Thermodynamic modeling of transcription: sensitivity analysis differentiates biological mechanism from mathematical model-induced effects

Spectroscopic analysis of 2-(5-mercapto-1,3,4-oxadiazol-2-yl)-6-methylquinolin-4-ol binding to blood plasma albumin

Binding of 2-(5-mercapto-1,3,4-oxadiazol-2-yl)-6-methylquinolin-4-ol (C1), a biologically active substance, to bovine blood plasma albumin (BSA) at 293, 298, and 303 K was studied using fluorescence (steady state, synchronous, excitation/emission matrix) and FT-IR spectroscopy methods. The experimental results showed that C1 causes fluorescence quenching of BSA through both static and dynamic quenching mechanisms. The thermodynamic parameters, enthalpy and entropy change, for the static quenching were calculated to be - 35.73 kJ mol-1 and - 35.34 J mol-1 K-1, which indicated that hydrogen bonding and van der Waals interactions were the predominant intermolecular forces regulating C1-BSA interactions. Distance between donor and acceptor (2.14, 2.26, and 2.30 nm) depending on the temperature, obtained from intrinsic Förster resonance energy transfer calculations, revealed the static quenching mechanism of BSA fluorescence in 0-3.0 × 10-5 mol/dm3 concentration range of C1. The micro-environmental and conformational changes in BSA structure, established by synchronous, excitation/emission matrices and FT-IR spectra showed the changes in the BSA secondary structure.

Graphical abstract

Fitting thermodynamic-based models: Incorporating parameter sensitivity improves the performance of an evolutionary algorithm

A detailed comprehension of transcriptional regulation is critical to understanding the genetic control of development and disease across many different organisms. To more fully investigate the complex molecular interactions controlling the precise expression of genes, many groups have constructed mathematical models to complement their experimental approaches. A critical step in such studies is choosing the most appropriate parameter estimation algorithm to enable detailed analysis of the parameters that contribute to the models. In this study, we develop a novel set of evolutionary algorithms that use a pseudo-random Sobol Set to construct the initial population and incorporate parameter sensitivities into the adaptation of mutation rates, using local, global, and hybrid strategies. Comparison of the performance of these new algorithms to a number of current state-of-the-art global parameter estimation algorithms on a range of continuous test functions, as well as synthetic biological data representing models of gene regulatory systems, reveals improved performance of the new algorithms in terms of runtime, error and reproducibility. In addition, by analyzing the ability of these algorithms to fit datasets of varying quality, we provide the experimentalist with a guide to how the algorithms perform across a range of noisy data. These results demonstrate the improved performance of the new set of parameter estimation algorithms and facilitate meaningful integration of model parameters and predictions in our understanding of the molecular mechanisms of gene regulation.

Sensitivity analysis methods in the biomedical sciences

Sensitivity analysis is an important part of a mathematical modeller's toolbox for model analysis. In this review paper, we describe the most frequently used sensitivity techniques, discussing their advantages and limitations, before applying each method to a simple model. Also included is a summary of current software packages, as well as a modeller's guide for carrying out sensitivity analyses. Finally, we apply the popular Morris and Sobol methods to two models with biomedical applications, with the intention of providing a deeper understanding behind both the principles of these methods and the presentation of their results.

Relative sensitivity analysis of the predictive properties of sloppy models

Commonly among the model parameters characterizing complex biological systems are those that do not significantly influence the quality of the fit to experimental data, so-called “sloppy” parameters. The sloppiness can be mathematically expressed through saturating response functions (Hill’s, sigmoid) thereby embodying biological mechanisms responsible for the system robustness to external perturbations. However, if a sloppy model is used for the prediction of the system behavior at the altered input (e.g. knock out mutations, natural expression variability), it may demonstrate the poor predictive power due to the ambiguity in the parameter estimates. We introduce a method of the predictive power evaluation under the parameter estimation uncertainty, Relative Sensitivity Analysis. The prediction problem is addressed in the context of gene circuit models describing the dynamics of segmentation gene expression in Drosophila embryo. Gene regulation in these models is introduced by a saturating sigmoid function of the concentrations of the regulatory gene products. We show how our approach can be applied to characterize the essential difference between the sensitivity properties of robust and non-robust solutions and select among the existing solutions those providing the correct system behavior at any reasonable input. In general, the method allows to uncover the sources of incorrect predictions and proposes the way to overcome the estimation uncertainties.

