1 Introduction
Computational modeling of biological systems is about describing quantitative relationships of biochemical reactions by systems of differential equations (Kitano, 2002). Knowledge of biological processes captured in such equations, when solutions to them match measurements made from the system of interest, help confirm our understanding of systems level function. Examples of such models include cell cycle progression (Chen et al., 2000), integrate and fire generation of heart pacemaker pulses (Zhang et al., 2000) and cellular behavior in synchrony with the circadian cycle (Leloup and Goldbeter, 2003). A particular appeal of modeling is that models can be interrogated with what if type questions to improve our understanding of the system, or be used to make quantitative predictions in domains in which measurements are unavailable.
A central issue in developing computational models of biological systems is setting parameters such as rate constants of biochemical reactions, synthesis and decay rates of macromolecules, delays incurred in transcription of genes and translation of proteins, and sharpness of nonlinear effects (Hill coefficient) are examples of such parameters. Parameter values are usually determined by conducting in vitro experiments (e.g. (Niedenthal et al., 1996; Wadsworth et al., 2001; Tseng et al., 2002; Wiedenmann et al., 2004)). When parameter values are not available from experimental measurements, modelers often resort to handtuning during the model development process and publish the range of values of a parameter required to achieve a match between model output and observed data. In dynamical systems characterized by variations over time, concentrations of different molecular species (proteins, metabolites etc.) may also be of interest. In this setting, we encounter two difficulties. First, parameters measured by in vitro experiments may not be good reflections of the in vivo reality. And, second, some parameters in a system may not be amenable to experimental measurements. These limitations motivate the need to infer parameters in a computational model based on inputoutput observations and recent literature on computational and systems biology has seen intense activity along these lines (Barenco et al., 2006; Vyshemirsky and Girolami, 2008; Jayawardhana et al., 2008; Henderson et al., 2010; Liu and Niranjan, 2012).
One way of setting parameters systematically is based in techniques for search and optimization. For example, Mendes and Kell (1998) compared several optimization based algorithms for estimating parameters along biochemical pathways, concluding that no single approach significantly outperforms other available approaches. Similar work on a developmental gene regulation circuit is described in FomekongNanfack et al. (2009), and on a spline approximation based method for learning the parameters of enzyme kinetic model in a cell cycle system is described in Zhan and Yeung (2011).
An alternate approach is the use of probabilistic Bayesian formulations to quantify uncertainties in the process of estimating parameters. Work described in Golightly and Wilkinson (2005); Dewar et al. (2010); Barenco et al. (2006); Vyshemirsky and Girolami (2008); Jayawardhana et al. (2008) fall into this category. With time varying or dynamical systems, some authors have pointed out advantages of sequential estimation models, formulating the problem as state and parameter estimation in a statespace modeling framework (Quach et al., 2007; Sun et al., 2008; Lillacci and Khammash, 2010)
. Kalman filtering and its variants, and nonparametric particle filtering have been applied, in this setting
(Yang et al., 2007; Liu and Niranjan, 2012).An approach that has attracted much interest recently is the method of Approximate Bayesian Computation (ABC) or likelihoodfree inference. While this approach has its roots in population genetics (Tavaré et al., 1997), where the likelihood is usually too complicated to write down in a computable form, it is also seen as a viable tool in systems biology parameter estimation problems ((Toni et al., 2009; Secrier et al., 2009)). In brief, the ABC approach assumes it is easier to simulate data from a model of interest than it is to compute and work with its likelihood under some assumed noise model. Hence a structured search could be carried out, in which one repeatedly simulates with parameter values sampled from a prior distribution, computes the error between simulated and observed data and decides to retain or reject a sample based on this error exceeding a threshold. Samples thus retained define a posterior density in the space of parameters to be estimated.
