Department of BioMedical Engineering, Eindhoven University of Technology, Den Dolech 2, Eindhoven, 5612 AZ, The Netherlands

Netherlands Consortium for Systems Biology, University of Amsterdam, Science Park 904, Amsterdam, 1098 XH, The Netherlands

Abstract

Background

The study of phenotype transitions is important to understand progressive diseases, e.g., diabetes mellitus, metabolic syndrome, and cardiovascular diseases. A challenge remains to explain phenotype transitions in terms of adaptations in molecular components and interactions in underlying biological systems.

Results

Here, mathematical modeling is used to describe the different phenotypes by integrating experimental data on metabolic pools and fluxes. Subsequently, trajectories of parameter adaptations are identified that are essential for the phenotypical changes. These changes in parameters reflect progressive adaptations at the transcriptome and proteome level, which occur at larger timescales. The approach was employed to study the metabolic processes underlying liver X receptor induced hepatic steatosis. Model analysis predicts which molecular processes adapt in time after pharmacological activation of the liver X receptor. Our results show that hepatic triglyceride fluxes are increased and triglycerides are especially stored in cytosolic fractions, rather than in endoplasmic reticulum fractions. Furthermore, the model reveals several possible scenarios for adaptations in cholesterol metabolism. According to the analysis, the additional quantification of one cholesterol flux is sufficient to exclude many of these hypotheses.

Conclusions

We propose a generic computational approach to analyze biological systems evolving through various phenotypes and to predict which molecular processes are responsible for the transition. For the case of liver X receptor induced hepatic steatosis the novel approach yields information about the redistribution of fluxes and pools of triglycerides and cholesterols that was not directly apparent from the experimental data. Model analysis provides guidance which specific molecular processes to study in more detail to obtain further understanding of the underlying biological system.

Background

Cardiovascular and metabolic diseases such as diabetes mellitus and metabolic syndrome are progressive in time

In systems biology mathematical modeling is applied to integrate different sources of experimental data of a phenotype and to investigate the complex interactions of underlying biological systems

In the present work, we propose a novel computational approach to analyze molecular adaptations in a biological system to overcome these problems. We use mathematical modeling to quantitatively integrate metabolic data of different phenotypes and subsequently exploit this mathematical framework to analyze which molecular processes have changed and are collectively responsible for the shift between phenotypes. This information is obtained by identifying the progression of necessary parameter changes required for the model to be consistent with the experimental data of these phenotypes. These changes in parameters reflect progressive adaptations at the transcriptome and proteome level, which occur at larger timescales than the metabolic processes. The approach involves consecutive steps of data gathering, model development, and parameter estimation, which will be discussed in detail. An advantage of our approach is that mathematical models containing processes at any timescale of interest can be used, while their long-term adaptations are captured by identifying necessary parameter changes. This enables us to study long-term aspects of short-term processes. Furthermore, in cases when the amount of information of the underlying biological system is limited, our approach could provide a means to describe adaptations in molecular processes without the necessity to develop detailed kinetic models of the modulating mechanisms. For instance, if one is interested in studying a metabolic pathway which is adapting due to activation of a signal transduction pathway, the modulating effects can be captured by identifying necessary changes in the metabolic pathway parameters rather than developing a mathematical model that includes an explicitly modeled signal transduction pathway. The approach, which is applicable to a multitude of biological systems, is demonstrated on the basis of a case involving the activation of the liver X receptor (LXR), a promising drug target for atherosclerotic therapies

The family of liver X receptors (LXR

Methods

Several theoretical sections are presented describing the methodology of the computational approach, which involves consecutive steps of data gathering, model development, and various parameter estimation steps.

