit method for approximating the solution of the initial value problem y' = f(x,y); y(x0) = y0 at x for a given step size h. For the midpoint method the derivative of y(x) is approximated by the symmetric difference y'(x) = ( y(x+h) - y(x-h) ) / 2h + O(h2).
modified midpoint method
Gragg's method
Gragg's modified midpoint method
The modified midpoint method is globally a second order method for approximating the solution of the initial value problem y' = f(x, y), y(x0) = y0, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of n substeps each of size h, h = H/n.
Bader-Deuflhard method
The Bader-Deuflhard method is an extrapolation method based on a semi-implicit discretization [http://identifiers.org/biomodels.kisao/KISAO_0000387]. It is a generalization of the Bulirsch-Stoer algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000379] for solving ordinary differential equations.
Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373-398, 1983.
semi-implicit midpoint rule
A semi-implicit version of the midpoint method that has an even error series [http://identifiers.org/biomodels.kisao/KISAO_0000381].
scaled preconditioned generalized minimal residual method
SPGMR
A scaled preconditioned version of 'generalized minimal residual algorithm' [http://identifiers.org/biomodels.kisao/KISAO_0000353]. For linear system Ax = b a preconditioner matrix P that approximates A is sought, for which linear system Px = b can be solved easily. Preconditioning is applied on the left only. Scaling is done using diagonal matrix D whose diagonal elements are weights w^i = rtol|y^i| +atol^i, where rtol is 'relative tolerance' [http://identifiers.org/biomodels.kisao/KISAO_0000209] and atol is 'absolute tolerance' [http://identifiers.org/biomodels.kisao/KISAO_0000211].
minimal residual method
MINRES
The 'minimal residual method' is an algorithm for the numerical solution of indefinite symmertic systems of linear equations.
Paige C.C., Saunders M.A. Solution of sparse indefinite systems of linear equations, SIAM Journal on Numerical Analysis (1975) Volume: 12, Issue: 4, Publisher: SIAM, Pages: 617-629.
quasi-minimal residual method
QMR
The QMR algorithm is a robust iterative solver for general nonsingular non-Hermitian linear systems. The method uses a robust implementation of the look-ahead Lanczos algorithm to generate basis vectors for the Krylov subspaces Kn(r0, A). The QMR iterates are characterized by a quasi-minimal residual property over Kn(r0, A).
Roland W. Freund and Noël M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. NUMERISCHE MATHEMATIK Volume 60, Number 1, 315-339.
biconjugate gradient stabilized method
Bi-CGSTAB
BiCGSTAB
An iterative method for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) [http://identifiers.org/biomodels.kisao/KISAO_0000358] and has faster and smoother convergence than the original BiCG.
Van der Vorst, H. A. (1992). Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing 13: 631-644.
ingenious conjugate gradients-squared method
CGS
A Lanczos-type method for nonsymmetric sparse linear systems. The method is based on a polynomial variant of the conjugate gradients algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000357]. Although related to the so-called bi-conjugate gradients (Bi-CG) algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000358], it does not involve adjoint matrix-vector multiplications, and the expected convergence rate is about twice that of the Bi-CG algorithm.
Peter Sonneveld. 1989. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 1 (January 1989), 36-52.
quasi-minimal residual variant of biconjugate gradient stabilized method
QMRCGSTAB
QMRCGSTAB is a quasi-minimal residual (QMR) variant of the Bi-CGSTAB algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000394] of van der Vorst for solving nonsymmetric linear systems. The motivation for the QMR variant is to obtain smoother convergence behavior of the underlying method.
T. F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto, and C. H. Tong. 1994. A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems. SIAM J. Sci. Comput. 15, 2 (March 1994), 338-347.
improved biconjugate gradient method
An 'improved biconjugate gradient method' branch contains algorithms which can be viewed as improvements over some of drawbacks of BCG [http://identifiers.org/biomodels.kisao/KISAO_0000358], such as (1) the need for matrix-vector multiplications with A^T (which can be inconvenient as well as doubling the number of matrix-vector multiplications compared to CG [http://identifiers.org/biomodels.kisao/KISAO_0000357] for each increase in the degree of the underlying Krylov subspace), (2) the possibility of breakdowns and (3) erratic convergence behavior.
transpose-free quasi-minimal residual algorithm
TFQMR
A version of CGS [http://identifiers.org/biomodels.kisao/KISAO_0000393] which 'quasi-minimizes' the residual in the space spanned by the vectors generated by the CGS iteration.
Roland W. Freund. 1993. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14, 2 (March 1993), 470-482.
preconditioning technique
Preconditioning is simply a means of transforming the original linear system into one which has the same solution, but which is likely to be easier to solve with an iterative solver.
Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM.
iterative method for solving a system of linear equations
homogeneousness of equation
Homogeneous equations are of the form Ly = 0, where the differential operator L is a linear operator and y is the unknown function.
symmetricity of matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
type of differential equation
Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM.
steady state root-finding method
Method for finding the root of a function (f(x) = 0).
Newton-type method
A method which attacks the solution of a nonlinear problem F(x) = 0 by solving a sequence of liner problems of the same kind. The solution of the system F(x)=0 can be interpreted as a steady state of a dynamic system x'(t)=F(x(t)). The Newton approach will only work if the fixed point [http://identifiers.org/biomodels.teddy/TEDDY_0000086] of the dinamic system is attractive [http://identifiers.org/biomodels.teddy/TEDDY_0000094].
Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
ordinary Newton method
A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which solves the general nonlinear problem F(x)=0 by applying successive linearization F'(x[k])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
simplified Newton method
A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which is characterized by keeping the initial derivative throughout the whole iteration: F'(x[0])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
Newton-like method
A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which is characterized by the fact that, in finite dimension, the Jacodian matrices are either replaced by some fixed 'close by' Jacobian F'(z) with z not equal to the initial guess x[0], or by some approximation so that: M'(x[0])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
inexact Newton method
iterative Newton method
truncated Newton method
For extremely large scale nonlinear problems the arising linear systems for the Newton corrections can no longer be solved directly ('exactly'), but must be solved iterativly ('inexactly) - which gives the name inexact Newton methods. The whole scheme then consists of an inner iteration (at Newton step k): F'(x[k])deltaxi[k]=-F(x[k])+ri[k], k=0,1,... xi[k+1]=x[k]+deltaxi[k], i=0,1,..,imax[k] in terms of residuals ri[k] and an outer iteration where, given x[0], the iterates are defined as x[k+1]=xi[k+1] for i=imax[k], k=0,1,...
exact Newton method
direct Newton method
Any of the finite dimensional Newton-type methods [http://identifiers.org/biomodels.kisao/KISAO_0000408] requires the numerical solution of the linear equations F'(x[k])deltax[k]=-F(x[k]). Whenever direct elimination methods are applicable, we speak of exact Newton methods.
maximum number of steps
maximum steps
The limit on number of internal steps before an output point.
Requested by Kristin Tøndel on Thursday, October 13, 2011 10:46:04 AM.
partial least squares regression method
PLSR
PLSR method
Multivariate regression method based on estimated latent variables. Related to Principal Component Analysis (PCA) and Principal Component Regression (PCR).
Wold S, Martens H, Wold H (1983) The multivariate calibration method in chemistry solved by the PLS method. In: Lecture notes in Mathematics, Matrix Pencils. Heidelberg: Springer-Verlag. pp. 286-293.
Requested by Kristin Tøndel on Thursday, October 13, 2011 11:13:17 AM.
hierarchical cluster-based partial least squares regression method
HC-PLSR
Multivariate regression method based on separating the observations into clusters and generating Partial Least Squares Regression (PLSR) [http://identifiers.org/biomodels.kisao/KISAO_0000416] models within each cluster. This local regression analysis is suitable for very non-linear systems. PLSR is a regression method based on estimated latent variables, related to Principal Component Analysis (PCA) and Principal Component Regression (PCR). Hierarchical cluster-based partial least squares regression method uses fuzzy C-means clustering, PLSR and Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) or Naive Bayes for classification of new observations to be predicted.
Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
Requested by Kristin Tøndel on Thursday, October 13, 2011 11:33:06 AM.
N-way partial least squares regression method
N-PLS
N-way PLSR
N-way partial least squares method
Multivariate regression method that can be used on N-way data. Based on Partial Least Squares Regression (PLSR) [http://identifiers.org/biomodels.kisao/KISAO_0000416], which is a regression method based on estimated latent variables. PLSR is related to Principal Component Analysis (PCA) and Principal Component Regression (PCR).
Bro R (1996) Multiway calibration. Multilinear PLS. J. Chemometrics 10: 47-6.
N-way toolbox website
metamodelling method
Deterministic dynamic models of complex biological systems contain a large number of parameters and state variables, related through nonlinear differential equations with various types of feedback. A metamodel of such a dynamic model is a statistical approximation model that maps variation in parameters and initial conditions (inputs) to variation in features of the trajectories of the state variables (outputs) throughout the entire biologically relevant input space.
number of partial least squares components
Parameter used by 'partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000416] describing number of PLS components to include in the regression analysis.
type of validation
Parameter of 'partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000416] describing how validation is performed. Possible values include cross-validation and test set validation.
number of N-way partial least squares regression factors
number of factors
Parameter of 'N-way partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000369] describing the number of factors to compute.
partial least squares regression-like method
Method for building regression models between independent and dependent variables.
mean-centring of variables
A boolean parameter of the 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417] specifying whether the variables were mean-centred prior to the regression analysis.
standardising of variables
A boolean parameter of the 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417] specifying whether the variables were standardised (divided by their standard deviations) prior to the regression analysis.
number of clusters
Parameter specifying the number of clusters used by C-means algorithm.
matrix for clusterization
A matrix to do the clustering in 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417].
clusterization parameter
Parameter used by algorithms performing clusterization.
variables preprocessing parameter
IDA-like method
Solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0. At each step, a Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408] leads to linear systems Jx=b, which are solved by one of five methods - two direct (dense or band; serial version only) and three Krylov [http://identifiers.org/biomodels.kisao/KISAO_0000354] (GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000353], BiCGStab [http://identifiers.org/biomodels.kisao/KISAO_0000392], or TFQMR [http://identifiers.org/biomodels.kisao/KISAO_0000396]).
CVODE-like method
Solves ODE initial value problems, in real N-space, written as y'=f(t,y), y(t0)=y0. It is capable for stiff and non-stiff systems and uses two different linear multi-step methods, namely the Adam-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] method and the backward differentiation formula [http://identifiers.org/biomodels.kisao/KISAO_0000288].
Higham-Hall method
RK5(4)7FEql
The equilibrium theory of Hall and Higham (1988) can be used to determine whether a Runge-Kutta algorithm will perform smoothly when stability restricts the stepsize. Higham-Hall method is a fifth order embedded Runge-Kutta method [http://identifiers.org/biomodels.kisao/KISAO_0000302], which behaves smoothly with respect to the standard type of stepsize controllers.
Higham, D.J. and Hall, G. (1990) Embedded Runge-Kutta formulae with stable equilibrium states. Journal of Computational and Applied Mathematics, 29 (1). pp. 25-33.
embedded Runge-Kutta 5(4) method
RK5(4)
An embedded Runge-Kutta integrator of order 5(4).
Dormand-Prince 8(5,3) method
This method is based on an 8(6) method by Dormand and Prince (i.e. order 8 for the integration and order 6 for error estimation) modified by Hairer and Wanner to use a 5th order error estimator with 3rd order correction.
Requested by Frank T. Bergmann on Thursday, November 29, 2012 9:51:58 AM.
flux balance analysis
FBA
Flux balance analysis is a mathematical approach for analyzing the flow of metabolites through a metabolic network.
Jeffrey D Orth, Ines Thiele and Bernhard O Palsson (2010) What is flux balance analysis? Nature Biotechnology 28, 245–248
Requested by Mark Moeller on Friday, January 25, 2013 11:11:30 AM.
COAST
controllable approximative stochastic reaction algorithm
An approximative algorithm for stochastic simulations of chemical reaction systems based on three different modeling levels: for small numbers of particles an exact [http://identifiers.org/biomodels.kisao/KISAO_0000236] stochastic [http://identifiers.org/biomodels.kisao/KISAO_0000104] model; for intermediate numbers an approximative [http://identifiers.org/biomodels.kisao/KISAO_0000237], but computationally more efficient stochastic [http://identifiers.org/biomodels.kisao/KISAO_0000104] model based on discrete Gaussian distributions; and for large numbers the deterministic [http://identifiers.org/biomodels.kisao/KISAO_0000103] reaction kinetics. In every simulation time step, the subdivision of the reaction channels into the three different modeling levels is done automatically, where all approximations applied can be controlled by a single error parameter for which an appropriate value can easily be found.
Holger Wagner, Mark Moeller, and Klaus Prank (2006) COAST: Controllable approximative stochastic reaction algorithm. J. Chem. Phys. 125, 174104
logical model simulation method
Qualitative (logical) models specify the evolution rules of their components. In each state a number of transitions are enabled. A 'logical model simulation method' guides the choice of the transitions processed at each step.
Daniel Machado et al. Modeling formalisms in Systems Biology. AMB Express 2011, 1:45
synchronous logical model simulation method
Qualitative (logical) models specify the evolution rules of their components. In the case of a synchronous updating all enabled transitions are processed simultaneously.
asynchronous logical model simulation method
Qualitative (logical) models specify the evolution rules of their components. It the case of an asynchronous updating all enabled transitions are performed independently: a state has as many successors as the number of transitions enabled in this state. An 'asynchronous logical model simulation method' specifies a rule to guide the choice of a unique transition at each step (for example random).
type of updating policy
A rule to guide the choice of a unique transition at each step used by an 'asynchronous logical model simulation method' [http://identifiers.org/biomodels.kisao/KISAO_0000450].
random updating policy
An updating policy that chooses a transition randomly.
ordered updating policy
An updating policy that chooses a transition in a definite way.
constant updating policy
An updating policy that chooses a transition in a constant way.
prioritized updating policy
An updating policy that chooses a transition in a prioritized way.
Requested by Andrew Miller on Tuesday, June 4, 2013 2:33:51 AM.
maximum step size
An upper limit, in the units of the bound variable over which a numerical integration is being performed, that an adaptive step size numerical integration algorithm should take.
Requested by Andrzej Kierzek on Thursday, April 24, 2014 12:40:35 PM Used to simulate systems involving reactions with propensities varying by many orders of magnitude.
maximal timestep method
Hybrid simulation algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000352] combining Gibson and Bruck algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000027] with the Gillespie tau-leap method [http://www.biomodels.net/kisao/KISAO#KISAO_0000039].
Puchalka J, Kierzek AM. Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophys J. 2004 Mar;86(3):1357-72.
maximal timestep
Key parameter of the 'maximal timestep method' [http://www.biomodels.net/kisao/KISAO#KISAO_0000468]. If Gillespie [http://www.biomodels.net/kisao/KISAO#KISAO_0000027] waiting time is longer than maximal timestep, slow reaction is not fired and tau-leap [http://www.biomodels.net/kisao/KISAO#KISAO_0000039] step is executed for fast reactions. Otherwise, slow reaction is fired and tau-leap [http://www.biomodels.net/kisao/KISAO#KISAO_0000039] is executed with shorter time step.
optimization algorithm
optimization method
An optimization algorithm tries to find the minumum or maximum of an arbitrary function. It takes a function of one or several variables and determines the values for the variables so that the function value is optimal.