Several enhancements to this basic scheme of Tavaré et al. (1997) have been developed. Beaumont et al. (2002)
used a weighted linear regression adjustment to improve the efficiency of search (ABCRegression) with interesting applications
(Hamilton et al., 2005). In systems biology, Toni et al. (2009) proposed a Sequential Importance Sampling (SIS) based ABC approach to infer the parameters of LotkaVolterra and repressilator models. This ABCSIS method was also utilized by Secrier et al. (2009) to estimate the parameters of mitogenactivated protein kinase (MAPK) signaling pathway (Krauss, 2006).In implementing ABC algorithms, an inherent compromise between computational efficiency and accuracy of inference has to be made. This compromise is in the form of the acceptance threshold (), also referred to as tolerance. Setting to a high value results in a large number of acceptances but the resulting inference will be inaccurate. Researchers use arbitrary userspecified approaches (Sisson et al., 2007; Beaumont et al., 2009; Toni et al., 2009) to deal with this issue, i.e. start with a large and progressively reduce in value as the algorithm converges. In an elegant piece of work recently, Del Moral et al. (2012) showed how such an scheduling approach can be adaptively set. For the ABC class of algorithms this adaptive technique is indeed a major breakthrough.
Chiachio et al. (2014)
demonstrate the effectiveness of this adaptive ABC method by estimating parameters of two stochastic benchmark models (the second order moving average process and the white noise corrupt linear oscillator).
Hainy et al. (2016) also used this algorithm to learn parameters in the spatial extremes model from the data on maximum annual summer temperatures from 39 sites in the Midwest region of the USA. In epidemiology, Lintusaari et al. (2016) utilized such likelihoodfree approach to identify of transmission dynamic models for infectious diseases.An additional point to consider, which we pursue in this work, is that in models of Systems Biology not all parameters behave in the same way. Gutenkunst et al. (2007) observe that in a large number of models, systems level behavior is not sensitive to some parameters while being critically sensitive to others. These are referred to as ‘sloppy’ and ‘stiff’ parameters, respectively. Over a wide range of values with sloppy parameters, system output varies very slightly, while even small changes to the setting of stiff parameters cause large perturbations in outputs. The ABC framework, with its analysis by synthesis structure, is ideally suited to exploit this observation as we demonstrate in this work. Additionally, we comment on the inherent compromise between accuracy and computational efficiency in choosing the number of particles and parallel filters for the adaptive ABCSMC algorithm.
2 Methods
The computational approach we propose, shown schematically in Fig. 1, is a combination of sensitivity analysis (Saltelli et al., 1999) and advanced ABCSMC (Del Moral et al., 2012). We choose a particular method of sensitivity analysis (global sensitivity analysis, discussed later), to group the unknown parameters of a model into stiff and sloppy parameters. The sloppy parameters are estimated by a coarse search, which in ABC algorithms is achieved by using large values for acceptance thresholds. Once these are set, stiff parameters are estimated with adaptive evaluation of acceptance thresholds over a finer range.
We now summarize the methods of ABCSMC with adaptive acceptance tolerance and the global sensitivity analysis techniques used in this work. We start with a succinct, tutorial type description of the ABCSMC and Sensitivity Analysis frameworks, and present implemetation details to Supplementary Material.
2.1 Adaptive ABCSMC
In parameter estimation problems, we seek a posterior distribution over parameters given observations, denoted . As noted in the Introduction, the basic idea in ABC algorithms is to sample the unknown from a prior distribution, , synthesize data from the model under study, , where is the initial condition and is the model, and accept as a sample for the posterior if the synthesized data (referred to pseudoobservations in the following descriptions) is close enough in some sense to the observations X. In its earliest form (Tavaré et al., 1997), the generated particle was accepted only if was identical to the observations X. It became immediately evident that this is an inefficient procedure because thousands of trials needed to be performed before accepting one of the generated particles. A modification to the scheme, introduced by Pitt and Shephard (1999) was to define a tolerance and accept particles when the discrepancy between and X was smaller than this. In our discussion with Systems Biology models, we will focus on
being a set of ordinary differential equations which can be numerically integrated, and use Euclidean distance between the pseudoobservations and the model output as measure of discrepancy.