Model development

The computational approach is developed to analyze progressively adapting biological systems that are modeled using a system of (non)linear ordinary differential equations (ODEs):

where _{0}. The vector of model outputs

which is described by a set of functions

Simulating the biological system of phenotype A

Once a network topology of molecular species and corresponding flux descriptions are defined, values for the kinetic parameters ^{-6 }to 10^{6}). For each parameter set a simulation to steady-state was carried out. Subsequently, the weighted sum of squared errors _{d }

where ^{A }
^{A }
^{A }
^{A }
_{d }
_{d }

where

Identification of molecular adaptations from phenotype A to phenotype B

Parameter estimation to describe phenotype B

The mathematical model together with the collection of acceptable parameter sets, represents the biological system of phenotype A. Molecular processes that are responsible for the transition of the biological system from phenotype A to phenotype B, are determined by identifying kinetic parameters that necessarily have to change in order to describe the biological system of phenotype B. A first approach could be to repeat the large-scale parameter estimation protocol, employed on phenotype A, for phenotype B. However, apart from being computationally expensive, comparing parameter sets from different phenotypes with each other is problematic, as they are obtained independently from each other. For instance, in the case when multiple separate minima exist, it would not be possible to know which realization of phenotype A is the reference for a specific realization of phenotype B. However, the fact that phenotype B originates from phenotype A could be used to address latter problem. The acceptable parameter sets from phenotype A could be used as initial values and reoptimized by once more applying a weighted non-linear least squares algorithm, minimizing _{d}

Iterative data integration and parameter estimation

Parameter adaptations describing a phenotype transition are often not unique. For instance, in order to increase a specific molecular concentration, corresponding production and degradation parameters can be changed in infinitely many different ways to accomplish this. Here, we assume that adaptations are minimal and proceed progressively in time. Therefore, the concept described in the previous section was extended to study progressively adapting biological systems, by defining artificial intermediate phenotypes. Hereto, the experimental data is interpolated from phenotype A to phenotype B in a number of steps. For instance, for a linear interpolation scheme this would imply ^{q }
^{A }
^{B}
^{A }
^{B }

Similar as in equation (3), for each parameter trajectory different realizations for ^{A }
^{B }

Regularization of parameter adaptation trajectories

It is assumed that adaptations are minimal and proceed progressively in time. Therefore, the parameter estimation protocol was extended to avoid needless change of parameters, hereby identifying minimal parameter adaptations that are necessary to describe a phenotype transition. To this end, _{d }
_{r }

where

Consistency of parameter adaptation trajectories

The identification of parameter adaptation trajectories was performed for each acceptable parameter set, which gives information about the possible dispersion of parameter trajectories due to kinetic variations between the different acceptable parameter sets. However, given the uncertainties arising from experimental data and parameter estimates, the reliability of individual parameter trajectories is also a relevant topic to explore; is an observed trajectory consistent or is its path just a coincidental result? Given a certain parameter trajectory, it is important to analyze how reliable and consistent its path is to eventually draw conclusions about potential molecular adaptations that could have taken place. Therefore, the protocol described above was extended by not only determining parameter trajectories from phenotype A to phenotype B, but also backwards from phenotype B to phenotype A. A backward trajectory is obtained by interpolating the data from phenotype B to phenotype A, whilst reoptimizing the parameters. The final values of the model states and parameters obtained from the forward trajectory are used as initial values to calculate the backward trajectory. Furthermore, the reference parameter set

Results

We presented a computational approach to analyze molecular adaptations in a biological system evolving through various phenotypes, which is generically applicable to different biological systems. In this section, the computational approach is demonstrated by applying it to a case of liver X receptor induced hepatic steatosis.

Experimental data

The acquisition of quantitative experimental data of different phenotypes is essential to gain insight in the progression of molecular adaptations in underlying biological systems. The available experimental data determines to a large extend the development of a mathematical model. The level of detail and precision at which certain biological processes can be integrated in a mathematical model, is determined by the selection of molecular species, as well as the type and quality of the measurements. With respect to the LXR case, several datasets of wild-type and T0901317 LXR activated C57BL/6J mice were obtained. Data was included containing measurements of hepatic triglyceride, free cholesterol, and cholesterylester levels, as well as plasma triglyceride, high-density-lipoprotein (HDL) cholesterol, total cholesterol, and free fatty acid levels in overnight-fasted mice

**Supplementary material**. Description of model equations, additional analyses, implementation details, and experimental data.

