mathematical modeling of biochemical network dynamics using ordinary differential equations ( odes ) has yielded impressive advances in our understanding of complex biological systems ( 1 ) . when constructing an ode model
, there is often a need to estimate kinetic rate constants based on time - series experimental measurements of molecular concentrations or enzyme activities . even when time - series experimental measurements of all species are available [ such as by using stable isotope labeling by amino acids in cell culture ( silac ) proteomics ( 2 ) ] , estimating the rate constants is still a difficult non - linear optimization problem ( 3 ) .
many widely used methods , collected into popular software packages such as complex pathway simulator ( copasi ) ( 4 ) , are applicable to this parameter estimation problem .
methods have traditionally been classified as local , global or hybrid global + local methods ( 5,6 ) .
the traditional application of parameter estimation for modeling network dynamics has been to small biochemical networks with sparse data sets . with the growing ease of measuring complete proteomes ( 7 ) and with the assembly of large network models ,
needs have expanded to include to data - rich approaches to parameter estimation , sometimes called spline - based collocation methods ( 5,810 ) .
spline - based collocation methods exploit complete or nearly complete data sets to interpolate directly the slopes of concentration over time , instead of relying on numerical simulations to compute the derivatives of the odes .
spline - based collocation methods have not previously been implemented in any parameter estimation web server .
the method of systematic parameter estimation for data - rich environments ( spedre ) ( 11 ) uses a spline - based collocation approach and coarse - grained discretization to provide approximate heuristic search of the global parameter space , with excellent scalability for large networks .
however , spedre is designed only for problems with low - degree networks ( few reactions per protein , but no limit on the number of proteins ) .
although there are many algorithms for parameter estimation [ reviewed in ( 12,13 ) ] , we are not aware of any parameter estimation web servers , other than copasi ( 14,15 ) and spedre .
parameter estimation in biological networks is challenging because the parameters are interdependent , which makes them impossible to estimate individually .
even if each parameter can take on only a fixed number of possible values , the possible combinations of these parameter values becomes astronomical , growing exponentially with respect to the number of reactions .
spedre exploits complete data sets to interpolate the slopes ( concentration change over time ) instead of simulating the whole system , and it exploits the low degree of the network to construct a linear number of sparsely connected low - dimensional subproblems .
the pipeline for spedre , illustrated in figure 1 , consists of five stages : input , pre - computing discretized tables for each subproblem , loopy belief propagation to merge the subproblems , local optimization to refine the coarse - grained solution and output . during the input stage
, spedre requires the user to provide the network connectivity , the reaction types ( michaelis  menten , mass action kinetics , etc . ) , the time - series measurements and some optional runtime settings . in the discretization stage
, spedre transforms the continuous range of rate constants into discrete bins , and it pre - computes lookup tables with discretized solutions to each ode . the next stage must construct a single system - wide parameter vector by looking up and merging the best parameter combinations from the low - dimensional subproblems .
message passing , a probabilistic network inference technique that computes probability distributions for the parameters and sends the distributions as messages across the edges of the network .
loopy belief propagation is an iterative heuristic that terminates by convergence or when the specified maximum number of iterations is reached . on termination ,
this set of bins provides a starting point for the post - processor to refine , using the levenberg 
in essence , spedre is a hybrid global + local optimization method , but unlike other hybrid global + local methods , the global portion ( called spedre - base ) does not use stochastic sampling .
the final output of spedre is a plot of the bins provided by loopy belief propagation , as well as a vector of the optimized rate parameters .
our previous work specified the spedre algorithm in detail ( 11 ) , whereas the current work aims to describe the web server interface . 