Multi-Response Parameter Interval Sensitivity and Optimization for the Composite Tape Winding Process

The composite tape winding process, which utilizes a tape winding machine and prepreg tapes, provides a promising way to improve the quality of composite products. Nevertheless, the process parameters of composite tape winding have crucial effects on the tensile strength and void content, which are closely related to the performances of the winding products. In this article, two different object values of winding products, including mechanical performance (tensile strength) and a physical property (void content), were respectively calculated. Thereafter, the paper presents an integrated methodology by combining multi-parameter relative sensitivity analysis and single-parameter sensitivity analysis to obtain the optimal intervals of the composite tape winding process. First, the global multi-parameter sensitivity analysis method was applied to investigate the sensitivity of each parameter in the tape winding processing. Then, the local single-parameter sensitivity analysis method was employed to calculate the sensitivity of a single parameter within the corresponding range. Finally, the stability and instability ranges of each parameter were distinguished. Meanwhile, the authors optimized the process parameter ranges and provided comprehensive optimized intervals of the winding parameters. The verification test validated that the optimized intervals of the process parameters were reliable and stable for winding products manufacturing.

Sig2GRN: a software tool linking signaling pathway with gene regulatory network for dynamic simulation

Background

Linking computational models of signaling pathways to predicted cellular responses such as gene expression regulation is a major challenge in computational systems biology. In this work, we present Sig2GRN, a Cytoscape plugin that is able to simulate time-course gene expression data given the user-defined external stimuli to the signaling pathways.

Methods

A generalized logical model is used in modeling the upstream signaling pathways. Then a Boolean model and a thermodynamics-based model are employed to predict the downstream changes in gene expression based on the simulated dynamics of transcription factors in signaling pathways.

Results

Our empirical case studies show that the simulation of Sig2GRN can predict changes in gene expression patterns induced by DNA damage signals and drug treatments.

Conclusions

As a software tool for modeling cellular dynamics, Sig2GRN can facilitate studies in systems biology by hypotheses generation and wet-lab experimental design.

Availability: http://histone.scse.ntu.edu.sg/Sig2GRN/

Quantitative perturbation-based analysis of gene expression predicts enhancer activity in early Drosophila embryo

Enhancers constitute one of the major components of regulatory machinery of metazoans. Although several genome-wide studies have focused on finding and locating enhancers in the genomes, the fundamental principles governing their internal architecture and cis-regulatory grammar remain elusive. Here, we describe an extensive, quantitative perturbation analysis targeting the dorsal-ventral patterning gene regulatory network (GRN) controlled by Drosophila NF-κB homolog Dorsal. To understand transcription factor interactions on enhancers, we employed an ensemble of mathematical models, testing effects of cooperativity, repression, and factor potency. Models trained on the dataset correctly predict activity of evolutionarily divergent regulatory regions, providing insights into spatial relationships between repressor and activator binding sites. Importantly, the collective predictions of sets of models were effective at novel enhancer identification and characterization. Our study demonstrates how experimental dataset and modeling can be effectively combined to provide quantitative insights into cis-regulatory information on a genome-wide scale.

DOI:http://dx.doi.org/10.7554/eLife.08445.001

DNA contains regions known as genes, which may be “transcribed” to produce the RNA molecules that act as templates for building proteins and regulate cell activity. Proteins called transcription factors can bind to specific sequences of DNA to influence whether nearby genes are transcribed. For example, so-called enhancer regions of DNA contain several binding sites for transcription factors, and this binding activates gene transcription. Little is known about how the transcription factor binding sites are organized in enhancer regions, which makes it difficult to use DNA sequence information alone to predict the regulation of genes.

A transcription factor called Dorsal controls the activity of a network of genes that plays a crucial role in the development of fruit fly embryos. Dorsal binds to the enhancer region of a gene called rhomboid, which has been well studied and is known to be a fairly typical example of an enhancer region.