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Computational model of hepatic lipid and plasma lipoprotein metabolism

A mathematical multi-compartment model was constructed, based on the available experimental data, which integrates metabolic processes involved in hepatic lipid metabolism, as well as plasma lipoprotein metabolism (Figure

Hepatic lipid and plasma lipoprotein metabolism

**Hepatic lipid and plasma lipoprotein metabolism**. The mathematical model has three compartments representing the liver, blood plasma, and peripheral tissues. The liver compartment includes reactions representing the production, utilization and storage of triglycerides and cholesterols, and the mobilization of these metabolites to the endoplasmic reticulum, where they are incorporated into nascent produced VLDL particles. The VLDL particles are secreted in the plasma compartment where they serve as nutrients for peripheral tissues. Remnant particles are taken up and cleared by the liver. The model furthermore includes the hepatic uptake of free fatty acids and reverse cholesterol transport via HDL. ABCA1, ATP-binding cassette transporter 1; ACAT, acyl-CoA: cholesterol acyltransferase; ApoB, apolipoprotein B; CE, cholesterylester; DGAT, diglyceride acyltransferase; ER, endoplasmic reticulum; FFA, free fatty acid; FC, free cholesterol; HDL, high-density-lipoprotein; HSL, hormone-sensitive lipase; IDL, intermediate density lipoprotein; LDL, low density lipoprotein; LDLr, low density lipoprotein receptor; LPL, lipoprotein lipase; MTP, microsomal triglyceride transfer protein; SR-B1, scavenger receptor class B1; TG, triglyceride; TGH, triglyceride hydrolase; VLDL, very low density lipoprotein; VLDLr, very low density lipoprotein receptor.

**SBML file**. Implementation of the mathematical model in SBML format.

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Simulating the wild-type mouse

A large-scale parameter estimation protocol was employed to capture multiple parameter sets that describe the experimental data of phenotype A (wild-type C57BL/6J mice). Mass isotopomer distribution analyses indicate that the metabolic fluxes are expected to be in the

Parameter scatter plots and predictions

**Parameter scatter plots and predictions**. Several parameters are very constrained by the data and have a well defined value (A and B), whereas others show a larger spread of possible values (D and E). The black dots represent the 10^{8 }initial sampled parameter values (individual dots not visible), whereas the red dots represent the 10^{4 }best parameter sets which were optimized. The resulting 2909 acceptable parameter sets that describe the experimental data are shown in green. Model predictions for the depositioning of hepatic triglycerides (C) and cholesterylesters (F) in cytoplasmic and endoplasmic reticulum fractions were obtained for all acceptable parameter sets. Note that only the total pools of triglyceride and cholesterylester were measured. The predictions for the triglyceride fractions are consistent, whereas the predictions for the cholesterylester fractions show a larger spread of possible outcomes.

Parameter adaptations from the wild-type to the LXR activated phenotype

Using the previously described techniques, an analysis was carried out to study the metabolic consequences of T0901317 induced LXR activation. It was assumed that metabolic adaptations upon LXR activation proceed linearly in time

Single-step versus multi-step optimization

**Single-step versus multi-step optimization**. A) An example of an acceptable parameter set describing the wild-type phenotype, which was not successfully optimized by single-step optimization to describe the LXR activated phenotype. This problem was circumvented by multi-step optimization. In the latter case, the parameter estimation is carried out in a step-wise fashion. Hereto, the experimental data is interpolated from the wild-type phenotype to the LXR activated phenotype. At each interpolation step the parameters are reoptimized in order to describe the newly interpolated data. This procedure is repeated until the final state representing phenotype B is reached. B, C) Two examples of corresponding parameter trajectories from the wild-type phenotype to the LXR activated phenotype, normalized by the initial parameters of the wild-type phenotype.

