Global sensitivity analysis of biological multiscale models

Plants are complex organisms that adapt to changes in their environment using an array of regulatory mechanisms that span across multiple levels of biological organization. Due to this complexity, it is difficult to predict emergent properties using conventional approaches that focus on single levels of biology such as the genome, transcriptome, or metabolome. Mathematical models of biological systems have emerged as useful tools for exploring pathways and identifying gaps in our current knowledge of biological processes. Identification of emergent properties, however, requires their vertical integration across biological scales through multiscale modeling. Multiscale models that capture and predict these emergent properties will allow us to predict how plants will respond to a changing climate and explore strategies for plant engineering. In this review, we (1) summarize the recent developments in plant multiscale modeling; (2) examine multiscale models of microbial systems that offer insight to potential future directions for the modeling of plant systems; (3) discuss computational tools and resources for developing multiscale models; and (4) examine future directions of the field.

Parameter estimation is a major challenge for mathematical modelling of biological systems. Given the uncertainties associated with model parameters, it is important to understand how sensitive the model output is to variations in parameter values. A local sensitivity analysis determines the model sensitivity to parameter variations over a localised region around the nominal parameter values, whereas a global sensitivity analysis (GSA) investigates the sensitivity over the entire parameter space. Using a T-cell receptor-activated Erk-MAPK signalling pathway model as an example, the authors present a comparative study of a variety of different sensitivity analysis techniques. These techniques include: local sensitivity analysis, existing GSA methods of partial rank correlation coefficient, Sobol's, extended Fourier amplitude sensitivity test, as well as a weighted average of local sensitivities and a new GSA method to extract global parameter sensitivities from a parameter identification routine. Results of this study revealed critical reactions in the signalling pathway and their impact on the signalling dynamics and provided insights into embedded regulatory mechanisms such as feedback loops in the pathway. From this study, a recommendation emerges for a general sensitivity analysis strategy to efficiently and reliably infer quantitative, dynamic as well as topological properties from systems biology models.

An approach to multiscale modelling with graph grammars.

