Identification of models of heterogeneous cell populations from population snapshot data
 Jan Hasenauer^{1}Email author,
 Steffen Waldherr^{1},
 Malgorzata Doszczak^{2},
 Nicole Radde^{1},
 Peter Scheurich^{2} and
 Frank Allgöwer^{1}
DOI: 10.1186/1471210512125
© Hasenauer et al; licensee BioMed Central Ltd. 2011
Received: 30 September 2010
Accepted: 28 April 2011
Published: 28 April 2011
Abstract
Background
Most of the modeling performed in the area of systems biology aims at achieving a quantitative description of the intracellular pathways within a "typical cell". However, in many biologically important situations even clonal cell populations can show a heterogeneous response. These situations require study of celltocell variability and the development of models for heterogeneous cell populations.
Results
In this paper we consider cell populations in which the dynamics of every single cell is captured by a parameter dependent differential equation. Differences among cells are modeled by differences in parameters which are subject to a probability density. A novel Bayesian approach is presented to infer this probability density from population snapshot data, such as flow cytometric analysis, which do not provide single cell time series data. The presented approach can deal with sparse and noisy measurement data. Furthermore, it is appealing from an application point of view as in contrast to other methods the uncertainty of the resulting parameter distribution can directly be assessed.
Conclusions
The proposed method is evaluated using artificial experimental data from a model of the tumor necrosis factor signaling network. We demonstrate that the methods are computationally efficient and yield good estimation result even for sparse data sets.
Background
The main goals of research in systems biology are the development of quantitative models of intracellular pathways and the development of tools to support the modeling process. Thereby, most of the available methods and models consider only a single "typical cell" whereas most experimental data used to calibrate the models are obtained using cell population experiments, e.g. western blotting. This yields problems in particular if the studied population shows a large celltocell variability. In such situations inferring a single cell model from cell population data can lead to biologically meaningless results. In order to understand the dynamical behavior of heterogeneous cell populations, it is crucial to develop cell population models, describing the whole population and not only a single individual [1–4].
This has already been realized by several authors, and it has been shown that stochasticity in biochemical reactions and unequal partitioning of cell material at cell division can lead to complex population dynamics [1–5], such as bimodal distributions. Besides these sources for heterogeneity also genetic and epigenetic differences have to be considered [6].
For the purpose of this paper, heterogeneity in populations is modeled by differences in parameter values and initial conditions of the model describing the single cell dynamics [4, 7, 8]. The network structure is assumed to be identical in all cells. The distribution of the parameter values within the cell population is described by a multivariate probability density function, which is part of the population model. This modeling framework is well suited for modeling genetic and epigenetic differences among cells [2, 4, 7].
In the following, the problem of estimating the probability density of the parameters is studied. Therefore, we employ population snapshot data (PSD), which provide single cell measurements at every time instance but which do not provide single cell time series data. A typical experimental setup which provides PSD is flow cytometric analysis. In general, PSD are a common data type in the experimental analysis of biological systems.
So far, there are not many methods available for the estimation of parameter distributions. In pharmacokinetic studies mixed effect models [9] are used frequently. Unfortunately, as in the problem we consider the number of individuals is very large (> 10^{4}) and the amount of information per individual very limited (often only one data point), these methods are computationally too demanding. Furthermore, as in this study we are particularly interested in intracellular signal transduction, also methods which purely focus on the population balance [10–12] cannot be employed. In [8, 13, 14] methods are proposed which can in principle deal with the problem at hand. There, the considered estimation problem has been formulated as a convex optimization problem. Unfortunately, these methods either require an extensive amount of measurement data [8, 13], and/or do not allow considering prior knowledge [8, 13, 14]. Additionally, no methods to evaluate the reliability of the estimates are provided.
In this paper a novel Bayesian approach [15, 16] for inferring the parameter density will be introduced. The approach is mainly based on the maximum likelihood methods presented in [13, 14], but can deal with sparse and noisy single cell data in addition to realistic measurement noise models. Furthermore, one may directly access the remaining uncertainty of the estimation result and the prediction uncertainties via the calculation of Bayesian confidence intervals [17, 18]. It is shown that the posterior distribution can be determined efficiently employing a parameterization of the parameter density in combination with commonly used Markov chain Monte Carlo (MCMC) sampling techniques [19].