Brinkhuis J, Tikhomirov V. Optimization: Insights and Applications Princeton Series in Applied Mathematics (illustrated), Princeton University Press, 2005.
local optimization algorithm
local optimiation method
A local optimization algorithm is an optimisation algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000470] that only finds a local optimum of a function. If several optima exist for the function, it usually depends on the starting values for the variables which optimum is found.
global optimization algorithm
global optimization method
A global optimization algorithm is an optimization algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000470] that tries to find the global optimum of a function. If a function has several minima/maxima with in the allowed range of variable values, the global minimum/maximum is the one with the smallest/largest function value.
Bayesian inference algorithm
A bayesian inference algorithm calculates a posterior probability distribution from a prior probability distribution and some additional evidence in the form of a likelyhood function.
George E. P. Box, George C. Tiao. Bayesian Inference in Statistical Analysis, Volume 40 of Wiley Classics Library. John Wiley & Sons (reprint, revised), 2011.
integration method
the integration method used by the solver
Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26
Requested in https://sourceforge.net/p/kisao/feature-requests/12/
iteration type
the type of iteration used by the solver
linear solver
the linear solver used by the solver during a Newton iteration
preconditioner
the preconditioner, if any, used by the solver when using a GMRES, BiCGStab or TFQMR linear solver during a Newton iteration.
upper half-bandwidth
the upper half-bandwidth value used by the Banded linear solver or preconditioner (a value between 0 and n-1 with n the number of ODEs/DAEs in the model).
lower half-bandwidth
the lower half-bandwidth value used by the Banded linear solver or preconditioner (a value between 0 and n-1 with n the number of ODEs/DAEs in the model).
interpolate solution
whether the solver returns an interpolated solution.
half-bandwith parameter
the parameter related to the half-bandwidth value used by the Banded linear solver or preconditioner.
Requested in https://sourceforge.net/p/kisao/feature-requests/13/
step size
the size of every step of the algorithm
Roadrunner
maximum order
Maximum order of method. For example, in Roadrunner it can be used for two parameters that one can set for deterministic runs: 'maximum_bdf_order' and 'maximum_adams_order'.
minimum step size
A lower limit, in the units of the bound variable over which a numerical integration is being performed, that a numerical integration algorithm with variable step size should take.
maximum iterations
iteration limit
For algorithms that iterate to a solution (steady state finders in particular), a limit on the number of iterations that should be performed.
The damping factor is a variable for at least some steady state algorithms: roadrunner allows you to set the minimum value for this.
minimum damping
seed
random seed
Random seed of a stochastic algorithm. Setting it allows one to reproduce their results while running the same algorithm on the same computer.
discrete event simulation algorithm
DES
Discrete Event Simulation algorithm refers to the simulation of systems whose (countable) discrete states change over time and are event-driven.
C.G. Cassandras & S. Lafortune "Introduction to Discrete Event Systems", Chapt 1 2nd Ed 2008. Springer
asynchronous updating policy
An updating policy where all enabled transitions (events) occur independently. Thus a state has as many successors as the number of transitions enabled in this state.
synchronous updating policy
An updating policy where all enabled transitions occur simultaneously. Thus a state will have at most one successor.
fully asynchronous updating policy
An updating policy where all enabled transitions occur either independently or (partially) simultaneously. (i.e. considering all possible combinations of enabled transitions). Thus a state has as many successors as the number of combinations of transitions enabled in this state.
random asynchronous updating policy
An updating policy where a single transition is picked randomly from the set of transitions enabled in this state. Thus a state will have at most one successor.
CVODES is a superset of CVODE [http://identifiers.org/biomodels.kisao/KISAO_0000019] and hence all options available to CVODE (with the exception of the FCVODE interface module) are also available for CVODES. Both integration methods (Adams-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] and BDF [http://identifiers.org/biomodels.kisao/KISAO_0000288]) and the corresponding nonlinear iteration methods, as well as all linear solver and preconditioner modules, are available for the integration of the original ODEs, the sensitivity systems, or the adjoint system.
CVODES
CVODES is a solver for stiff and nonstiff ODE systems (initial value problem) given in explicit form y’ = f(t,y,p) with sensitivity analysis capabilities (both forward and adjoint modes).
AMICI
KLU
"Clark Kent" LU factorization algorithm
KLU is a software package and an algorithm for solving sparse unsymmetric linear systems of equations that arise in circuit simulation applications. It relies on a permutation to Block Triangular Form (BTF), several methods for finding a fill-reducing ordering (variants of approximate minimum degree and nested dissection), and Gilbert/Peierls’ sparse left-looking LU factorization algorithm to factorize each block. The package is written in C and includes a MATLAB interface.
Davis, Timothy A., and Ekanathan Palamadai Natarajan. "Algorithm 907: KLU, a direct sparse solver for circuit simulation problems." ACM Transactions on Mathematical Software (TOMS) 37.3 (2010): 36.
Requested in https://sourceforge.net/p/kisao/feature-requests/31/
number of runs
The number of runs that a simulation should perform. Typically used to specify the number of runs for a stochastic simulation.
Ticket 32
dynamic flux balance analysis
DFBA
Dynamic flux balance analysis (DFBA) enables the simulation of dynamic biological systems by assuming organisms reach steady state rapidly in response to changes in the extracellular environment. DFBA couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches.
DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis. Gomez JA, Höffner K, Barton PI. BMC Bioinformatics. 2014 Dec 18;15:409.
DFBAlab
SOA-DFBA
SOA
static optimization approach dynamic flux balance analysis
Dynamic Flux Balance Analysis (DFBA) [http://identifiers.org/biomodels.kisao/KISAO_0000499] couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. The static optimization approach (SOA) uses the Euler forward method [http://identifiers.org/biomodels.kisao/KISAO_0000030], solving the embedded LPs at each time step. The FBA fluxes are assumed to be constant during the time step.
DOA-DFBA
DOA
dynamic optimization approach dynamic flux balance analysis
Dynamic Flux Balance Analysis (DFBA) [http://identifiers.org/biomodels.kisao/KISAO_0000499] couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. The dynamic optimization approach (DOA) discretizes the time horizon and optimizes simultaneously over the entire time period of interest by solving a nonlinear programming problem (NLP).
DA-DFBA
direct approach dynamics flux balance analysis
Dynamic Flux Balance Analysis (DFBA) [http://identifiers.org/biomodels.kisao/KISAO_0000499] couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. The direct approach (DA) includes the LP solver in the right-hand side evaluator for the ordinary differential equations (ODEs) and takes advantage of reliable implicit ODE integrators with adaptive step size for error control.
simulated annealing
Simulated annealing is an optimization algorithm first proposed by Kirkpatrick et al. 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!).
S. Kirkpatrick, J., C.D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science. 220. 671 - 680. 1983.
random search
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.
particle swarm
The particle swarm optimization method suggested by Kennedy and Eberhart is inspired by a flock of birds or a school of fish searching for food. Each particle has a position Xi and a velocity Vi in the parameter space. Additionally, it remembers its best achieved objective value O and position Mi. 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.
J. Kennedy and R. Eberhart. Particle Swarm Optimization. Proceedings of the Fourth IEEE International Conference on Neural Networks, Perth, Australia. 1942 - 1948. 1995.
genetic algorithm
The genetic algorithm (GA) 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).
T. Bäck, D.B. Fogel, and Z. Michalewicz. Handbook of evolutionary computation. Oxford: IOP Publishing/Oxford University Press. 1997.
genetic algorithm SR
genetic algorithm with stochastic ranking
T. Runarsson and X. Yao. Stochastic ranking for constrained evolutionary optimization. EEE Transactions on Evolutionary Computation. 4. 284 - 294. 2000.
evolutionary programming
Evolutionary programming (EP) 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).
evolutionary strategy
SRES
evolutionary strategies with stochastic ranking
Evolutionary Strategies with Stochastic Ranking (SRES) is similar to Evolutionary Programming. 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.
truncated Newton
The Truncated Newton method is a sophisticated variant of the Newton optimization method. 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 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.