The Sequential Monte Carlo (SMC) method has recently been applied within the ABC setting (Sisson et al., 2007; Toni et al., 2009; Beaumont et al., 2009), in which, instead of synthesis from a single sample drawn from the prior being tested for acceptance of the sample, a population of samples is drawn, and progressively perturbed and synthesis from them tested for acceptance. At any stage, particles are drawn by sampling from the population of the previous stage in proportion to weights associated with them and perturbed via a transition kernel . Pseudoobservations are then synthesized from the underlying model for the new population of samples: . Particle is accepted and weighted if the discrepancy between synthetic data and true dataset X is lower than the current tolerance . To preserve diversity of particles, i.e. when {, resampling is also applied (Doucet et al., 2001). This population based approach is shown to result in more effective exploration of the space. Additionally, as mentioned before, the acceptance criterion is gradually reduced sequence of tolerance thresholds at Sisson et al. (2007); Beaumont et al. (2009); Toni et al. (2009), in an arbitrary, userdefined way.
Del Moral et al. (2006) introduced a SMC sampler which constructs (referred to integer factor) Markov kernels being operated in parallel. In the series of ABC developments, Del Moral et al. (2012) extended this concept to the ABC setting and formed an adaptive algorithm to determine the tolerance schedule . The weight calculation of this SMC sampler is given as
(1) 
where are the pseudoobservations, and is the subblock of pseudoobservations ( is equivalent to aforementioned ) can be interpreted as the synthetic outputs generated by the parameter particle at iteration . and are the dimension of states in system and the number of data points in output, respectively. is an indicator function that returns one if the discrepancy between pseudoobservation and data X is less than the tolerance , zero otherwise. A is the discrepancy, and which is measured by an Euclidean distance in this work.
By taking advantage of distribution of particles in the parallel filters, ABCSMC is able to adaptively select the current tolerance level . The idea behind this automatic scheme is to determine an appropriate reduction of the tolerance level based on the proportion of particles surviving under the current tolerance. If a large amount of particles remain ‘alive’, it implies the acceptance criterion is relatively loose and it is safe to make a jump for the next tolerance level. In contrast, if the ratio of ‘alive’ particles is low, this means that particles are less likely to describe the posterior and therefore, a tiny movement should be considered.
Given the proportion of ‘alive’ particles by
(2) 
the next tolerance level is chosen by making such proportion equals to a given percentage of the current tolerance , where is the tolerance reduction factor. The search of is achieved by fzero function in Matlab via setting the starting point to .
To solve ODEs in use of running filters parallel, the initial conditions are set to , where , is ‘true’ initial values for generating observations and is the scaling factor to control consistency/inconsistency of initial conditions across filters. Additionally, for moving particles to explore the parameter space, similarly to the previous work (Liu and Niranjan, 2012), we use the kernel smoothing with shrinkage as parameter evolution (Liu and West, 2001), instead of random walk kernel. The details and pseudocode of ABCSMC can be found in Section 2 of Supplementary materials accompanying the online version of this paper.