To understand the regulatory information encoded in the DNA sequences of enhancers, Sayal, Dresch et al. have now used a technique called perturbation analysis to investigate the interactions that are likely to occur between Dorsal and other transcription factors as they bind to the rhomboid enhancer. This technique involves systematically mutating the enhancer to remove different combinations of transcription factor binding sites and quantitatively investigating the effect this has on gene activity. A large set of mathematical models were then trained using this data and shown to correctly predict the activity of a range of other gene regulatory regions. The collective predictions of the models identified new enhancer regions and revealed details about how different types of transcription factor binding sites are arranged within enhancers.

As we enter an era where the DNA sequences of entire human populations are increasingly accessible, we would like to know the functional significance of changes in gene regulatory regions. Sayal, Dresch et al. show that the regulatory properties of specific control proteins are accessible by employing quantitative experiments and mathematical models. Similar studies will be required to learn how mutations found across the genome may alter gene expression, leading to better diagnosis and treatment of disease.

DOI:http://dx.doi.org/10.7554/eLife.08445.002

A Systematic Ensemble Approach to Thermodynamic Modeling of Gene Expression from Sequence Data

To understand the relationship between an enhancer DNA sequence and quantitative gene expression, thermodynamics-driven mathematical models of transcription are often employed. These "sequence-to-expression" models can describe an incomplete or even incorrect set of regulatory relationships if the parameter space is not searched systematically. Here, we focus on an enhancer of the Drosophila gene ind and demonstrate how a systematic search of parameter space can reveal a more comprehensive picture of a gene's regulatory mechanisms, resolve outstanding ambiguities, and suggest testable hypotheses. We describe an approach that generates an ensemble of ind models; all of these models are technically acceptable solutions to the sequence-to-expression problem in light of wild-type data, and some represent mechanistically distinct hypotheses about the regulation of ind. This ensemble can be restricted to biologically plausible models using requirements gleaned from in vivo perturbation experiments. Biologically plausible models make unique predictions about how specific ind enhancer sequences affect ind expression; we validate these predictions in vivo through site mutagenesis in transgenic Drosophila embryos.

A global sensitivity analysis approach for morphogenesis models

BACKGROUND

Morphogenesis is a developmental process in which cells organize into shapes and patterns. Complex, non-linear and multi-factorial models with images as output are commonly used to study morphogenesis. It is difficult to understand the relation between the uncertainty in the input and the output of such 'black-box' models, giving rise to the need for sensitivity analysis tools. In this paper, we introduce a workflow for a global sensitivity analysis approach to study the impact of single parameters and the interactions between them on the output of morphogenesis models.

RESULTS

To demonstrate the workflow, we used a published, well-studied model of vascular morphogenesis. The parameters of this cellular Potts model (CPM) represent cell properties and behaviors that drive the mechanisms of angiogenic sprouting. The global sensitivity analysis correctly identified the dominant parameters in the model, consistent with previous studies. Additionally, the analysis provided information on the relative impact of single parameters and of interactions between them. This is very relevant because interactions of parameters impede the experimental verification of the predicted effect of single parameters. The parameter interactions, although of low impact, provided also new insights in the mechanisms of in silico sprouting. Finally, the analysis indicated that the model could be reduced by one parameter.

CONCLUSIONS

We propose global sensitivity analysis as an alternative approach to study the mechanisms of morphogenesis. Comparison of the ranking of the impact of the model parameters to knowledge derived from experimental data and from manipulation experiments can help to falsify models and to find the operand mechanisms in morphogenesis. The workflow is applicable to all 'black-box' models, including high-throughput in vitro models in which output measures are affected by a set of experimental perturbations.