P.E. Gill, W. Murray, and M.H. Wright. Practical Optimization. London, Academic Press. 1981.
steepest descent
Steepest descent 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.
D.B. Fogel, L.J. Fogel, and J.W. Atmar. Meta-evolutionary programming. 25th Asiloma Conference on Signals, Systems and Computers. IEEE Computer Society, Asilomar . 540 - 545. 1992.
praxis
Praxis is a direct search method that searches for the minimum of a nonlinear function without requiring (or attempting to calculate) derivatives of that function. Praxis was developed by Brent after the method proposed by Powell. 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.
P.R. Brent. A new algorithm for minimizing a function of several variables without calculating derivatives. In Algorithms for minimization without derivatives, (Englewood Cliffs, NJ: Prentice-Hall, Inc.). 117 - 167. 1973.
NL2SOL
adaptive nonlinear least-squares algorithm
The NL2SOL method is based on an adaptive nonlinear least-squares algorithm, devised by Dennis and colleagues. For problems with large number of residuals, this algorithm is known to be more reliable than Gauss-Newton or Levenberg-Marquardt method and more efficient than the secant or variable metric algorithms that are intended for general function minimization.
J. E. Dennis, D. M. Gay, and R. E. Welsch. An adaptive nonlinear least-squares algorithm. ACM Trans. Math. Softw.. 7(3). 348 - 368. 1981.
Nelder-Mead
simplex method
This method also known as the simplex method is due to Nelder and Mead. 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.
J. A. Nelder and R. Mead. A simplex method for function minimization. Computer Journal. 7. 308 - 313. 1965.
Levenberg-Marquardt
Levenberg-Marquardt is a gradient descent method. It is a hybrid between the steepest descent and the Newton methods. 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.
K. Levenberg. A method for the solution of certain nonlinear problems in least squares. Quart. Appl. Math.. 2. 164 - 168. 1944.
D.W. Marquardt. An algorithm for least squares estimation of nonlinear parameters. SIAM Journal. 11. 431 - 441. 1963.
Hooke&Jeeves
Hooke and Jeeves method
Hooke-Jeeves method
method of Hooke and Jeeves
The method of Hooke and Jeeves 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.
R. Hooke and T. A. Jeeves. "Direct search" solution of numerical and statistical problems. Journal of the Association for Computing Machinery. 8. 212 - 229. 1961.
number of generations
The parameter is a positive integer value to determine the number of generations the evolutionary algorithm shall evolve the population.
evolutionary algorithm parameter
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.
true
evolutionary algorithm
An optimisation algorithm that mimics evolution and is based on reproduction and selection.
simulated annealing parameter
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.
cooling factor
Rate by which the temperature is reduced from one cycle to the next, given by the formula: Tnew=Told*"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.
partitioned leaping method
Multiscale simulation approach for modeling stochasticity in chemical reaction networks. The approach seamlessly integrates exact-stochastic and "leaping" methodologies into a single partitioned leaping algorithmic framework. The technique correctly accounts for stochastic noise at significantly reduced computational cost, requires the definition of only three modelindependent parameters and is particularly well-suited for simulating systems containing widely disparate species populations.
Harris LA, Clancy P. A "partitioned leaping" approach for multiscale modeling of chemical reaction dynamics. J Chem Phys. 125 (14), 144107 (2006).
stop condition
A condition upon which a simulation should terminate.
flux variability analysis
FVA
Method for determining the minimum and maximum flux of each reaction that satisfies the flux constraints of the flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model.
COBRApy
Mahadevan R, Schilling CH. The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. Metab Eng. 2003;5(4):264-276.
geometric flux balance analysis
gFBA
geometric FBA
Method for determining the central flux distribution among all flux distributions that satisfy the constraints of the flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model.
Smallbone K, Simeonidis E. Flux balance analysis: a geometric perspective. J Theor Biol. 2009;258(2):311-315.
parsimonious enzyme usage flux balance analysis (minimum sum of absolute fluxes)
pFBA
parsimonious FBA
parsimonious flux balance analysis
Method for determining the smallest flux distribution among all flux distributions that satisfy the constraints of the flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model.
CBMPy
Lewis NE, Hixson KK, Conrad TM, et al. Omic data from evolved E. coli are consistent with computed optimal growth from genome-scale models. Mol Syst Biol. 2010;6:390.
parallelism
Number of parallel processes to use.
fraction of optimum
Fraction of the optimum solution which must be maintained.
loopless
Whether to return only loopless flux solutions.
pFBA factor
Maximum permissible sum of absolute fluxes.
reactions
FVA algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000526] parameter: reactions to compute the variablity of.
JRK
Odespy
SciPy
deSolve
DVODE
Real-valued Variable-coefficient Ordinary Differential Equation solver, with fixed-leading-coefficient implementation
VODE provides implicit Adams method (for non-stiff problems) and a method based on backward differentiation formulas (BDF) (for stiff problems).
GillesPy2
Peter N. Brown, George D. Byrne & Alan C. Hindmarsh. VODE: A Variable-Coefficient ODE Solver. SIAM J. Sci. and Stat. Comput., 10(5), 1038–1051 (1988).
ZVODE
Complex-valued Variable-coefficient Ordinary Differential Equation solver, with fixed-leading-coefficient implementation
ZVODE provides implicit Adams method (for non-stiff problems) and a method based on backward differentiation formulas (BDF) (for stiff problems).
MATLAB
Octave
explicit Runge-Kutta method of order 3(2)
RK23
RK23 uses the Bogacki-Shampine pair of formulas [1]. The error is controlled assuming accuracy of the second-order method, but steps are taken using the third-order accurate formula (local extrapolation is done). A cubic Hermite polynomial is used for the dense output.
P. Bogacki & L. F. Shampine. A 3(2) pair of Runge - Kutta formulas. Applied Mathematics Letters 2(4), 321-325 (1989).
JModelica
safety factor on new step selection
safe
safety
minimum factor to change step size by
dfactor
fac1
Minimum factor to increase/decrease step size by in one step. The new step-size is chosen subject to the restriction fac1 <= current step-size / old step-size <= fac2.
maximum factor to change step size by
fac2
ifactor
Maximum factor to increase/decrease step size by in one step. The new step-size is chosen subject to the restriction fac1 <= current step-size / old step-size <= fac2.
beta parameter for stabilized step size control
beta
correction step should use internally generated full Jacobian
MF=22
with Jacobian
Specifies whether the iteration method of the ODE solver’s correction step is chord iteration with an internally generated full Jacobian or functional iteration with no Jacobian. Option is only considered when the user has not supplied a Jacobian function and has not indicated (by setting either upper or lower band) that the Jacobian is banded.
stability limit detection flag
Flag to activate stability limit detection.
IDAS
implicit differential-algebraic solver with sensitivity analysis
IDAS solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0 with sensitivity analysis.
SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. Alan C. Hindmarsh, Peter N. Brown, Keith E. Grant, Steven L. Lee, Radu Serban, Dana E. Shumaker & Carol S. Woodward. ACM Transactions on Mathematical Software 31(3), 363-396 (2005).
include sensitivity variables in error control mechanism
errconS
Specifies whether sensitivity variables are included or not in the error control mechanism.
convex optimization algorithm
Optimization of a convex function over a convex set. Convex optimization is subclass of global optimization because conveness gaurantees that each local optimum is a global optimum.