To illustrate the properties of the proposed methods, a mathematical model of the tumor necrosis factor (TNF) pathway [20] is analyzed using artificial experimental data.
Methods
Problem statement
Cell population model
with state variables , output variables , and parameters . The vector field is Lipschitz continuous and the functions and are continuous. If for example the concentration is measured via flow cytometry, we would have , where c is a proportionality factor. The index i specifies the individual cells within the population. The parameters θ^{(i)}can be kinetic constants, e.g. reaction rates or binding affinities.
Note that interactions among individual cells influencing the analyzed pathway are not allowed. This is a restriction but indeed fulfilled in many in vitro lab experiments.
Measurement data and noise
In this paper we consider experimental setups where measurement data are obtained in the form of population snapshots, e.g. via flow cytometry. Population snapshots are taken at different times t_{ j }, and the j th snapshot contains measurements for the output y of M_{ j } cells. Due to experiment setup, it can be assumed that any cell is present at most in one snapshot.
The cells in the individual snapshots are referenced through index sets: snapshot j contains all cells from the index set , with M_{0} = 0. Thereby, in which is the number of snapshots.
in which is the total number of measured cells.
We emphasize that experimental setups are considered in which cells are not tracked over time. These setups are very common in studies on the population scale. Classical examples for measurement techniques yielding such data are flow cytometric analysis and cytometric fluorescence microscopy. These measurement techniques allow to determine protein concentrations within single cells. As the population is well mixed when the measurement is performed and no cell is measured more than once, the individual single cell measurements are independent. This independence of and (respectively and , i_{1} ≠ i_{2}, holds if both cells are measured during one snapshot as well as if the cells are measured within different snapshots .
for all j = {×, +}, k = 1, ..., m. This noise model allows the study of relative and absolute measurement noise and describes the commonly seen noise distributions in biological experiments [21].
Problem formulation
As mentioned previously, when studying heterogeneous populations the density of the parameters Θ is in general unknown but necessary to gain an indepth understanding of the population dynamics. Therefore, we are concerned with the problems:
Problem 1 Given the measurement data , the cell population model Σ_{pop}, and the noise model (8), infer the parameter density Θ*.
Problem 2 Given the measurement data , the cell population model Σ_{pop}, and the noise model (8), determine the uncertainty of the estimated parameter density Θ*.
Unfortunately, the number of cells considered in a standard lab experiment is on the order of 10^{4} to 10^{7}. Thus, simulating the population model (2) is computationally expensive. Furthermore, it is hard from a theoretical point of view to deal with ensemble models such as (2). Densitybased descriptions of the population dynamics are far more appealing for solving Problem 1 and 2. Therefore, in the next section a density description of the population is introduced.
Densitybased modeling of cell populations
in which is an arbitrary subset of the output space. Hence, y^{(i)}(t) can be interpreted as a random variable which is distributed according to ϒ(yt,Θ).
are used to conserve the property that all variables are positive. The positive definite matrix H is used to select the width of the kernel , and is chosen using the available rule of thumb described in [23]. The selection of H is crucial for achieving a good approximation of the probability density, and depends strongly on S.
Approaches similar to the one we use to approximate ϒ(yt,Θ) are employed in [13, 14], in [8] with a naive density estimator and in [24] in the context of cell migration.
Estimation of the parameter density
In the previous section an approach to determine the output density ϒ within the cell population for a given parameter density Θ is presented. Based on this an approach for estimating Θ from the available data is developed next.
Bayes' theorem for heterogeneous cell populations
in which , with y(t^{(i)}, θ) being the output at time t^{(i)}for a cell having parameters θ. This reformulation of (16) is possible as ϒ directly depends on Θ. This step simplifies the evaluation of tremendously.
Based on (15) and (17), the calculation of the posterior probability for a given probability density of the parameters Θ is possible. Unfortunately, the inference problem nevertheless cannot be solved directly, as Θ is element of a function space, and hence further steps are necessary.