Global sensitivity analysis of a dynamic model for gene expression in Drosophila embryos

It is well known that gene regulation is a tightly controlled process in early organismal development. However, the roles of key processes involved in this regulation, such as transcription and translation, are less well understood, and mathematical modeling approaches in this field are still in their infancy. In recent studies, biologists have taken precise measurements of protein and mRNA abundance to determine the relative contributions of key factors involved in regulating protein levels in mammalian cells. We now approach this question from a mathematical modeling perspective. In this study, we use a simple dynamic mathematical model that incorporates terms representing transcription, translation, mRNA and protein decay, and diffusion in an early Drosophila embryo. We perform global sensitivity analyses on this model using various different initial conditions and spatial and temporal outputs. Our results indicate that transcription and translation are often the key parameters to determine protein abundance. This observation is in close agreement with the experimental results from mammalian cells for various initial conditions at particular time points, suggesting that a simple dynamic model can capture the qualitative behavior of a gene. Additionally, we find that parameter sensitivites are temporally dynamic, illustrating the importance of conducting a thorough global sensitivity analysis across multiple time points when analyzing mathematical models of gene regulation.

Sequence-based model of gap gene regulatory network

BACKGROUND

The detailed analysis of transcriptional regulation is crucially important for understanding biological processes. The gap gene network in Drosophila attracts large interest among researches studying mechanisms of transcriptional regulation. It implements the most upstream regulatory layer of the segmentation gene network. The knowledge of molecular mechanisms involved in gap gene regulation is far less complete than that of genetics of the system. Mathematical modeling goes beyond insights gained by genetics and molecular approaches. It allows us to reconstruct wild-type gene expression patterns in silico, infer underlying regulatory mechanism and prove its sufficiency.

RESULTS

We developed a new model that provides a dynamical description of gap gene regulatory systems, using detailed DNA-based information, as well as spatial transcription factor concentration data at varying time points. We showed that this model correctly reproduces gap gene expression patterns in wild type embryos and is able to predict gap expression patterns in Kr mutants and four reporter constructs. We used four-fold cross validation test and fitting to random dataset to validate the model and proof its sufficiency in data description. The identifiability analysis showed that most model parameters are well identifiable. We reconstructed the gap gene network topology and studied the impact of individual transcription factor binding sites on the model output. We measured this impact by calculating the site regulatory weight as a normalized difference between the residual sum of squares error for the set of all annotated sites and for the set with the site of interest excluded.

CONCLUSIONS

The reconstructed topology of the gap gene network is in agreement with previous modeling results and data from literature. We showed that 1) the regulatory weights of transcription factor binding sites show very weak correlation with their PWM score; 2) sites with low regulatory weight are important for the model output; 3) functional important sites are not exclusively located in cis-regulatory elements, but are rather dispersed through regulatory region. It is of importance that some of the sites with high functional impact in hb, Kr and kni regulatory regions coincide with strong sites annotated and verified in Dnase I footprint assays.

Multiscale modeling of dorsoventral patterning in Drosophila

The role of mathematical models of signaling networks is showcased by examples from Drosophila development. Three models of consecutive stages in dorsoventral patterning are presented. We begin with a compartmental model of intracellular reactions that generates a gradient of nuclear-localized Dorsal, exhibiting constant shape and dynamic amplitude. A simple thermodynamic model of equilibrium binding explains how a spatially uniform transcription factor, Zelda, can act in combination with a graded factor, Dorsal, to cooperatively regulate gene expression borders. Finally, we formulate a dynamic and stochastic model that predicts spatiotemporal patterns of Sog expression based on known patterns of its transcription factor, Dorsal. The future of coupling multifarious models across multiple temporal and spatial scales is discussed.

Spurious transcription factor binding: non-functional or genetically redundant?

Transcription factor binding sites (TFBSs) on the DNA are generally accepted as the key nodes of gene control. However, the multitudes of TFBSs identified in genome-wide studies, some of them seemingly unconstrained in evolution, have prompted the view that in many cases TF binding may serve no biological function. Yet, insights from transcriptional biochemistry, population genetics and functional genomics suggest that rather than segregating into 'functional' or 'non-functional', TFBS inputs to their target genes may be generally cumulative, with varying degrees of potency and redundancy. As TFBS redundancy can be diminished by mutations and environmental stress, some of the apparently 'spurious' sites may turn out to be important for maintaining adequate transcriptional regulation under these conditions. This has significant implications for interpreting the phenotypic effects of TFBS mutations, particularly in the context of genome-wide association studies for complex traits.