CPLEX
CVXOPT
ConvOpt
GLPK
Gurobi
Mosek
OptLang
SoPlex
XPRESS
linear programming
Method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
quadratic programming
Process of solving a quadratic optimization problem.
non-linear programming
Process of solving an optimization problem where some of the constraints or the objective function are nonlinear.
Dantzig's simplex algorithm
Approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem.
primal-dual interior point method
The Interior Point method approximates the constraints of a linear programming model as a set of boundaries surrounding a region.
Sanjay Mehrotra. On the Implementation of a Primal-Dual Interior Point Method. SIAM Journal on Optimization, 2(4), 575–601(1991).
Optimization method such as the revised simplex method [http://identifiers.org/biomodels.kisao/KISAO_0000550] or primal-dual interior point method [http://identifiers.org/biomodels.kisao/KISAO_0000551].
optimization solver
Optimization solver such as CPLEX, GLPK, or Gurobi.
RAVEN
parsimonius flux balance analysis (minimum number of active fluxes)
A technique for selecting a parsimonious flux distribution which has a minimal number of active fluxes.
absolute quadrature tolerance
Absolute error tolerance of the adjoint solution.
relative quadrature tolerance
Relative error tolerance of the adjoint solution.
absolute steady-state tolerance
Absolute error tolerance of the steady-state.
relative steady-state tolerance
Relative error tolerance of the steady-state.
initial step size
Initial time step size.
LSODA/LSODAR hybrid method
Automatically use LSODA or LSODAR as apropriate for the given problem. Use LSODA if the problem has no roots. Use LSODAR if the problem has roots.
Pedro Mendes, Stefan Hoops, Sven Sahle, Ralph Gauges, Joseph Dada & Ursula Kummer. Computational modeling of biochemical networks using COPASI. Methods in Molecular Biology 500, 17–59 (2009).
Frank T. Bergmann, Stefan Hoops, Brian Klahn, Ursula Kummer, Pedro Mendes Jürgen Pahle & Sven Sahle. COPASI and its applications in biotechnology. Journal of Biotechnology 261, 215–220 (2017).
Stefan Hoops, Sven Sahle, Ralph Gauges, Christine Lee, Jürgen Pahle, Natalia Simus, Mudita Singhal, Liang Xu, Pedro Mendes & Ursula Kummer. COPASI—a complex pathway simulator. Bioinformatics 22 (24), 3067–3074 (2006).
Sven Sahle, Ralph Gauges, Jürgen Pahle, Natalia Simus, Ursula Kummer, Stefan Hoops, Christine Lee, Mudita Singhal, Liang Xu & Pedro Mendes. Simulation of biochemical networks using COPASI—a complex pathway simulator. Proceedings of the 2006 Winter Simulation Conference, Monterey, CA, USA, 1698–1706 (2006).
Pahle hybrid Gibson-Bruck Next Reaction method/Runge-Kutta method
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. The partitioning of the biochemical network can change dynamically during the simulation. The reaction probabilities of the stochastic subnet are approximated as constant during one stochastic step. A 4th-order Runge-Kutta method is used to numerically integrate the deterministic part of the system. The stochastic subnet is simulated by the Gibson-Bruck Next Reaction Method.
Pahle hybrid Gibson-Bruck Next Reaction method/LSODA method
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. The partitioning of the biochemical network can change dynamically during the simulation. The reaction probabilities of the stochastic subnet are approximated as constant during one stochastic step. The deterministic subnet is integrated with LSODA. The stochastic subnet is simulated by the Gibson-Bruck Next Reaction Method.
Pahle hybrid Gibson-Bruck Next Reaction method/RK-45 method
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. The partitioning of the biochemical network can change dynamically during the simulation. The reaction probabilities of the stochastic subnet are approximated as constant during one stochastic step. The deterministic subnet is integrated with RK-45. The stochastic subnet is simulated by the Gibson-Bruck Next Reaction Method.
stochastic Runge-Kutta method
Technique for the approximate numerical solution of a systems of stochastic differential equations (SDEs). The method is a generalisation of the Runge-Kutta method for ordinary differential equations to stochastic differential equations.
absolute tolerance for root finding
Absolute error tolerance for root finding.
stochastic second order Runge-Kutta method
RI5
Technique for the second order approximate numerical solution of a systems of stochastic differential equations (SDEs). The method is a generalisation of the Runge-Kutta method for ordinary differential equations to stochastic differential equations.
Andreas Rößler. Second order Runge-Kutta methods for Itô stochastic differential equations. SIAM Journal Numerical Analysis 47 (3), 1713–1738 (2009).
force physical correctness
Indicates whether to force physical correctness.
NLEQ1
Newton-type method for solveing non-linear (NL) equations (EQ)
numerical solution of nonlinear (NL) equations (EQ) especially designed for numerically sensitive problems
Damped Newton-algorithm with rank strategy for systems of highly nonlinear equations. Global Newton method with error oriented convergence criterion; arbitrary selection of direct linear equation solver.
Ulrich Nowak & Lutz Weimann. A family of Newton codes for systems of highly nonlinear equations - algorithm, implementation, application. Konrad-Zuse-Zentrum für Informationstechnik Berlin 91-10 (1991).
Peter Deuflhard. Newton methods for nonlinear problems. Affine Invariance and Adaptive Algorithms (2004).
NLEQ2
Damped Newton-algorithm with rank strategy for systems of highly nonlinear equations. Global Newton method with error oriented convergence criterion; QR-decomposition with subcondition number estimate.
auto reduce tolerances
Whether to automatically reduce tolerances.
absolute tolerance adjustment factor
How much to adjust the absolute tolerance.
level of superimposed noise
noise level
Standard deviation of the Gaussian noise which is added to each prediction.
BoolNet
probabilistic logical model simulation method
Qualitative (logical) models specify the evolution rules of their components. Probabilistic networks allow for specifying more than one transition function per variable/gene. Each of these functions has a probability to be chosen, where the probabilities of all functions for one variable sum up to 1. Transitions are performed synchronously by choosing one transition function for each gene according to their probabilities and applying them to the current state.
species transition probabilities
Probability of each species to be chosen for the next state transition.
hybrid tau-leaping method
A continuously coupled hybrid deterministic/stochastic simulation algorithm for biochemical networks. Biochemical species are classified as continuous, discrete, or switch. Tau-leaping is used to simulate stochastic species, and LSODA or another ODE integration method is used to simulate continuous species. Switch species are dynamically classified as either continuous or discrete at each timestep depending on a user defined error tolerance.
Drawert B et al. GillesPy2
quadratic MOMA
MOMA
Quadratic Minimization of Metabolic Adjustment
minimization of metabolic adjustment
Minimization of metabolic adjustment (MOMA) is an extension of FBA for the prediction of flux distributions in gene knockouts. MOMA employs quadratic programming to identify the closest point (in terms of its Euclidean distance) in the permissible flux space of the knockout to the wild-type flux vector by solving the optimization problem Min sum((fluxAi - fluxBi)^2) + sum(fluxAi)^(fluxMinimizationWeight) + sum(fluxBi)^(fluxMinimizationWeight)
COBRA Toolbox
Optflux
Segre, D., Vitkup, D., & Church, G. M. (2002). Analysis of optimality in natural and perturbed metabolic networks. Proceedings of the National Academy of Sciences, 99(23), 15112-15117.
flux minimization weight
The degree to which minimization of the sum of fluxes should be taken into account in Minimization of Metabolic Adjustment (MOMA) which solvers the optimization problem Min sum((fluxAi - fluxBi)^2) + sum(fluxAi)^(fluxMinimizationWeight) + sum(fluxBi)^(fluxMinimizationWeight)
nested algorithm
nested method
subalgorithm
A nested algorithm of an algorithm
linear MOMA
Linear Minimization of Metabolic Adjustment
Linear minimization of metabolic adjustment (MOMA) is an extension of FBA for the prediction of flux distributions in gene knockouts. Linear MOMA employs linear programming to identify the closest point (in terms of its L1 norm) in the permissible flux space of the knockout to the wild-type flux vector by solving the optimization problem Min sum(|fluxAi - fluxBi|)
Szappanos, B., Kovács, K., Szamecz, B., Honti, F., Costanzo, M., Baryshnikova, A., ... & Andrews, B. J. (2011). An integrated approach to characterize genetic interaction networks in yeast metabolism. Nature genetics, 43(7), 656-662.