asymptotic analysis of the underlying spedre - base algorithm ( 11 ) reveals attractive properties of the method because the time complexity scales exponentially with the network degree , but it scales efficiently ( polynomially ) with the number of species , the number of reactions and the size of the data set .
correspondingly , the method scales well on biological pathways with a bounded number of reactions per species .
dense networks with hub - like species have high network degree and are unsuitable for the spedre algorithm .
the spedre - base algorithm was implemented in c++ with an interface to copasi ( version 4.6 , build 32 ) for the levenberg  marquardt algorithm .
the web server version of spedre was implemented using the opal toolkit , as introduced in ( 18 ) . to display the customizable bin plot of spedre results , google chart api ( google inc . )
spedre requires two main inputs from users : the concentrations of all the molecular species and the biochemical reactions .
the concentrations of the molecular species would come from proteomic experiments , from computational hypotheses that merge published sub - networks or from any source that specifies the abundance of every species , at a set of discrete timepoints .
this requirement for input of all species is highly restrictive , but it allows spedre to focus on data - rich problems , which are a specialized but growing segment of parameter estimation work .
the concentrations must be specified in a comma separated value ( csv or txt ) file .
the file format is compatible with time - course simulation output from the copasi software , where the first line specifies the headers ( time and species names ) .
the first column of each additional row specifies the time value , with the species levels in the remaining columns .
if a data point is found to be invalid , the relevant time point is removed from the system before performing rate constant estimation .
if there is greater incompleteness or sparsity in the available measurements , users should use the copasiweb ( 14 ) service instead .
the biochemical reactions must be specified as an xml file in copasi_ml format ( 4 ) , which can be obtained from sbml format using a link to the conversion service in copasiweb ( 14 ) .
rate laws with predefined reaction types , which are necessary for constructing the correct types of kinetic parameters .
spedre is currently restricted to the following reaction types :  michaelis - menten catalysis ,  mass action ( irreversible) and  enzyme simple [ rate_constant  ( enzyme )  ( substrate ) ] .
in other words , a reversible reaction must be represented using two separate reactions , one in each direction ; reactions with high - order combinations must be re - expressed as a series of subreactions .
future work may automate the conversion process . the smbl conversion service provided at copasiweb ( 14 ) , may disrupt the names of the reaction types (  rate laws ) , in which case users must change the names of the reaction types manually .
the web server homepage provides descriptions and illustrations for several published pathways , including the akt pathway ( 19 ) , the mapk pathway ( 20 ) and a pathway of actin filament assembly - disassembly ( 21 ) . also available
is a spectrum of artificial networks ( circular or tree - shaped ) with widely varying sizes .
a set of default parameters can be modified using the submission form , if users wish to adjust how spedre executes .
the main options are the number of bins for discretizing the parameter ranges , and the number of iterations for loopy belief propagation .
the upper and lower bounds of the rate constants can be specified globally , or individual rate constants can override the upper bound and lower bound if these are specified in the network connectivity input file .
the maximum number of iterations can be set to zero if users wish to perform a standalone local search .
the anticipated error rate is an option for theoretical calculations , and it allows users to add gaussian noise to the observed data . another option for specialized users , called  samples per voxel , allows each voxel of parameter space ( each set of parameter bins ) to be evaluated by sampling multiple random points in the voxel , instead of using the voxel midpoint .
an example execution based on the mapk cascade is shown in figure 2 . using the web interface
, users can submit input files that follow the specified formats ( figure 2 , top box ) , and spedre performs the computation task while simultaneously displaying the execution page ( figure 2 , middle box ) . 

different execution specifications may result in different runtime performance , and some jobs may require several hours to complete .
users may wish to bookmark the location of the output page for a later visit .
a status page is linked to the execution page and shows the percentage of the overall task that has been completed .
users are advised to consult the asymptotic analysis of spedre - base algorithm ( 11 ) when adjusting the spedre execution options beyond the default values . on completion
, spedre returns an estimated range for each rate constant , in the form of a bin plot , as shown in figure 2 ( bottom box ) .
a bin plot is a visual representation of the resulting voxel in high - dimensional space , which gives users an impression of the exponential number of possible combinations of rate constants , even in a coarsely discretized search space .
finally , the bin midpoints are used as a starting point for local optimization , and the refined set of parameters are output as a vector of floating point numbers . the bin plot in figure 2
was generated using the google chart api ( google inc . ) , which imposes certain constraints , including the maximum url length of 2048 characters ( for plot formatting ) and maximum plot size of 300 000 pixels .
users may encounter cluttered plots for large network size ( > 30 rate constants ) .
as the api is actively developed with large user base across the industry , this current limit will be overcome with new updates of the api .
table 1 shows the web server s performance on five test cases using noiseless and 20% noise data . the weighted sum - of - square error ( sse )
represents how well a model with estimated parameters could fit the input time - series data . 