Statistical method for estimation of the predictive power of a gene circuit model

In this paper, a specific aspect of the prediction problem is considered: high predictive power is understood as a possibility to reproduce correct behavior of model solutions at predefined values of a subset of parameters. The problem is discussed in the context of a specific mathematical model, the gene circuit model for segmentation gap gene system in early Drosophila development. A shortcoming of the model is that it cannot be used for predicting the system behavior in mutants when fitted to wild type (WT) data. In order to answer a question whether experimental data contain enough information for the correct prediction we introduce two measures of predictive power. The first measure reveals the biologically substantiated low sensitivity of the model to parameters that are responsible for correct reconstruction of expression patterns in mutants, while the second one takes into account their correlation with the other parameters. It is demonstrated that the model solution, obtained by fitting to gene expression data in WT and Kr⁻ mutants simultaneously, and exhibiting the high predictive power, is characterized by much higher values of both measures than those fitted to WT data alone. This result leads us to the conclusion that information contained in WT data is insufficient to reliably estimate the large number of model parameters and provide predictions of mutants.

In silico evolution of the hunchback gene indicates redundancy in cis-regulatory organization and spatial gene expression

Biological development depends on the coordinated expression of genes in time and space. Developmental genes have extensive cis-regulatory regions which control their expression. These regions are organized in a modular manner, with different modules controlling expression at different times and locations. Both how modularity evolved and what function it serves are open questions. We present a computational model for the cis-regulation of the hunchback (hb) gene in the fruit fly (Drosophila). We simulate evolution (using an evolutionary computation approach from computer science) to find the optimal cis-regulatory arrangements for fitting experimental hb expression patterns. We find that the cis-regulatory region tends to readily evolve modularity. These cis-regulatory modules (CRMs) do not tend to control single spatial domains, but show a multi-CRM/multi-domain correspondence. We find that the CRM-domain correspondence seen in Drosophila evolves with a high probability in our model, supporting the biological relevance of the approach. The partial redundancy resulting from multi-CRM control may confer some biological robustness against corruption of regulatory sequences. The technique developed on hb could readily be applied to other multi-CRM developmental genes.

In vivo neuron-wide analysis of synaptic vesicle precursor trafficking

During synapse development, synaptic proteins must be targeted to sites of presynaptic release. Directed transport as well as local sequestration of synaptic vesicle precursors (SVPs), membranous organelles containing many synaptic proteins, might contribute to this process. Using neuron-wide time-lapse microscopy, we studied SVP dynamics in the DA9 motor neuron in Caenorhabditis elegans. SVP transport was highly dynamic and bi-directional throughout the entire neuron, including the dendrite. While SVP trafficking was anterogradely biased in axonal segments prior to the synaptic domain, directionality of SVP movement was stochastic in the dendrite and distal axon. Furthermore, frequency of movement and speed were variable between different compartments. These data provide evidence that SVP transport is differentially regulated in distinct neuronal domains. It also suggests that polarized SVP transport in concert with local vesicle capturing is necessary for accurate presynapse formation and maintenance. SVP trafficking analysis of two hypomorphs for UNC-104/KIF1A in combination with mathematical modeling identified directionality of movement, entry of SVPs into the axon as well as axonal speeds as the important determinants of steady-state SVP distributions. Furthermore, detailed dissection of speed distributions for wild-type and unc-104/kif1a mutant animals revealed an unexpected role for UNC-104/KIF1A in dendritic SVP trafficking.

Global parameter estimation for thermodynamic models of transcriptional regulation