ROOM
Regulatory on/off minimization of metabolic flux changes
Constraint-based algorithm for predicting the metabolic steady state after gene knockouts which aims to minimize the number of significant flux changes (hence on/off) with respect to the wild type.
Shlomi, T., Berkman, O., & Ruppin, E. (2005). Regulatory on/off minimization of metabolic flux changes after genetic perturbations. Proceedings of the national academy of sciences, 102(21), 7695-7700.
BKMC
Boolean Kinetic Monte-Carlo
Boolean kinetic Monte-Carlo
The Boolean kinetic Monte Carlo method (BKMC) is a natural generalization of the asynchronous Boolean simulation method, with a direct probabilistic interpretation. In the BKMC framework, the dynamics is parameterized by a biological time and the order of update is noisy, which is less strict than priority classes introduced in GINsin. A BKMC model is specified by logical rules as in regular Boolean models but with a more precise information: a numerical rate is added for each transition of each node.
MaBoSS
Stoll, G., Viara, E., Barillot, E., & Calzone, L. (2012). Continuous time Boolean modeling for biological signaling: application of Gillespie algorithm. BMC systems biology, 6(1), 1-18.
Spatiocyte method
Lattice-based stochastic particle simulation method for biochemical reaction and diffusion processes.
E-Cell 4
Arjunan, S., & Tomita, M. (2009). A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation. Nature Precedings, 1-1.
minimum order
Minimum order of method.
initial order
Initial order of method.
TOMS731
Moving-grid interface for systems of one-dimensional time-dependent partial differential equations.
Blom, J. G., & Zegeling, P. A. (1994). Algorithm 731: a moving-grid interface for systems of one-dimensional time-dependent partial differential equations. ACM Transactions on Mathematical Software (TOMS), 20(2), 194-214.
Gibson-Bruck next reaction algorithm with indexed priority queue
SBSCL
IMEX
Implicit-Explicit Runge-Kutta method
implicit-explicit Runge-Kutta method
Method for solving stiff and imaginary ODE problems
BioUML
flux sampling
Method for sampling fluxes from the null space of a flux balance analysis model
ACB flux sampling method
Artificial centering boundary flux sampling method
Heirendt, L., Arreckx, S., Pfau, T., Mendoza, S. N., Richelle, A., Heinken, A., ... & Magnusdóttir, S. (2019). Creation and analysis of biochemical constraint-based models using the COBRA Toolbox v. 3.0. Nature protocols, 14(3), 639-702.
ACHR flux sampling method
artificial centering hit-and-run flux sampling method
Kaufman, D. E., & Smith, R. L. (1998). Direction choice for accelerated convergence in hit-and-run sampling. Operations Research, 46(1), 84-95.
mdFBA
metabolic dilution flux balance analysis
Benyamini, T., Folger, O., Ruppin, E., & Shlomi, T. (2010). Flux balance analysis accounting for metabolite dilution. Genome biology, 11(4), R43.
dynamic rFBA
dynamic regulatory flux balance analysis
rFBA
regulatory flux balance analysis
Method for predicting the dynamics of metabolic fluxes under patterns of the regulation of gene expression
Covert, M. W., Schilling, C. H., & Palsson, B. (2001). Regulation of gene expression in flux balance models of metabolism. Journal of theoretical biology, 213(1), 73-88.
Minimization of Metabolic Adjustment
minimization of metabolic adjustment (MOMA) is an extension of FBA for the prediction of flux distributions in gene knockouts. MOMA identifies the closest point in the permissible flux space of the knockout to the wild-type flux vector by solving an optimization problem.
order
Order of method
Method for predicting metabolic fluxes under patterns of the regulation of gene expression
srFBA
SR-FBA
steady-state regulatory flux balance analysis
Method for predicting steady-state metabolic fluxes under patterns of the regulation of gene expression
Shlomi, T., Eisenberg, Y., Sharan, R., & Ruppin, E. (2007). A genome‐scale computational study of the interplay between transcriptional regulation and metabolism. Molecular systems biology, 3(1), 101.
Numeric value specifying the desired tolerance the user wants to achieve. A smaller value means that the prediction is calculated more accurately.
MLB
hybrid Gibson - Milstein method
A hybrid stochastic method partitions the system into subsets of fast and slow reactions and approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the Gibson algorithm. Fixed time step Milstein is used for approximate numerical solution of CLE.
H.Salis, V. Sotiropoulos and Y. Kaznessis,'Multiscale Hy3S: Hybrid stochastic simulation for supercomputers', BMC Bioinformatics 7:93 (2006).
hybrid Gibson - Euler-Maruyama method
hybrid adaptive Gibson - Milstein method
number of trials
trials
Number of multiple trials (e.g., of a scatter search method).
minimum species threshold for continuous approximation
Epsilon
Minimum number of molecules of both reactant and product species required for approximation as a continuous Markov process.
minimum reaction rate for continuous approximation
Lambda
lambda
Minimum reaction rate required for approximation as a continuous Markov process.
MSR tolerance
Multiple slow reactions tolerance
multiple slow reactions tolerance
Maximum allowed effect of executing multiple slow reactions per numerical integration of the SDEs.
SDE tolerance
Maximum allowed value of the drift and diffusion errors.
Stochastic differential equation tolerance
hierarchical stochastic simulation algorithm
hSSA
Fast, memory-efficient method for stochastic simulation of hierarchically organized models, such as a model of a cellular population where each cell in the population is represented by the same species and reactions.
Watanabe, L. H., & Myers, C. J. (2014). Hierarchical stochastic simulation algorithm for SBML models of genetic circuits. Frontiers in bioengineering and biotechnology, 2, 55.
hierarchical Fehlberg method
hODE
hierarchical ODE integration method
hierarchical ordinary differential equation integration method
Method for continuous simulation of hierarchically organized models, such as a model of a cellular population where each cell in the population is represented by the same species and reactions.
hierarchical flux balance analysis
Hierarchical FBA
hFBA
hierarchical FBA
Method for constraint-based simulation of hierarchically organized models, such as a model of a cellular population where each cell in the population is represented by the same species and reactions.
embedded Runge-Kutta Prince-Dormand (8,9) method
RK8PD
An embedded Runge-Kutta integrator of order 8(9).
composite-rejection stochastic simulation algorithm
Slepoy, A., Thompson, A. P., & Plimpton, S. J. (2008). A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. The journal of chemical physics, 128(20), 05B618.
incremental stochastic simulation algorithm
iSSA
Performs local averaging over small time-intervals to compute statistics on typical behavior.