The Matlab code of proposed method is available at: https://github.com/brianliu2/ABCSMCSA
2.2 Extended Fourier Amplitude Sensitivity Test
The extended Fourier amplitude sensitivity test (eFAST) (Saltelli et al., 1999)
is one of the global sensitivity analysis techniques, which is derived by decomposing variance and claimed to be applicable for analyzing the nonlinear and nonmonotonic systems. It has been successfully deployed to analyze the property of parameters in TCRactivated MAPK signalling pathway model
(Zheng and Rundell, 2006) and thermodynamics gene expression model (Dresch et al., 2010). The algorithm initially partitions the total variance of the data, evaluating what fraction of the variance can be determined by variations in the parameter of interest. This quantity, known as the sensitivity index, defines the property of stiff or sloppy, is calculated as(3) 
The variance is calculated with respect to sets of synthesis which are generated by solving the model associated with samples of parameters. In order to draw samples for parameters, Saltelli (2002)
derived a heuristic sinusoidal function, given as
(4) 
where is the
matrix for providing random phaseshifting and which follows uniformly distributed
. specifies the number of samples drawn from the function . is the dimension of parameters in system. in Eqn. (2.2) defines aequally spaced vector where each interval is equally distributed from
to . In this scheme, it is crucial to properly determine the frequency vector , in which the highest value referred to maximum frequency is assigned to the parameter under study. is known as the interference factor and acts as the remover for numerical amplitude from superposing of waves. From the empirical investigation (Saltelli, 2002; Marino et al., 2008b), it is usually used as 4 or 6 (we use 4 in this work). Other frequencies in are set in the range with a regular increment , where .This strategy means that the sensitivity of underlying parameter is assessed by picking the samples with the highest frequency , while the samples for the rest of the parameters are selected with the complementary frequencies . This process is repeated until the samples of each parameter is drawn with highest frequency once. An illustrative example of this cycling process is shown in Fig. 2.
Consequently, the parameter can be seen as the most sensitive when which has the largest normalized variance, i.e. its variation causes the highest uncertainty. In addition, for any sample of parameter drawn from the unit hypercube , it has to be rescaled to realvalued space as such the model evaluation can be carried out. Hence, a noninformative prior covered whole range of parameter value is required. In this work, we consider the suggestion from Marino et al. (2008a), where authors claimed that eFAST artificially yields small but nonzero sensitivity indexes for parameters, and thus a ‘dummy’ parameter which does not really exist in model and has no influence on behaving dynamics in any way, is introduced in eFAST to absorb such artifacts. The complete description of eFAST including the mathematical derivation and pseudocode are presented in Section 3 of Supplementary materials.
3 Results and Discussion
We demonstrate the effectiveness of the proposed method using three models of sustained periodic oscillations and and one model of transient response to shock. These are models used by previous authors, including ourselves, enabling comparisons to be made. We give details of implementation in Section 6 of Supplementary materials. Crucial results from the heat shock and repressilator systems are discussed here and results from the remaining two systems are given in Supplementary materials.
The heat shock model We consider the case of estimating all six parameters of this model simultaneously. This is the most difficult case considered in our previous work (Liu and Niranjan, 2012), in which particle filters were able to correctly estimate four of the six unknown parameters, and failed in the remaining two ( and ). Fig. 3.A shows results of sensitivity analysis of one of the states (see equations of the system in Supplementary materials). Of these, parameters , and , to which is most sensitive to, are treated as stiff and the remainder as sloppy. Posterior distributions over estimates of the stiff parameters, using the proposed method, are shown in Fig. 3.C and the corresponding sloppy ones are given in Fig. S1(c). While the posterior means of and are estimated to a high level of accuracy, is not. This is explained by the sensitivity results in which we note that, though we took all these three to be stiff parameters, the state is not as sensitive to as it is to and . Hence, relatively poor estimates of will still lead to accurate reconstruction of the model outputs.
In Fig. 3.B, we show two responses synthesized from the posterior mean values of particle distributions from a classic particle filtering approach (as considered in our previous work, Liu and Niranjan (2012)) and the ABCSMC method proposed here. We note that in this difficult problem of estimating all six parameters of the model, ABCSMC is able to successfully find solutions of parameters from which the underlying state could be accurately generated for the state in the heat shock model. We observe the same to be true for the other two states, and as well (graphs shown in Fig. S1.(a) of Supplementary materials).