The parameter trajectories were regularized according to equation (6) to avoid needless change of parameters. A potential risk of regularization, as always with multi-objective optimization, is that for a low _{d }
_{r }
_{d }
_{r }
_{d}

Effect of regularization strength

**Effect of regularization strength**. A) Model error _{d }_{r }

Regularization of parameter trajectories

**Regularization of parameter trajectories**. Three examples of parameter trajectories from the wild-type phenotype to the LXR activated phenotype obtained without regularization (blue dashed) and with regularization using

The uncertainty associated with parameter trajectories was investigated, among other things, by repeatedly calculating forward and backward trajectories. Figure

(In)consistency of parameter adaptation trajectories

**(In)consistency of parameter adaptation trajectories**. Three examples of back and forward parameter trajectories from the wild-type phenotype to the LXR activated phenotype, using a hundred repetitions.

The parameter adaptation trajectories were determined for all acceptable parameter sets and used to determine how the fluxes of triglycerides and cholesterols change in time from the wild-type phenotype to the LXR activated phenotype. The data interpolation was carried out in a hundred steps and the back and forward flux trajectories were determined using a hundred repetitions. Figure

Flux adaptations upon T0901317 induced LXR activation

**Flux adaptations upon T0901317 induced LXR activation**. Flux trajectories from the wild-type phenotype (green) to the LXR activated phenotype (red). The data interpolation was carried out in a hundred steps and the back and forward flux trajectories were determined using a hundred repetitions. Molecular fluxes (A-C, E-H) are given in mM/h, whereas the triglyceride concentrations presented in (D) are given in mM.

As described in previous sections, several parameters are not constrained by the data and show a large spread of possible outcomes. This makes the calculation of consistent quantitative trajectories impossible. Nonetheless, relative changes with respect to the initial values of the wild-type phenotype could still provide useful information, e.g., to determine ranges of feasible fold inductions of molecular concentrations and fluxes, and to discriminate between different possible scenarios. The latter could be used to generate new hypotheses and to guide the design of new experiments. An example is depicted in Figure

Adaptations in cholesterol metabolism upon T0901317 induced LXR activation

**Adaptations in cholesterol metabolism upon T0901317 induced LXR activation**. Adaptations in metabolic processes/components involved in hepatic cholesterol metabolism, normalized by the values of corresponding wild-type phenotype. The green dots represent the wild-type phenotype, whereas the blue and black dots represent the LXR activated phenotype. Solutions corresponding to an increased HDL-CE synthesis/uptake rate are colored blue, whereas solutions corresponding to a decreased HDL-CE synthesis/uptake rate are colored black. The ellipses were calculated by means of principal component analysis (PCA) and contain 95% of the corresponding solutions. ABCA1, ATP-binding cassette transporter 1; ABCG5, ATP-binding cassette transporter G5; SR-B1, scavenger receptor class B1.

Discussion

To improve our understanding of progressive diseases such as diabetes mellitus and metabolic syndrome, the study of phenotype transitions is important. A challenging task is to explain the progression of phenotype transitions in terms of molecular adaptations in underlying biological systems. Here, we propose a novel computational approach to analyze biological systems evolving through various phenotypes and to predict which molecular processes are responsible for the transition. We presented a case involving the activation of the liver X receptor, a promising drug target for atherosclerotic therapies.

Parameter adaptation trajectories during phenotype transitions: strengths and limitations

A large-scale parameter estimation protocol was employed to capture multiple parameter sets describing the biological system of phenotype A. A collection of 2909 acceptable parameter sets were obtained that describe the experimental data. A substantial fraction of the optimized parameter sets were not acceptable. These parameter sets did either not describe the experimental data or displayed unphysiologically high fluxes for any of the reactions. It appeared that the latter criterion contributed significantly to the rejection of parameter sets. The efficiency of sampling acceptable parameter sets could potentially be improved by including the rejection criteria in the optimization objective function. Note, that the computational approach is not restricted to the parameter estimation protocol presented here. Various parameter optimization methods exist that minimize the difference between experimental data and corresponding model simulations, e.g., trust-region optimization methods, simulated annealing, and genetic algorithms