Event Abstract Back to Event Statistical uncertainty and sensitivity analysis of intracellular signaling models - through approximate Bayesian computation and variance based global sensitivity analysis Olivia Eriksson1, 2*, Alexandra Jauhiainen1, 3, Sara Maad Sasane4, Anu G Nair5, Carolina Sartorius4 and Jeanette Hellgren Kotaleski2, 5, 6 1 Contributed equally to this work, Sweden 2 Stockholm University, Department of Numerical Analysis and Computer Science, Sweden 3 AstraZeneca AB R&D, Early Clinical Biometrics, Sweden 4 Lund University, Centre for Mathematical Sciences, Sweden 5 KTH Royal Institute of Technology, School of Computer Science and Communication, Sweden 6 Karolinska Institute, Department of Neuroscience, Sweden Computational dynamical models describing intracellular signaling pathways of nerve cells are increasing in size and complexity as more information is obtained from experiments. Models describing protein interactions within neurons can contain over hundred protein reactions [1, 2]. These models are built from knowledge about the interaction topology, inferred from e.g. gene knock-out experiments, as well as from experimental quantitative data describing the input-output relationship of the observed system. The quantitative data are often sparse as compared to the size of the system, and translating the experimental information into dynamical models often results in large uncertainties in the parameter (e.g reaction rates) values. This uncertainty is partly due to the sparseness in experimental time series data (practical unidentifiability), but it is also an intrinsic feature of the modeled system (structural unidentifiability) and could reflect the possibility for biological variation [3]; in many biological systems the same function can be achieved in partly different ways due to degeneracy and redundancy. Here we investigate the extent of this parameter uncertainty, in a model describing calcium (Ca)-dependent activation of Calmodulin (CaM), Protein phosphatase 2B (PP2B) and Ca/CaM-dependent protein kinase II (CamKII) [4], which is an intracellular pathway of importance to synaptic plasticity. We presume that the structure of the model is correct and then characterize the uncertainty in the parameters based on data. This is done by sampling the subspace of parameter values which has an equally good fit to data (corresponding to the “viable space” in [5]), through a statistical approach known as Approximate Bayesian Computation (ABC) with Markov Chain Monte Carlo (MCMC) [6]. ABC has been proposed for parameter estimation in system biology models previously, albeit in slightly different settings, e.g. [7, 8], and can be applied to a wide range of problems. In order to characterize the viable space efficiently with several experimental datasets, we employ multivariate probability distributions called copulas as a part of the ABC sampling. The result is a multivariate posterior distribution for the parameters, which represents how much they are constrained by the data (i.e. the uncertainty) as illustrated in Figure 1 (pairwise plots of viable space for four parameters indicating constraints and dependencies before and after calibration to data). The next step is to look into how the parameter uncertainty is translated to uncertainty of the predictions made from the model. We also investigate the role and importance of different parameters, with respect to different model outputs including the predictions, by means of global sensitivity analysis [9]. Figure 2 illustrates the sensitivity of one of the outputs calculated by a method based on the decomposition of the variance [10, 11, 12], showing different types of sensitivity measures. Finally, we explore how the sensitivity of different subsets of parameters change as different parts of the viable space are investigated, which is shown in Figure 3, where sensitivity profiles are clustered into groups with different sensitivity characteristics. Figure 1 Figure 2 Figure 3 Acknowledgements This research was supported by funding from the Swedish e-Science Research Centre and the Human Brain Project References [1] Gutierrez-Arenas O, Eriksson O, Hellgren Kotaleski J (2014) Segregation and Crosstalk of D1 Receptor-Mediated Activation of ERK in Striatal Medium Spiny Neurons upon Acute Administration of Psychostimulants. PLoS Comput Biol 10, e1003445. [2] Nair AG, Gutierrez-Arenas O, Eriksson O, Vincent P, and Hellgren Kotaleski J (2015) Sensing Positive versus Negative Reward Signals through Adenylyl Cyclase-Coupled GPCRs in Direct and Indirect Pathway Striatal Medium Spiny Neurons. J. Neuroscience 35, 14017-14030. [3] Marder E, Goeritz ML, Otopalik AG (2015) Robust circuit rhythms in small circuits arise from variable circuit components and mechanisms. Curr Opin Neurobiol 31, 156-163. [4] Nair AG, Gutierrez-Arenas O, Eriksson O, Jauhiainen A, Blackwell KT and Kotaleski, JH (2014). Modeling intracellular signaling underlying striatal function in health and disease. Progress in molecular biology and translational science, 123, 277 [5] Zamora-Sillero E, Hafner M, Ibig A, Stelling J, and Wagner, A. (2011). Efficient characterization of high-dimensional parameter spaces for systems biology. BMC systems biology 5, 142. [6] Marjoram P, Molitor J, Plagnol V, and Tavare S (2003) Markov chain Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. U.S.A. 100, 15324–15328. [7] Toni T, Welch D, Strelkowa N, Ipsen A, and Stumpf MP. Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J R Soc Interface, 6, 187–202. [8] Liepe J, Kirk P, Filippi S, Toni T, Barnes CP, and Stumpf MP. A framework for parameter estimation and model selection from experimental data in systems biology using approximate Bayesian computation. Nat Protoc, 9, 439–456. [9] Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S Global sensitivity analysis: the primer. In Wiley 2008 New York, NY:Wiley. [10] Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Math. Comput. Simul. 55, 271–280. [11] Saltelli A (2002) Making best use of model evaluations to compute sensitivity indices. Comp Phys Comm 145, 280 – 297. [12] Halnes G, Ulfhielm E Ljunggren EE, Kotaleski JH and Rospars JP (2009) Modelling and sensitivity analysis of the reactions involving receptor, G-protein and effector in vertebrate olfactory receptor neurons. Journal of computational neuroscience 27, 471-491. Keywords: sensitivity analysis, Approximate Bayesian Computation, copulas, intracellular signalling, Dynamical Modeling, global sensitivity analysis, Markov chain Monte Carlo, variance decomposition Conference: Neuroinformatics 2016, Reading, United Kingdom, 3 Sep - 4 Sep, 2016. Presentation Type: Investigator presentations Topic: Computational neuroscience Citation: Eriksson O, Jauhiainen A, Maad Sasane S, Nair A, Sartorius C and Hellgren Kotaleski J (2016). Statistical uncertainty and sensitivity analysis of intracellular signaling models - through approximate Bayesian computation and variance based global sensitivity analysis. Front. Neuroinform. Conference Abstract: Neuroinformatics 2016. doi: 10.3389/conf.fninf.2016.20.00014 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 31 May 2016; Published Online: 18 Jul 2016. * Correspondence: PhD. Olivia Eriksson, Contributed equally to this work, Stockholm, Sweden, olivia@kth.se Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski Google Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski Google Scholar Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski PubMed Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Nonlinear dynamics and chaotic systems have been of great interest to many scientists and engineers over the past two decades. Many nonlinear systems are a result of mathematical models of physical and biological systems that possess inherent nonlinear properties. In this paper, we focus our attention on a second-degree nonlinear differential equation with a sinusoidal forcing function and constant coefficients; such equations, which are commonly employed for mathematical modeling of biological systems usually possess inherent nonlinear properties. We show that the second-degree nonlinear differential equation system could be aperiodic and therefore the long-term behavior is not predictable. However, in the case of the increased frequency of the sinusoidal input, the system shows somewhat periodic, convergent behavior. The simulations demonstrate how small, seemingly insignificant changes to the parameters (e.g., initial conditions) can dramatically alter the system response.