Repressilator system The deterministic repressilator system is constructed as a synthetic gene regulatory circuit which can sustain oscillations by the mutual repression of gene transcription (Elowitz and Leibler, 2000). This system consists of six differential equations, from three pairs of mRNA and protein, and has four parameters. This system was analysed by Toni et al. (2009) to demonstrate their ABCSIS approach.
While we have set the posterior means of a coarse search to sloppy parameters ( and ), ABCSMC is capable to find accurate solutions of stiff parameters, given allocated extra computational budget. Posterior distributions of a stiff () and sloppy () parameters from our proposed method and its predecessor ABCSIS are shown in Fig.4.BE. We note that, with an identical tight acceptance condition , both methods are adequate in estimating stiff parameters to a relatively high level of accuracy. However, ABCSIS shows a distinguishable advantage in identifying sloppy parameters, whereas our proposed method fails to achieve convergence to the correct estimate. Reconstruction of model behavior, using the posterior mean values from both methods, tells that the failure of finding solutions for sloppy parameters does not hugely hamper the effectiveness in characterizing dynamics. The complete results are shown as color figure in Fig. S2S3 of Supplementary materials. In addition, we also tested how sensitivity analysis affects ABCSMC by setting all possible combinations of stiffness/sloppiness to parameters, we found that at lease one stiff parameter is correctly chosen, ABCSMC is able to find solutions. Results are given in Table S7.
In order to show how ABCSIS is influenced by the suboptimal userspecific algorithmic setting, we compare the computational expense between ABCSIS and our proposed method using identical initial and final tolerances. Since ABCSMC on average requires 140 iterations to reach , we therefore reduce the tolerance from down to using 140 steps at regular intervals. The algorithm efficiency is quantified by counting the number of model evaluations, at which most of computational budget is spent, to reach final tolerance . Fig. 4.F clearly shows that the number of evaluations made by ABCSIS is approximately seven times greater than ABCSMC. This is because the suboptimal chosen transition kernel and tolerance schedule make ABCSIS expend considerable computation on searching for acceptable particles.
Other systems We further consider delaydriven oscillatory system and cell cycle system, for which the complete descriptions and results are given in Section 4.3 and Section 4.4 of Supplementary materials.
In the delaydriven oscillatory example, for testing the general advantage of proposed method, we use the nonsophisticated ABCMCMC (Marjoram et al., 2003) instead of ABCSMC in this example. From the results given in Fig. S4, it is easy to observe that the original ABCMCMC poorly performs with the coarse acceptance criterion, while it can produce the accurate inference by using a small tolerance . When the similar precision of inferences is achieve, as expected, the proposed method greatly outperforms in efficiency (the comparison is shown in Fig. S4.F).
The cell cycle system, with 12 free parameters in the model, is a relatively high dimensional system. When particles are sampled in this space, we found that for a significant fraction of the samples, the ODE solver failed to terminate. With the proposed method, because sensitivity analysis identifies the sloppy parameters which have been set to the posterior means of a coarse search, the dimensionality is reduced and this issue gets circumvented.
However, since the state in this cell cycle has no influential parameter in behaving dynamics (sensitivity analysis with respect to is given in Fig. S8), therefore, a favorable computational budget allocation for state can only be provided by associating the higherorder sensitivity index (influence on outputs caused by the interaction between parameters).
Effect of and In running ABCSMC algorithms, we would ideally like to choose large values for and for achieving high accuracy of exploring the posterior density. Naturally, this will increase the computational complexity of the implementation. How does one tradeoff spending a given computational budget across (the number of parallel filters) and (the number of particles per filter)? The choice is likely to be problem dependent, but an empirical exploration, using the repressillator system, over three sets of values for these = 20 and = 2, 000; = 200 and = 200; and = 2, 000 and = 20, keeping their product at 40, 000, is shown in Fig. 5. The corresponding accuracies of inference are given in Table 1.