Deciphering the mechanisms involved in gene regulation holds the key to understanding the control of central biological processes, including human disease, population variation, and the evolution of morphological innovations. New experimental techniques including whole genome sequencing and transcriptome analysis have enabled comprehensive modeling approaches to study gene regulation. In many cases, it is useful to be able to assign biological significance to the inferred model parameters, but such interpretation should take into account features that affect these parameters, including model construction and sensitivity, the type of fitness calculation, and the effectiveness of parameter estimation. This last point is often neglected, as estimation methods are often selected for historical reasons or for computational ease. Here, we compare the performance of two parameter estimation techniques broadly representative of local and global approaches, namely, a quasi-Newton/Nelder-Mead simplex (QN/NMS) method and a covariance matrix adaptation-evolutionary strategy (CMA-ES) method. The estimation methods were applied to a set of thermodynamic models of gene transcription applied to regulatory elements active in the Drosophila embryo. Measuring overall fit, the global CMA-ES method performed significantly better than the local QN/NMS method on high quality data sets, but this difference was negligible on lower quality data sets with increased noise or on data sets simplified by stringent thresholding. Our results suggest that the choice of parameter estimation technique for evaluation of gene expression models depends both on quality of data, the nature of the models [again, remains to be established] and the aims of the modeling effort.

Post-GWAS: where next? More samples, more SNPs or more biology?

The power of genome-wide association studies (GWAS) rests on several foundations: (i) there is a significant amount of additive genetic variation, (ii) individual causal polymorphisms often have sizable effects and (iii) they segregate at moderate-to-intermediate frequencies, or will be effectively 'tagged' by polymorphisms that do. Each of these assumptions has recently been questioned. (i) Why should genetic variation appear additive given that the underlying molecular networks are highly nonlinear? (ii) A new generation of relatedness-based analyses directs us back to the nearly infinitesimal model for effect sizes that quantitative genetics was long based upon. (iii) Larger effect causal polymorphisms are often low frequency, as selection might lead us to expect. Here, we review these issues and other findings that appear to question many of the foundations of the optimism GWAS prompted. We then present a roadmap emerging as one possible future for quantitative genetics. We argue that in future GWAS should move beyond purely statistical grounds. One promising approach is to build upon the combination of population genetic models and molecular biological knowledge. This combined treatment, however, requires fitting experimental data to models that are very complex, as well as accurate capturing of the uncertainty of resulting inference. This problem can be resolved through Bayesian analysis and tools such as approximate Bayesian computation-a method growing in popularity in population genetic analysis. We show a case example of anterior-posterior segmentation in Drosophila, and argue that similar approaches will be helpful as a GWAS augmentation, in human and agricultural research.

Using evolutionary computations to understand the design and evolution of gene and cell regulatory networks

This paper surveys modeling approaches for studying the evolution of gene regulatory networks (GRNs). Modeling of the design or 'wiring' of GRNs has become increasingly common in developmental and medical biology, as a means of quantifying gene-gene interactions, the response to perturbations, and the overall dynamic motifs of networks. Drawing from developments in GRN 'design' modeling, a number of groups are now using simulations to study how GRNs evolve, both for comparative genomics and to uncover general principles of evolutionary processes. Such work can generally be termed evolution in silico. Complementary to these biologically-focused approaches, a now well-established field of computer science is Evolutionary Computations (ECs), in which highly efficient optimization techniques are inspired from evolutionary principles. In surveying biological simulation approaches, we discuss the considerations that must be taken with respect to: (a) the precision and completeness of the data (e.g. are the simulations for very close matches to anatomical data, or are they for more general exploration of evolutionary principles); (b) the level of detail to model (we proceed from 'coarse-grained' evolution of simple gene-gene interactions to 'fine-grained' evolution at the DNA sequence level); (c) to what degree is it important to include the genome's cellular context; and (d) the efficiency of computation. With respect to the latter, we argue that developments in computer science EC offer the means to perform more complete simulation searches, and will lead to more comprehensive biological predictions.

A primer on thermodynamic-based models for deciphering transcriptional regulatory logic

A rigorous analysis of transcriptional regulation at the DNA level is crucial to the understanding of many biological systems. Mathematical modeling has offered researchers a new approach to understanding this central process. In particular, thermodynamic-based modeling represents the most biophysically informed approach aimed at connecting DNA level regulatory sequences to the expression of specific genes. The goal of this review is to give biologists a thorough description of the steps involved in building, analyzing, and implementing a thermodynamic-based model of transcriptional regulation. The data requirements for this modeling approach are described, the derivation for a specific regulatory region is shown, and the challenges and future directions for the quantitative modeling of gene regulation are discussed.