The main focus of this thesis is the uncertainty propagation (UP) and global sensitivity analysis (GSA) in complex hydrogeological numerical models. Various methods are presented with applications on numerical models for the groundwater flow and mean lifetime expectancy (MLE) in the scope of Andra's (French National Radioactive Waste Management Agency) project for the geological disposal of high-level and intermediate-level long-lived radioactive wastes in a highly impermeable layer from Callovo-Oxfordian age (COX) in France. <br> A state of the art is provided for the theory of uncertainty propagation and for the methodologies of sensitivity analyses in a broad sense. Methods for UP are provided from 2-levels Factorial Designs to quasi-random samplings. GSA techniques encompass screening methods such as the Morris Measures and the Derivative-based Global Sensitivity Measures (DGSM), and also the so-called Sobol' indices based upon the variance decomposition of the response of interest. Meta-modelling techniques such as polynomial regression and Polynomial Chaos Expansions (PCE) meta-models are employed as surrogate models for UP and GSA purpose at negligible computational-costs. A comparison of GSA techniques upon various complex analytical test-functions was undertaken with the purpose of determining a relevant method to be employed in the context of “screening” out unimportant variables in computer-intensive, high-dimensional models. <br> A numerical model of groundwater flow and lifetime expectancy is employed for assessing the effect of uncertain advection-dispersion parameters and their spatial distributions upon the MLE from a target zone inside the domain. The model is a 2-dimensions synthetic cross-section of the eastern region of the Paris Basin (Meuse/Haute-Marne sector). This model was used as an exploratory tool for sensitivity analysis methods applied upon numerous sets of uncertain hydrodynamic and dispersion parameters in 15 layers. The uncertainty characterizing the permeability-porosity values in aquifer formations encompassing the COX have proved to add much of variability to the MLE calculated from the target zone. The model also served at exploring the effect of the spatial variability of permeability-porosity parameters in two main aquifer sequences on the groundwater flow rates and MLE in the model. The variabilities of the output responses are mainly due to the uncertainty upon the means and variances of the permeability-porosity distributions, as well as the longitudinal correlation lengths, in each sequence. <br> Then, a 3-dimensions high-definition hydrogeological model representing the Meuse/Haute-Marne sector in the eastern region of the Paris Basin is a more comprehensive numerical model incorporating realistic geometries, fractures, heterogeneities and discontinuities encountered on field. A sensitivity analysis of the MLE from a given zone located in the middle of the COX layer was performed by perturbing the hydraulic conductivities and porosities values in fourteen hydrogeological formations. The uncertain permeability-porosity parameters in the aquifer formations from Bathonian in the Dogger sequence, and Rauracian-Sequanian in the Oxfordian sequence, have strong and rather non-linear effects on the variability of the output response of interest. <br> The methodologies for UP and GSA employed in the present thesis have proved to be very efficient when applied to large hydrogeological models of groundwater flow and MLE. In particular, quasi-random sampling methods offer a flexible frame for providing the uncertainty distribution of the output response of interest at low computational costs. Screening techniques provide a fast estimation for the overall contribution of each input variable to the variability of the output. Meta-modelling techniques such as PCE proved highly accurate in individualising the low- and high-order effects of each input variable upon the output response of interest.

We present a global sensitivity analysis that quantifies the impact of parameter uncertainty on model outcomes. Specifically, we propose variance‐decomposition‐based Sobol' indices to establish an importance ranking of parameters and univariate effects to determine the direction of their impact. We employ the state‐of‐the‐art approach of constructing a polynomial chaos expansion of the model, from which Sobol' indices and univariate effects are then obtained analytically, using only a limited number of model evaluations. We apply this analysis to several quantities of interest of a standard real‐business‐cycle model and compare it to traditional local sensitivity analysis approaches. The results show that local sensitivity analysis can be very misleading, whereas the proposed method accurately and efficiently ranks all parameters according to importance, identifying interactions and nonlinearities. Computational techniques uncertainty quantification sensitivity analysis polynomial chaos expansion C60 C63

A global sensitivity analysis of a multiscale computational model of microvascular flow is presented. A total of 140 simulations have been completed and analyzed varying 6 input parameters and considering their effects on 7 output variables. Interestingly, the vascular network topology has been found as a determinant factor for both vasculature-related and interstitium-related quantities. Regarding the firsts, the vascular network topology has obtained a score of 5.5/6 and 6/6 for average and spatial distribution respectively (where 6 is the maximum and 1 is the minimum). On the other hand, considering interstitium-related quantities, the score is 4/6 and 5/6 for average and spatial distribution respectively. These results suggest that the network topology has a significant influence on the outcome of the computational analysis.