Winstead, C., Madsen, C., & Myers, C. (2010, May). iSSA: An incremental stochastic simulation algorithm for genetic circuits. In Proceedings of 2010 IEEE International Symposium on Circuits and Systems (pp. 553-556). IEEE.
implicit 4th order Runge-Kutta method at Gaussian points
RK4IMP
stochastic simulation algorithm with normally-distributed next reaction times
NMC
implementation
An implementation of an algorithm. For example, simulation tools can this parameter to differentiate among C, Python, and Java implementations of the same algorithms and allow investigators to select one of these specific implementations through SED-ML.
fully-implicit regular grid finite volume method with a variable time step
semi-implicit regular grid finite volume method with a fixed time step
IDA-CVODE hybrid method
Meta algorithm which chooses between IDA and CVODE depending on the problem to be solved. CVODE is used for ordinary differential equation (ODE) systems. IDA is used for differential-algebraic equation (DAE) systems.
bunker
A variant of the stochastic simulation algorithm (SSA) in which the time to the next reaction is equated to the mean inter-event time (inverse of the sum of the propensitites of the reactions) rather than sampled from a distribution parameterized by this mean inter-event time. In this method, the next reaction time is deterministic rather than stochastic as in SSA.
emc-sim
A variant of the stochastic simulation algorithm (SSA) in which the time to the next reaction is a constant equal to 1 time unit. In this method, the next reaction time is deterministic rather than stochastic as in SSA.
parsimonius flux balance analysis
A technique for selecting a flux distribution which is parsimonious by some metric, such as a solution which has the minimal number of active fluxes or a solution which has the smallest sum of active fluxes.
stochastic simulation leaping method
flux balance method
flux balance problem
method for solving a system of linear equations
method for solving a system of linear equations Example system of equations: 3x + 2y - z = 0 2x - 2y + 4z = 0 -x + 1/2y - z = 0
dense direct solver
denese
band direct solver
banded
diagonal approximate Jacobian solver
diagonal
modelling and simulation algorithm parameter value
A value of a parameter of an algorithm
null
none
general steady state method
A method looking for a steady state of a dynamic system.
iterative root-finding method
Iterative method for finding the root of a function (f(x) = 0).
functional iteration root-finding method
Iterative method for finding the root of a function f(x) given by y^{n(m+1)} = h_{n} β_{n,0} f(t_{n}, y^{n(m)}) + a_n This method only involves evaluations of f. This method is suitable for non-stiff functions.
Roger Peng. Advanced Statistical Computing.
computational function
A mathematical function such as the calculation of a minimum, maximum, or mean of a set of values.
scaled property
unscaled property
primary property
A primary output of a simulation.
derived property
An output of a simulation which can be derived from its primary outputs.
level
A level such as of a qualitative variable.
flux
A rate through an volume such as of a reaction of a constraint-based models.
lower bound
A lower bound on an estimate of a quantity.
bound
An upper or lower bound on an estimate of a quantity.
minimum flux
Minimum possible flux such as computed by flux variability analysis (FVA, KISAO_0000526).
upper bound
An upper bound on an estimate of a quantity.
maximum flux
Maximum possible flux such as computed by flux variability analysis (FVA, KISAO_0000526).
objective value
Value of an objective function such as of a constraint-based model.
propensity
Tendency of an event such as of the firing of a reaction in the stochastic simulation algorithm (SSA, KISAO_0000029).
derivative
Rate of change of a variable with respect to another variable.
step
Iteration such as along a pseudo timecourse of a logical simulation.
shadow price
Change, per infinitesimal unit of the constraint, in the optimal value of the objective function of an optimization problem obtained by relaxing the constraint.
sensitivity
The sensitivity of a variable to another variable, such as the derivative a variable with respect to another.
reduced costs
The amount by which an objective function coefficient would have to improve before it would be possible for a corresponding variable to assume a positive value in the optimal solution.
concentration rate
Rate of a process relative to a volume, such as the rate of a reaction in molar^-1 s^-1.
particle number rate
Rate of a process in extensive/absolute units such as mole reactions per second.
amount rate
rate of a process in extensive/absolute units such as reactions per second.
rate
Speed at which a process is occuring such as the temporal rate of a chemical reaction,
use adaptive time steps
Whether an algorithm should use adaptive or fixed time steps.
sequential logical simulation method
Qualitative (logical) models specify the evolution rules of their components. In the case of a sequential updating, nodes are updated sequentially in a pre-determined deterministic order.
logical model analysis method
A method for analyzing a logical model, such as finding its fixed points.
Naldi MDD logical model stable state search method
Naldi Multi-valued Decision Diagram stable state search method
Efficient method for determining the stable states of a regulatory graph using a Multi-valued Decision Diagram (MDD) representation of the logical functions.
Aurélien Naldi, Denis Thieffry and Claudine Chaouiya. Decision Diagrams for the Representation and Analysis of Logical Models of Genetic Networks. Computational Methods in Systems Biology 2007, 4695.
logical model stable state search method
Method for determining the stable states of a regulatory graph.
logical model trap space identification method
Method for determining the trap spaces, or stable motifs or symbolic stable states, of a regulatory graph,
Klarner ASP logical model trap space identification method
Klarner Answer Set Programming logical model trap space identification method
Optimization-based method rooted in answer set programming (ASP) for computing the trap spaces of a regulatory graph.
Hannes Klarner, Alexander Bockmayr, and Heike Siebert. Computing maximal and minimal trap spaces of Boolean networks. Natural Computing 14.4 (2015): 535-544.
BDD logical model trap space identification method
Binary Decision Diagram logical model trap space identification method
Method for determining the trap spaces of a regulatory graph using a Binary Decision Diagram (BDD).
Method for solving Volterra equations. Uses the trapezoidal rule.
Second order backward implicit product Euler scheme
XPP
Peter Linz. Numerical methods for Volterra integral equations of the first kind. Computer Journal 12 (4): 393–397 (1969).
Peter Linz. Analytical and Numerical Methods for Volterra Equations. 1985.
maximum number of iterations for root finding
Jacobian epsilon
memory size
Maximum number of points to store in memory, such as in the second order backward implicit product Euler scheme.
Numerical Recipes in C "stiff" Rosenbrock method
Implementation of the Kaps and Rentrop using the parameters recommended by Shampine and Watts.
Shampine, L.F., and Watts, H.A. 1977, in Mathematical Software III, J.R. Rice, ed. (New York: Academic Press), pp. 257–275; 1979, Applied Mathematics and Computation, vol. 5, pp. 93–121.
stiff
William H. Press, Saul A. Teukolsky, William T. Vetterling & Brian P. Flannery. Numerical Recipes in C. The Art of Scientific Computing, 2nd Edition, 1992. Chapter 16. Pages 739-741.
Resource Balance Analysis
RBA
RBApy
A. Goelzer, V. Fromion, G. Scorletti. Cell design in bacteria as a convex optimization problem. Automatica, 47(6):1210-1218 (2011).
use multiple steps
Whether to perform a multiple time step simulation.
use stiff method
Specifies whether the integrator attempts to solve stiff equations.
Numerical Recipes in C "quality-controlled Runge-Kutta" method
rkqs
Cash-Karp method with step size adjustment.
William H. Press, Saul A. Teukolsky, William T. Vetterling & Brian P. Flannery. Numerical Recipes in C. The Art of Scientific Computing, 2nd Edition, 1992. Chapter 16. Pages 719.
skip reactions that produce negative species amounts
Parameter which instructs a simulation tool to skip reactions that would result in negative amounts of species.
presimulate
Whether a model should be presimulated prior to analysis.
Broyden method
Family of Quasi-Newton methods for finding roots in k variables originally described by C. G. Broyden in 1965.
degree of linearity
The degree of linearity of a system.
maximum number of steps for presimulation
Maximum number of steps to take in presimulating a model prior to analysis.
maximum number of steps for approximation
Maximum number of steps to take in approximating an analysis.
maximum time for approximation
Maximum amount of time to spend approximating an analysis.
duration
Length of time to simulate.
maximum time
Maximum amount of wall time for an operation.
allow approximation
Whether to find an approximate solution if an exact solution could not be found.
relative tolerance for approximation
Relatative tolerance for an alternative approximate solution to an exact solution which could not be found.
number of steps per output
Number of simulation steps between each simulation output.
BioNetgen
biological state optimization method
A method for computing the optimal state of a biological system according to a particular objective.