Evaluating thermodynamic models of enhancer activity on cellular resolution gene expression data

With the advent of high throughput sequencing and high resolution transcriptomic technologies, there exists today an unprecedented opportunity to understand gene regulation at a quantitative level. State of the art models of the relationship between regulatory sequence and gene expression have shown great promise, but also suffer from some major shortcomings. In this paper, we identify and address methodological challenges pertaining to quantitative modeling of gene expression from sequence, and test our models on the anterior-posterior patterning system in the Drosophila embryo. We first develop a framework to process cellular resolution three-dimensional gene expression data from the Drosophila embryo and create data sets on which quantitative models can be trained. Next we propose a new score, called 'weighted pattern generating potential' (w-PGP), to evaluate model predictions, and show its advantages over the two most common scoring schemes in use today. The model building exercise uses w-PGP as the evaluation score and adopts a systematic strategy to increase a model's complexity while guarding against over-fitting. Our model identifies three transcription factors--ZELDA, SLOPPY-PAIRED, and NUBBIN--that have not been previously incorporated in quantitative models of this system, as having significant regulatory influence. Finally, we show how fitting quantitative models on data sets comprising a handful of enhancers, as reported in earlier work, may lead to unreliable models.

TWO-LAYER MATHEMATICAL MODELING OF GENE EXPRESSION: INCORPORATING DNA-LEVEL INFORMATION AND SYSTEM DYNAMICS

High-throughput genome sequencing and transcriptome analysis have provided researchers with a quantitative basis for detailed modeling of gene expression using a wide variety of mathematical models. Two of the most commonly employed approaches used to model eukaryotic gene regulation are systems of differential equations, which describe time-dependent interactions of gene networks, and thermodynamic equilibrium approaches that can explore DNA-level transcriptional regulation. To combine the strengths of these approaches, we have constructed a new two-layer mathematical model that provides a dynamical description of gene regulatory systems, using detailed DNA-based information, as well as spatial and temporal transcription factor concentration data. We also developed a semi-implicit numerical algorithm for solving the model equations and demonstrate here the efficiency of this algorithm through stability and convergence analyses. To test the model, we used it together with the semi-implicit algorithm to simulate a Drosophila gene regulatory circuit that drives development in the dorsal-ventral axis of the blastoderm-stage embryo, involving three genes. For model validation, we have done both mathematical and statistical comparisons between the experimental data and the model's simulated data. Where protein and cis-regulatory information is available, our two-layer model provides a method for recapitulating and predicting dynamic aspects of eukaryotic transcriptional systems that will greatly improve our understanding of gene regulation at a global level.

Deciphering the transcriptional cis-regulatory code

Information about developmental gene expression resides in defined regulatory elements, called enhancers, in the non-coding part of the genome. Although cells reliably utilize enhancers to orchestrate gene expression, a cis-regulatory code that would allow their interpretation has remained one of the greatest challenges of modern biology. In this review, we summarize studies from the past three decades that describe progress towards revealing the properties of enhancers and discuss how recent approaches are providing unprecedented insights into regulatory elements in animal genomes. Over the next years, we believe that the functional characterization of regulatory sequences in entire genomes, combined with recent computational methods, will provide a comprehensive view of genomic regulatory elements and their building blocks and will enable researchers to begin to understand the sequence basis of the cis-regulatory code.

Conclusions via unique predictions obtained despite unidentifiability--new definitions and a general method

It is often predicted that model-based data analysis will revolutionize biology, just as it has physics and engineering. A widely used tool within such analysis is hypothesis testing, which focuses on model rejections. However, the fact that a systems biology model is non-rejected is often a relatively weak statement, as such models usually are highly over-parametrized with respect to the available data, and both parameters and predictions may therefore be arbitrarily uncertain. For this reason, we formally define and analyse the concept of a core prediction. A core prediction is a uniquely identified property that must be fulfilled if the given model structure is to explain the data, even if the individual parameters are non-uniquely identified. It is shown that such a prediction is as strong a conclusion as a rejection. Furthermore, a new method for core prediction analysis is introduced, which is beneficial for the uncertainty of specific model properties, as the method only characterizes the space of acceptable parameters in the relevant directions. This avoids the curse of dimensionality associated with the generic characterizations used by previously proposed methods. Analysis on examples shows that the new method is comparable to profile likelihood with regard to practical identifiability, and thus generalizes profile likelihood to the more general problem of observability. If used, the concepts and methods presented herein make it possible to distinguish between a conclusion and a mere suggestion, which hopefully will contribute to a more justified confidence in systems biology analyses.