Dimensionality reduction for surrogate model construction for global sensitivity analysis: Comparison between active subspace and local sensitivity analysis

Virus infection dynamics is governed by the processes on multiple scales: on the whole organism level, tissue level, and intracellular level. In this paper, we develop a multi-scale multi-compartment model of HIV infection in a simplified setting and the computational methods for numerical realization of the model. The multiscale model describes the processes from various scales and of different nature (cell motility, virus diffusion, intracellular virus replication). Intracellular replication model is based on a Markov chain with time-inhomogeneous propensities that depend on the extracellular level of virions. Reaction diffusion equations used to model free virion diffusion in the lymphoid tissue have moving sources, which are determined by the positions of the infected cells (immune cell motility model) and the rate of virion secretion from them (intracellular model). Immune cell motility model parameterizes the intercellular interaction forces, friction and the stochastic force of active cell motility. Together, this allows for a proper description of the intracellular stochasticity that propagates across multiple scales. A hybrid discrete-continuous stochastic-deterministic algorithm for simulation of the multiscale model based on the uniformization Monte Carlo method is implemented.

Sensitivity analysis is a vital tool in numerical modeling to identify important parameters and processes that contribute to the overall uncertainty in model outputs. We developed a new sensitivity analysis method to quantify the relative importance of uncertain model processes that contain multiple uncertain parameters. The method is based on the concepts of Bayesian networks (BNs) to account for complex hierarchical uncertainty structure of a model system. We derived a new set of sensitivity indices using the methodology of variance‐based global sensitivity analysis with the Bayesian inference. The framework is capable of representing the detailed uncertainty information of a complex model system using BNs and affords flexible grouping of different uncertain inputs given their characteristics and dependency structures. We have implemented the method on a real‐world biogeochemical model at the groundwater‐surface water interface within the Hanford Site's 300 Area. The uncertainty sources of the model were first grouped into forcing scenario and three different processes based on our understanding of the complex system. The sensitivity analysis results indicate that both the reactive transport and groundwater flow processes are important sources of uncertainty for carbon‐consumption predictions. Within the groundwater flow process, the structure of geological formations is more important than the permeability heterogeneity within a given geological formation. Our new sensitivity analysis framework based on BNs offers substantial flexibility for investigating the importance of combinations of interacting uncertainty sources in a hierarchical order, and it is expected to be applicable to a wide range of multiphysics models for complex systems.

Global Sensitivity Analysis of Arrhythmic Risk Biomarkers in Cardiac Single Cell Models

Sensitivity analyses are frequently used during the design of engineering systems to qualify and quantify the effect of parametric variation in the performance of a system. Two primary types of sensitivity analyses are generally used: local and global. Local analyses, generally involving derivative-based measures, have a significantly lower computational burden than global analyses but only provide measures of sensitivity around a nominal point. Global analyses, generally performed with a Monte Carlo sampling approach, and variation-based measures provide a complete description of sensitivity but incur a large computational burden and require information regarding the distributions of the design parameters in a concept. Local analyses are generally suited to the early stages of design when parametric information is limited, and a large number of concepts must be evaluated (necessitating a light computational burden). Global analyses are more suited to the later stages of design when more information about parametric distributions is available and fewer concepts are under consideration. Current derivative-based local approaches provide a different and incompatible set of measures than a global variation-based analysis. This makes a direct comparison of local to global measures ill posed. To reconcile local and global sensitivity analyses, a hybrid local variation-based sensitivity (HyVar) approach is presented. This approach has a similar computational burden to a local approach but produces measures or percentage contributions. The HyVar approach is directly comparable to global variation-based approaches. In this paper, the HyVar sensitivity analysis method is developed in the context of a functional based behavioral modeling framework. An example application of the method is presented along with a summary of results produced from a more comprehensive example.

This paper presents a global sensitivity analysis based strategy for uncertainty quantification of the operating revenue and the carbon dioxide emissions in a crude oil distillation column. Aspen HYSYS is used for rigorous simulation of the column. A two-stage approach, which is executed by an interface between MATLAB and HYSYS, is implemented in this study. A multiplicative dimensional reduction method is applied to first identify which factors have the most influential to the model outputs, e.g. the operating revenue and the CO2 emissions. In the second stage, the uncertainty quantification regressed by Gaussian prediction regression method is exploited. As a result, there is a good agreement between the predicted results by the Gaussian prediction regression method with that of the conventional Quasi-Monte Carlo approach, which shows that the computational efforts was reduced significantly compared to the conventional study (about less than 10 times). Furthermore, the Kernel points estimated by the Gaussian prediction regression method are generated to highlight influential factors identified in the first stage.

Sensitivity analysis and calibration of phenomenological models for seismic analyses