Parameterization of parameter density
The ansatz functions are themselves probability densities with . The weighting vector is denoted by , where n_{ φ } is the number of ansatz functions and to guarantee that Θ_{ φ } is a probability density. The weightings φ_{ j } can be interpreted as parameters determining the probability density Θ_{ φ } and are for the remainder also called density parameters.
Note that the method presented in the following is independent of the choice of ansatz functions. Nevertheless, a clever choice of the ansatz functions may improve the approximation of the true parameter density dramatically. In this work, the ansatz functions are chosen to be multivariate Gaussians.
in which ϒ (yt, Λ_{ j }) is the output density obtained for single cell parameters distributed according to Λ_{ j }. This representation of the response is possible as the output density fulfills the superposition principle with respect to the parameter distribution Θ_{ φ }. This reformulation has the advantage that computing the output density for arbitrary density parameters φ only requires the nonrecurring computation of the responses ϒ (yt, Λ_{ j }) and summation of those.
Reformulation of posterior probability
in which θ^{(k)}is drawn from Λ_{ j }, θ^{(k)}~ Λ_{ j }, and S_{ c } is the total size of the Monte Carlo sample {θ^{(k)}}_{ k }. If Λ_{ j } allows for an efficient drawing of samples, the computational cost of determining is reasonable, requiring S_{ c } simulations of the single cell model (1).
in which is the unnormalized posterior probability. Sampling from and will yield the same distribution of sample members. Furthermore, and have the same extrema.
Computation of maximum posterior probability estimate
Given the simplified unnormalized posterior probabilities one important question is which parameter density Θ_{ φ } maximizes . This is the most likely parameter density for the measured data and the prior knowledge.
This minimization problem can for concave p(Θ_{ φ }) be solved rather efficiently, as in such case (31) is a convex optimization problem [25]. For this problem solvers exist which guarantee convergence to the global optimum in polynomial time, e.g the interior point method [25].
Uncertainty of parameter density
In the previous section a method is presented which allows the computation of the maximum posterior probability estimate . As measurement data are limited and noise corrupted this estimate will not be the true parameter density. Hence, the uncertainty of the parameter density has to be evaluated.
Sampling of posterior probability density
In order to analyze the uncertainty of the estimate, a sample of the posterior probability density is generated. This is possible, as the unnormalized posterior probability of a distribution can be evaluated efficiently given (24)  (28). In this work the sampling is performed with a classical MetropolisHastings method [19]. Also Gibbs or slice sampling approaches may be employed. Compared to importance and rejection sampling these methods are well suited as they do not require the selection of an appropriate proposal density, a task which is difficult in this case.
The main point of concern when using MCMC sampling for the problem at hand is that the prior probability and the posterior probability respectively are only nonzero on a (n_{ φ } 1) dimensional subset of the density parameter space (28). This is due to the fact that the sum over the elements of φ has to be one for Θ_{ φ } being a probability density. If a standard proposal step was used, the acceptance rate would have been zero.
 1.Draw proposals from the (n_{ φ } 1)dimensional reduced proposal density T_{ r },(32)
 2.Determine such that ,(33)
In this work, the reduced proposal density is chosen to be a multivariate normal distribution, , with covariance matrix .
The distinction of the two cases is hereby crucial to ensure that only probability densities which are greater than zero for all are accepted.
By combining update and acceptance step one obtains an algorithm which draws a sample of weighting vectors , or respectively parameter densities , from the posterior distribution. The number of sample members is thereby S_{ φ }. The pseudo code for the routine is given in Algorithm 1. In particular, the facts that

the conditional probabilities are only computed once in the beginning, and that

every evaluation of the acceptance probability p_{ a }requires only a small number of algebraic operations,
ensure hereby an efficient sampling. Without this reformulation the integral defining the conditional probability would have to be computed in each update step. The resulting computational effort would be very large.
Algorithm 1 Sampling of posteriori distribution
Require: data , prior p(Θ_{ φ }), model p(yθ ), ansatz functions , initial point φ^{0}.