Catching transcriptional regulation by thermostatistical modeling

Gene expression is frequently regulated by multiple transcription factors (TFs). Thermostatistical methods allow for a quantitative description of interactions between TFs, RNA polymerase and DNA, and their impact on the transcription rates. We illustrate three different scales of the thermostatistical approach: the microscale of TF molecules, the mesoscale of promoter energy levels and the macroscale of transcriptionally active and inactive cells in a cell population. We demonstrate versatility of combinatorial transcriptional activation by exemplifying logic functions, such as AND and OR gates. We discuss a metric for cell-to-cell transcriptional activation variability known as Fermi entropy. Suitability of thermostatistical modeling is illustrated by describing the experimental data on transcriptional induction of NFκB and the c-Fos protein.

Sensitivity analysis approaches applied to systems biology models

With the rising application of systems biology, sensitivity analysis methods have been widely applied to study the biological systems, including metabolic networks, signalling pathways and genetic circuits. Sensitivity analysis can provide valuable insights about how robust the biological responses are with respect to the changes of biological parameters and which model inputs are the key factors that affect the model outputs. In addition, sensitivity analysis is valuable for guiding experimental analysis, model reduction and parameter estimation. Local and global sensitivity analysis approaches are the two types of sensitivity analysis that are commonly applied in systems biology. Local sensitivity analysis is a classic method that studies the impact of small perturbations on the model outputs. On the other hand, global sensitivity analysis approaches have been applied to understand how the model outputs are affected by large variations of the model input parameters. In this review, the author introduces the basic concepts of sensitivity analysis approaches applied to systems biology models. Moreover, the author discusses the advantages and disadvantages of different sensitivity analysis methods, how to choose a proper sensitivity analysis approach, the available sensitivity analysis tools for systems biology models and the caveats in the interpretation of sensitivity analysis results.

The unconserved groucho central region is essential for viability and modulates target gene specificity

Groucho (Gro) is a Drosophila corepressor required by numerous DNA-binding repressors, many of which are distributed in gradients and provide positional information during development. Gro contains well-conserved domains at its N- and C-termini, and a poorly conserved central region that includes the GP, CcN, and SP domains. All lethal point mutations in gro map to the conserved regions, leading to speculation that the unconserved central domains are dispensable. However, our sequence analysis suggests that the central domains are disordered leading us to suspect that the lack of lethal mutations in this region reflects a lack of order rather than an absence of essential functions. In support of this conclusion, genomic rescue experiments with Gro deletion variants demonstrate that the GP and CcN domains are required for viability. Misexpression assays using these same deletion variants show that the SP domain prevents unrestrained and promiscuous repression by Gro, while the GP and CcN domains are indispensable for repression. Deletion of the GP domain leads to loss of nuclear import, while deletion of the CcN domain leads to complete loss of repression. Changes in Gro activity levels reset the threshold concentrations at which graded repressors silence target gene expression. We conclude that co-regulators such as Gro are not simply permissive components of the repression machinery, but cooperate with graded DNA-binding factors in setting borders of gene expression. We suspect that disorder in the Gro central domains may provide the flexibility that allows this region to mediate multiple interactions required for repression.

Mathematical modeling of gene expression: a guide for the perplexed biologist

The detailed analysis of transcriptional networks holds a key for understanding central biological processes, and interest in this field has exploded due to new large-scale data acquisition techniques. Mathematical modeling can provide essential insights, but the diversity of modeling approaches can be a daunting prospect to investigators new to this area. For those interested in beginning a transcriptional mathematical modeling project, we provide here an overview of major types of models and their applications to transcriptional networks. In this discussion of recent literature on thermodynamic, Boolean, and differential equation models, we focus on considerations critical for choosing and validating a modeling approach that will be useful for quantitative understanding of biological systems.