NRMSE  

2000  20  6.412.71  1.920.20  1.010.20  1.040.34 
200  200  6.170.72  1.210.04  1.210.10  1.110.10 
20  2000  5.200.45  1.110.04  1.030.05  0.990.02 
As seen in the graphs, smaller values of are computationally less expensive. This is due to lower value of
causing a higher probability of particles being ‘killed’ and having a zero weight. As as consequence, decrements taken in
are smaller slowing convergence to its target . For instance, considering the large example, i.e. = 2000, evaluating importance using Eqn.1, the zero weight barely appears. Intuitively, the greater nonzero proportion of the weight vector implies that most particles fulfill the current acceptance criterion, therefore, a large decrement should be taken for the next tolerance level. From the perspective of accuracy, when gaining a larger number of particles, the diversity of realizations increases and better performance in terms of precision is naturally expected. This conclusion is verified by the comparison of Normalized Root Mean Squared Error (NRMSE) which is a measurement to quantify the quality of prediction and its expression is given as:(5) 
From the results shown in Table 1, the higher value generally delivers the better inference, where = 2000 performs best among all considered settings.
Why use eFAST for sensitivity analysis? We have chosen eFAST, a global method, for sensitivity analysis in this work. The search carried out by this method, systematically exploring a wide range in the parameter space is computational expensive. An alternate method of quantifying parameter sensitivities, particularly in the context of particle filtering, adopted by Toni et al. (2009)
is based on principal component analysis (PCA) of the particle distribution, and to use the principal directions as indicators of sloppy and stiff directions of parameter combinations. Parameters that are well aligned to the directions may show the declared stiff and sloppy. However, this simplistic approach is inadequate for our purpose for two reasons. Firstly, we wish to identify sloppy and stiff parameters of the models before running the particle filters, hence PCA on the distribution of a converged solution is not very useful. Secondly, and somewhat more importantly, PCA is a good representation of data that is multivariate Gaussian. If we expect the distribution of particles to be multivariate Gaussian, we would not be running particle filtering anyway, because Kalman filter type updates of posterior means and covariances are the known optimal solutions. In order to confirm this, we took the distributions of particles, after convergence, of all the four models considered and tested them for multivariate Gaussianity using HenzeZirkler (multivariate normality) test
(Henze and Zirkler, 1990). Such test is carried out upon measuring the nonnegative functional distance between two distributions. If data of interest are distributed as multivariate normal, the test statistic is approximately lognormally distributed. The function
hzTest from R package MVN (Korkmaz et al., 2014) is used in this work. From the test, we found three of them to be nonGaussian. This confirms that the choice of global variance based sensitivity analysis is the correct one for our analysis.4 Conclusion
In this work, we proposed an inference method for analyzing Systems Biology models that couples sensitivity analysis and approximate Bayesian computation. Our proposed method is particularly advantageous in difficult settings of estimating all (or most of) the parameters of a model from noisy observations, because it strikes a balance between accuracy and efficiency. The method exploits the fact that all parameters in model have different significance in characterizing model dynamics in terms of their sensitivities. By resynthesis data from models with estimated parameters, we show that the values of parameters that are more critical (stiff parameters) need to be determined with care, while the sloppy parameters need not be estimated to high precision. To facilitate such inference, we have proposed a three stage strategy in which a selective computational budget allocation is implemented via sensitivity analysis, in which the sloppy group is estimated by a coarse search followed the reestimation of stiff parameters to tighter error tolerances.
We have demonstrated the effectiveness of the proposed approach on three systems of oscillatory behavior and one of transient response. The results show that the introduction of favorable inference strategy allows to reduce the dimension of unknown parameters, and paves a potential way to tackle the complex problems. Additionally, the used ABCSMC has attracted much interest due to its adaptivity in determining tolerance schedule and the ‘no rejection’ of particles allows to boost the efficiency via parallel computing. In the simple problem, e.g. the delaydriven oscillatory system, with performing similarly in accuracy, the proposed inference method expends much less computational cost than the existed ABC methods.
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