Calculation of conditional probabilities employing p(yθ ).
Initialize the Markov Chain with φ^{0}.
for k = 1 to S_{ φ }do
Given φ^{ i } propose φ^{k+1}from proposal density T (φ^{k+1}φ^{ k }).
Calculate posterior probability .
Generate uniformly distributed random number r ∈ [0,1].
if r <p_{ a }(φ^{k+1}φ^{ k }) then
Accept proposed parameter vector φ^{k+1}.
else
Restore previous parameter vector, φ^{k+1}= φ^{ k }.
end if
end for
end
Bayesian confidence intervals
The sample generated by Algorithm 1 contains information about the shape of the posterior density . This information can be employed to determine the Bayesian confidence intervals, also called credible intervals.
In this work an approach is presented which employs the percentile method [17] to analyze the uncertainty of Θ_{ φ }. The 100αth percentile of a random variable r is the value below which 100α % of the observations fall. Accordingly, the 100(1α)th percentile interval of r is defined as . The Bayesian confidence interval is frequently defined as the 95th percentile interval [18]. Thus, the parameter is contained in with a probability of 95% given the measurement data and the prior knowledge.
As the sample is given, an approximation of and can be obtained by studying the percentiles of the sample [26]. For instance the nearest rank method or linear interpolation between closest ranks can be used to determine .
Predictions of output density
is determined.
This sample can be used to approximate the prediction confidence interval . As the population model has to be simulated only n_{ φ } times, this calculation is computationally tractable.
Results and discussion
To illustrate the properties of the proposed methods, artificial measurement data of a cell population responding to a tumor necrosis factor (TNF) stimulus will be analyzed. For illustration purposes, we consider a case where only one parameter is distributed in a first step. In a second step, we show that the method is also applicable in the case of multiparametric heterogeneity.
In multicellular organisms, the removal of infected, malfunctioning, or no longer needed cells is an important issue. Therefore, multicellular organisms developed different mechanisms to externally enforce cell death. Thereby the signaling molecule TNF is one of the key players.
TNF can bind to specific death receptors in the cell membrane and is able to induce programmed cell death, also called apoptosis, via the activation of the caspase cascade. On the other hand, it promotes cell survival via the inflammatory response, specifically activation of the NFκ B pathway [27]. The proportion of the activation of these two signaling pathways decides about the fate of the single cell. In the following a simple model for the caspase and NFκ B activation is studied which reproduces the main features of the single cell response to a TNF stimulus.
Model of TNF signaling pathway
Nominal parameter values for the TNF signaling model (41).
i  1  2  3  4  5 

a _{ i }  0.6  0.2  0.2  0.5  
b _{ i }  0.4  0.7  0.3  0.5  0.4 
It is known from experiments that the cellular response to a TNF stimulus is highly heterogeneous within a clonal cell population. Some cells die, others survive. The reasons for this heterogeneous behavior are unclear, but of great interest for biological research in TNF signaling, e.g. the use of TNF or related molecules as anticancer agent.
To understand the biological process at the physiological and biochemical level it is crucial to consider this cellular heterogeneity, using for example cell population modeling. Here, we model heterogeneity at the cell level via differences in the parameters b_{3} and a_{4}. The parameter b_{3} describes the inhibitory effect of NFκ B via the C3a inhibitor XIAP onto the C3 activity, and the parameter a_{4} models the activation of Iκ B via NFκ B. Studies showed that these two interactions show large celltocell variability [4, 7, 28].
Univariate parameter density
Given the ansatz functions Λ_{ j } (45) the conditional probabilities of observing are determined using importance sampling, according to (25). This computation takes about three minutes, on a standard personal computer using a single CPU. Thereby, 32% of the computation time are required for the simulation of the individual cells y(t, θ^{(j)}) for individual parameter values θ^{(j)}, and 59% for the evaluation of the conditional probability . The rest is spent on pre and postprocessing. Subsequently, MCMC sampling is performed to obtain a sample of the prior (with σ^{2} = 0.05), of the conditional, and of the posterior probability distribution. The sample has S_{ φ } = 10^{6} members and the generation takes only four minutes. The computation is very fast, as the proposed approach simplified the evaluation of the conditional probability to a matrix vector multiplication. Note, that all steps during the computation of the conditional probabilities and the MCMC sampling can be parallelized, yielding a tremendous speedup for more complex models.