Enzyme Cost Minimization
ECM
For a given metabolic network model, the Enzyme Cost Minimization method determines plausible metabolite and enzyme concentrations (which determine thermodynamic forces and enzyme catalytic rates). Fluxes, enzyme kinetic constants, admissible metabolite concentration ranges, and enzyme cost weights (e.g. enzyme molecular masses) are given as input data. The method applies the principle that high enzyme (or enzyme plus metabolite) concentrations must be avoided, and maximizes a weighted sum of enzyme and metabolite concentrations.
Metabolic Pathway Analysis
Elad Noor, Avi Flamholz, Arren Bar-Even, Dan Davidi, Ron Milo & Wolfram Liebermeister. The Protein Cost of Metabolic Fluxes: Prediction from Enzymatic Rate Laws and Cost Minimization. PLoS Comput Biol 12 (11): e1005167 (2016).
Max-min Driving Force method
MDF
For a given metabolic network model, the MDF method determines plausible metabolite concentrations and thermodynamic forces. Flux directions, equilibrium constants (or equivalently, standard Gibbs free energies of reactions) and admissible metabolite concentration ranges are given as input data. The method applies the principle that low thermodynamic forces must be avoided, and maximizes the minimum thermodynamic force across the entire network.
Elad Noor, Arren Bar-Even, Avi Flamholz, Ed Reznik, Wolfram Liebermeister & Ron Milo. Pathway thermodynamics highlights kinetic obstacles in central metabolism. PLOS Computational Biology 10(2): e1003483 (2014).
type of system described
mathematical system
biological system
metabolic system
cellular system
biochemical system
lps
ODE solver
ordinary differential equation solver
An ODE solver is the general category of packages such as CVODE-like methods [http://identifiers.org/biomodels.kisao/KISAO_0000433] or Livermore solvers [http://identifiers.org/biomodels.kisao/KISAO_0000094] that solve systems of ordinary differential equations.
The children parameters of this term are applied when the parent general term is implented as the more-specific value of this term. For example: a 'parameters for' term might be used as a child of an 'ODE Solver' ([http://identifiers.org/biomodels.kisao/KISAO_0000694]), have a value of 'KISAO_0000019' (CVODE)', and have a child term 'use stiff method' of 'true' ([http://identifiers.org/biomodels.kisao/KISAO_0000671])
LPS
steady state problem
stochastic differential equation solver
An SDE solver is the general category of packages that provide stochastic solutions to a system of differential equations with propensities.
The set of coordinates for all particles of these entities.
differential algebraic equation solver
A DAE solver is the general category of packages such as IDA-like methods [http://identifiers.org/biomodels.kisao/KISAO_0000432] that solve systems of differential algebraic equations (DAEs). DAEs are a superset of ODEs that may additionally contain algebraic equations or 'fast' reactions.
systems property
A systems-level property of an entire model or simulation.
concentration control coefficient matrix (unscaled)
The unscaled concentration control coefficient matrix. The dimensions are species by reactions.
control coefficient (scaled)
A scaled control coefficient of any dependent element (such as a reaction or a floating species) with respect to an independent element (such as a global parameter or boundary species).
control coefficient (unscaled)
An unscaled control coefficient of any dependent element (such as a reaction or a floating species) with respect to an independent element (such as a global parameter or boundary species).
elasticity matrix (unscaled)
The unscaled elasticity matrix. The dimensions are reactions by species.
elasticity coefficient (unscaled)
An unscaled elasticity coefficient of any reaction with respect to an independent element (such as a global parameter or boundary species).
elasticity matrix (scaled)
The scaled elasticity matrix. The dimensions are reactions by species.
elasticity coefficient (scaled)
A scaled elasticity coefficient of any reaction with respect to an independent element (such as a global parameter or boundary species).
reduced stoichiometry matrix
The reduced stoichiometry matrix. The dimensions are species by reactions.
reduced Jacobian matrix
The reduced Jacobian matrix. The dimensions are species by species.
reduced eigenvalue matrix
The reduced eigenvalue matrix of a model. The dimensions are species by two, where the first column is the real part of the eigenvalues, and the second column is the imaginary part of the eigenvalues.
stoichiometry matrix
full stochiometry matrix
The (full) stoichiometry matrix. The dimensions are species by reactions.
Jacobian matrix
full Jacobian matrix
The (full) Jacobian matrix. The dimensions are species by species.
Eigenvalue matrix
full eigenvalue matrix
The (full) eigenvalue matrix of a model. The dimensions are species by two, where the first column is the real part of the eigenvalues, and the second column is the imaginary part of the eigenvalues.
flux control coefficient matrix (unscaled)
The unscaled flux control coefficient matrix. The dimensions are reactions by reactions.
flux control coefficient matrix (scaled)
The scaled flux control coefficient matrix. The dimensions are reactions by reactions.
link matrix
The link matrix of a model.
kernel matrix
The Kernel matrix of a model.
L0 matrix
The L0 matrix of a model.
Nr matrix
The Nr matrix of a model.
model and simulation property characteristic
A property of a variable of a model or simulation.
intensive property
An intensive variable such as a concentration or temperature.
extensive property
An extensive variable such as an amount, particle number, or mass.
aggregation function
A function that aggregates a set of results, reducing its dimension(s). Examples include functions that compute minima or maxima of sets of values.
LPS, MK
mean ignoring NaN
The mean (average) of a set of values, ignoring NaN entries.
standard deviation ignoring NaN
The standard deviation of a set of values, ignoring NaN entries.
standard error ignoring NaN
The standard error of a set of values, ignoring NaN entries.
maximum ignoring NaN
The maximum value of a set of values, ignoring NaN entries.
minimum ignoring NaN
The minimum value of a set of values, ignoring NaN entries.
maximum
The maximum of a set of values. If the values contain NaN the maximum is NaN.
model and simulation property
A variable of a model or simulation.
time
The implied time variable of the model state.
rate of change
The rate of change of one variable with respect to a second variable.
concentration control coefficient matrix (scaled)
The scaled concentration control coefficient matrix. The dimensions are species by reactions.
amount
particle number
The extensive quantity particle number, or, the molar amount of the entity multiplied by Avogadro's number.
concentration
The intensive quantity concentration, or, the amount of the entity with respect to the entity in which it resides.
temperature
minimum
The minimum of a set of values. If the values contain NaN the minimum is NaN.
mean
The mean of a set of values. If the values contain NaN the mean is NaN.
standard deviation
The standard deviation of a set of values. If the values contain NaN the standard deviation is NaN.
standard error
The standard error of a set of values. If the values contain NaN the standard deviation is NaN.
sum ignoring NaN
The sum of a set of values, ignoring Nan entries.
sum
The sum of a set of values. If the values contain NaN the sum is NaN.
product ignoring NaN
The product of a set of values, ignoring Nan entries.
product
The product of a set of values. If the values contain NaN the product is NaN.
cumulative sum ignoring NaN
The cumulative sum of a set of values, ignoring Nan entries.
cumulative sum
The cumulative sum of a set of values. If the values contain NaN the cumulative sum is NaN.
cumulative product ignoring NaN
The cumulative product of a set of values, ignoring Nan entries.
cumulative product
The cumulative product of a set of values. If the values contain NaN the cumulative product is NaN.
count ignoring NaN
The number of non-zero elements of a set of values, ignoring Nan entries.
count
The number of non-zero elements of a set of values. If the values contain NaN the count is NaN.
length ignoring NaN
The number of elements of a set of values, ignoring Nan entries.
length
The number of elements of a set of values.
median ignoring NaN
The median of a set of values, ignoring Nan entries.
median
The median of a set of values. If the values contain NaN the median is NaN.
variance ignoring NaN
The variance of a set of values, ignoring Nan entries.
variance
The variance of a set of values. If the values contain NaN the variance is NaN.