Parameter trajectories describing the phenotype transition were determined by interpolating the data between phenotypes. The data interpolation was carried out in a hundred steps. We have performed several tests by using different numbers of steps. Performing more than a hundred steps did not change the results significantly. The computational approach allows free choice of interpolation schemes. Hence, when information is available about the progressive nature of certain biological processes, this information could be incorporated in the interpolation scheme. Furthermore, the computational approach could be used to explore different possible transition scenarios by employing a variety of different interpolation schemes. The latter could be useful when sufficient information about the transition characteristics is lacking, e.g., to test hypotheses about the feasibility of specific transitions with respect to the available experimental data. In this work we focused on diseases that arise progressively, e.g., hepatic steatosis, diabetes type 2, and metabolic syndrome. However, some diseases arise abruptly like in case of diabetes type 1. In latter cases it could be relevant to explore switch-like interpolation schemes and investigate whether the computational model can exhibit bistable behavior ^{-/- }

Metabolic adaptations upon T0901317 induced LXR activation

A computational model of hepatic lipid and plasma lipoprotein metabolism was developed to study the metabolic consequences of LXR activation. We were able to quantitatively integrate data of wild-type and LXR activated C57BL/6J mice into a consistent model and identified trajectories of parameter adaptations, describing the change in phenotype. The presented model predictions are in good agreement with experimental observations by other groups and contribute to the current understanding of LXR activation. The VLDL-TG production rate increases about 2.6 times upon LXR activation, as predicted by the model and experimentally measured

Mathematical modeling of progressively adapting biological systems

Mathematical modeling is well suited for integrating different sources of experimental data for a certain phenotype and allows investigating of the complex interactions of underlying biological systems. A mathematical model can be used to obtain thorough understanding of a biological system, e.g. by investigating its complex behavior in response to various stimuli. However, simulating and predicting long-term progressive adaptations is challenging. The multiscale nature of progressively adapting biological systems complicates the development of predictive computational models. As such, one of the central and formidable challenges in systems biology is to develop multiscale computational models and methods that can be used to study molecular mechanisms underlying progressive diseases

The computational approach presented here was employed to study the metabolic consequences of LXR activation, which displays several of the aforementioned issues. For example, the underlying biological system contains processes at timescales ranging from seconds to hours, whereas the phenotypical characteristics develop at a timescale ranging from days to weeks in mice. Our approach has as advantage that it can readily deal with large differences in timescales. For instance, long-term changes in short-term processes could be studied by explicitly modeling the short-term processes, whereas the long-term modulation could be captured by identifying necessary parameter changes. This implies that molecular adaptations could be described without the necessity to develop detailed kinetic models of the modulating mechanisms. This is a major advantage, e.g., for the LXR case, as LXRs modulate a wide range of heavily interlinked complex metabolic processes and signal transduction pathways of which the kinetics and molecular mechanisms are not well understood.

Conclusions

The study of phenotype transitions is important to understand disease progression. We developed a novel computational approach to analyze molecular adaptations in a biological system evolving through various phenotypes, which is generically applicable to different biological systems. For the case of liver X receptor induced hepatic steatosis the novel approach yields information about the redistribution of fluxes and pools of triglycerides and cholesterols that was not directly apparent from the experimental data. The collection of parameter and corresponding flux trajectories give a broad overview of key-processes that are involved in the phenotype transition and how they potentially change in time. Model analysis provides guidance which specific molecular processes to study in more detail to obtain further understanding of the underlying biological system. The main findings are that both the VLDL-TG and VLDL-CE production rates are increased, as well as the uptake of triglycerides through lipolysis. The liver plays a major role in the re-uptake of lipoproteins and hereby prevents plasma hyperlipidemia. The increased triglyceride fluxes are especially stored in cytosolic fractions, rather than in endoplasmic reticulum fractions.

Authors' contributions

CT developed the mathematical model, performed the computational analysis, and wrote the paper. JV contributed to the computational analysis, improved the computational efficiency of the software, and revised the paper. PH supervised the study and revised the paper. NR supervised the study, contributed to the computational analysis, and revised the paper. All authors read and approved the final manuscript.

Acknowledgements

Research was funded by the Netherlands Consortium for Systems Biology (NCSB). We gratefully thank Aldo Grefhorst, Maaike Oosterveer, Barbara Bakker, and Albert Groen for useful discussions.