The results of the sampling are illustrated in Figure 7 using parallel coordinates [29]. From this representation of it can be seen that after the learning processes most of the density parameters still show large uncertainties. The uncertainty in the posterior distribution is a lot smaller than the uncertainty in the likelihood function, due to the stabilization via the prior. Note that the visualization also uncovers pronounced correlations between some parameters, e.g. φ_{10} and φ_{11} are negatively correlated for . This indicates that the model of the density of b_{3} is overparameterized with respect to the data. Thus, the number of ansatz functions could be reduced while still achieving a good fit.
To analyze the uncertainty of Θ_{ φ } in more detail the sample is employed to determine the 80%, 90%, 95%, and 99% Bayesian confidence intervals. The results are depicted in Figure 8. It can be seen that the confidence intervals for some values of b_{3} are rather small, indicating that the data contain many information about these regions. Unfortunately, in particular for b_{3} > 0.6 the confidence intervals are very wide showing that the parameter density in this area cannot be inferred precisely. But, although the amount of data is limited and the uncertainty with single φ_{ i }'s may be large, the posterior distribution of Θ_{ φ } already shows key properties of the true parameter density, e.g. the bimodal shape, which has not been provided as prior information. This bimodal shape is also seen in the likelihood function, but there the uncertainties are larger than in the posterior probability distribution.
It is obvious that, although the parameter distributions show large uncertainties, the predictions are rather certain. This is indicated by tight confidence intervals. Furthermore, the mean predictions and the predictions with highest posterior probability agree well with the true output distribution , for measured output C3a and predicted output NFκ B. The small prediction uncertainties can be explained to be sloppiness [30] of the density parameters φ_{ i } parametrizing the distribution of b_{3}. A detailed analysis indicates (not shown here) that the number of ansatz function can be decreased, still ensuring a good approximation of the distribution of b_{3}.
Multivariate parameter density
In many biological systems several cellular parameters are heterogeneous and different cellular concentrations can be measured [7]. Therefore, we show in this section that the proposed method can also be employed to estimated multivariate parameter densities from multidimensional outputs. Furthermore, the influence of the choice of the prior on the estimation result is analyzed.
Conclusions
In this paper a Bayesian approach for inferring the parameter density for heterogenous cell populations with single cell resolution from population data is presented. We show that the proposed model can deal with limited and noisy measurement data as well as realistic noise models. The method utilizes a parameterization of the parameter density which, in combination with a reformulation of the conditional probability, allows a computationally efficient evaluation of the posterior probability. Compared to other methods for cell populations this approach does not rely on the approximation of the measured population response using density estimators.
For sampling from the posterior probability the MetropolisHastings algorithm is used. Here it has been adapted to be applicable to the considered constraint problem. Using this sampling strategy a sample from the posterior probability density is determined. This sample is employed to compute Bayesian confidence intervals for the parameter distribution, as well as for the model predictions. Also summary statistics like mean parameter density and mean predicted output density can easily be determined. For concave prior distributions we could even show that the calculation of the parameter density for the highest posterior probability is a convex problem.
The properties of the proposed scheme are evaluated using artificial data of a TNF signal transduction model. It could be shown that the proposed method yields good estimation results for a realistic experimental setup. Furthermore, although the remaining uncertainties are large, the predictions showg only small uncertainty indicating sloppiness of parameters.
Concerning the choice of the prior distribution it could be shown that the regularizing effect is beneficial if only little data is available. On the other hand, if the amount of available data increases, informative but not carefully chosen priors slow down the convergence.
Declarations
Acknowledgements
The authors acknowledge financial support from the German Federal Ministry of Education and Research (BMBF) within the FORSYSPartner program (grant nr. 0315280A and D), from the German Research Foundation within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart, and from Center Systems Biology (CSB) at the University of Stuttgart.
Authors’ Affiliations
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