table 1.performance of the spedre web service on a series of test cases : synthetic networks ( circular network , tree network with random branch ) and biological networks ( pi3k / akt cascade , mapk cascade and actin filament assembly / disassembly pathway)test caseweighted sse ( unitless)spedre - base run - time ( s)total spedre run - time ( s)1 .
actin filament assembly / disassembly1.37468.00468.51weighted sse ( objective function value ) : sum - of - square error , weighted by mean square of each species concentrations across all time points , between simulated and given time series .
spedre was executed with lower bound = 0 , upper bound = 1 , logarithmically spaced binning with five bins , maximum number of iterations = 5 , gaussian noise = 0 and number of samples per voxel = 5 .
performance of the spedre web service on a series of test cases : synthetic networks ( circular network , tree network with random branch ) and biological networks ( pi3k / akt cascade , mapk cascade and actin filament assembly / disassembly pathway ) weighted sse ( objective function value ) : sum - of - square error , weighted by mean square of each species concentrations across all time points , between simulated and given time series .
spedre was executed with lower bound = 0 , upper bound = 1 , logarithmically spaced binning with five bins , maximum number of iterations = 5 , gaussian noise = 0 and number of samples per voxel = 5 .
these test cases used coarse discretization to run quickly at the expense of accuracy . for the pi3k / akt cascade and mapk cascade ,
( a good match with data means the parameterized model gives plausible explanations of the data , but alternative models or parameters may also exist ) .
the runtime of spedre base ( i.e. spedre without levenberg  marquardt ) was close to the total spedre runtime , indicating that the hybrid global - local approach incurs low additional runtime cost compared with global search alone .
the runtime measures also provide an empirical demonstration that spedre runtime scales efficiently with the size of the input network and poorly with the degree of the network .
specifically , the high - degreed actin filament assembly / disassembly pathway has only 14 species and 25 reactions , whereas the low - degreed circular network has 80 species and 80 reactions ; yet , execution on the latter network completes 23 times faster than the former .
spedre has been implemented as a web - based service for performing rate constant estimation on biochemical networks , such as cell signaling pathways and metabolic networks .
spedre uses a spline - based collocation approach , requiring extensive data as input and providing efficient coverage of enormous parameter spaces .
the computational power of a web server makes it suitable for intensive rate constant estimation jobs .
the server has dynamic display of the bin plot , as shown in figure 2 ( bottom box ) , which is customizable using javascript .
spedre performs preprocessing of user inputs to eliminate missing or invalid data points from the data file . in the scenarios involving a dense network or other features that violate the requirements of spedre
this service is not predictive because measured rate constants are not yet available for pathways of significant size . for users who wish to address the accuracy of parameter estimation as a purely mathematical problem , artificial data sets are available and
singapore - mit alliance iup [ r-154 - 001 - 348 - 646 to l.t.k . and j.k.w .
] ; mechanobiology institute ( to l.t.k . ) . funding for open access charge :
 cell signaling pathways and metabolic networks are often modeled using ordinary differential equations ( odes ) to represent the production / consumption of molecular species over time . regardless whether a model is built de novo or adapted from previous models , there is a need to estimate kinetic rate constants based on time - series experimental measurements of molecular abundance . for data - rich cases such as proteomic measurements of all species 
 , spline - based parameter estimation algorithms have been developed to avoid solving all the odes explicitly . 
 we report the development of a web server for a spline - based method . 
 systematic parameter estimation for data - rich environments ( spedre ) estimates reaction rates for biochemical networks . as input , it takes the connectivity of the network and the concentrations of the molecular species at discrete time points . 
 spedre is intended for large sparse networks , such as signaling cascades with many proteins but few reactions per protein . 
 if data are available for all species in the network , it provides global coverage of the parameter space , at low resolution and with approximate accuracy . 
 the output is an optimized value for each reaction rate parameter , accompanied by a range and bin plot . 
 spedre uses tools from copasi for pre - processing and post - processing . 
 spedre is a free service at http://ltklab.org/spedre . 
INTRODUCTION
PROCESSING METHOD
INPUT
OUTPUT
PERFORMANCE
DISCUSSION
FUNDING