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1.  Model-Based Analysis for Qualitative Data: An Application in Drosophila Germline Stem Cell Regulation 
PLoS Computational Biology  2014;10(3):e1003498.
Discovery in developmental biology is often driven by intuition that relies on the integration of multiple types of data such as fluorescent images, phenotypes, and the outcomes of biochemical assays. Mathematical modeling helps elucidate the biological mechanisms at play as the networks become increasingly large and complex. However, the available data is frequently under-utilized due to incompatibility with quantitative model tuning techniques. This is the case for stem cell regulation mechanisms explored in the Drosophila germarium through fluorescent immunohistochemistry. To enable better integration of biological data with modeling in this and similar situations, we have developed a general parameter estimation process to quantitatively optimize models with qualitative data. The process employs a modified version of the Optimal Scaling method from social and behavioral sciences, and multi-objective optimization to evaluate the trade-off between fitting different datasets (e.g. wild type vs. mutant). Using only published imaging data in the germarium, we first evaluated support for a published intracellular regulatory network by considering alternative connections of the same regulatory players. Simply screening networks against wild type data identified hundreds of feasible alternatives. Of these, five parsimonious variants were found and compared by multi-objective analysis including mutant data and dynamic constraints. With these data, the current model is supported over the alternatives, but support for a biochemically observed feedback element is weak (i.e. these data do not measure the feedback effect well). When also comparing new hypothetical models, the available data do not discriminate. To begin addressing the limitations in data, we performed a model-based experiment design and provide recommendations for experiments to refine model parameters and discriminate increasingly complex hypotheses.
Author Summary
We developed a process to quantitatively fit mathematical models using qualitative data, and applied it in the study of how stem cells are regulated in the fruit fly ovary. The available published data we collected are fluorescent images of protein and mRNA expression from genetic experiments. Despite lacking quantitative data, the new process makes available quantitative model analysis techniques to reliably compare different models and guide future experiments. We found that the current consensus regulatory model is supported, but that the data are indeed insufficient to address more complex hypotheses. With the quantitatively fit models, we evaluated hypothetical experiments and estimated which future measurements should best refine or test models. The model fitting process we have developed is applicable to many biological studies where qualitative data are common, and can accelerate progress through more efficient experimentation.
doi:10.1371/journal.pcbi.1003498
PMCID: PMC3952817  PMID: 24626201
2.  The importance of geometry in mathematical models of developing systems 
Understanding the interaction between the spatial variation of extracellular signals and the interpretation of such signals in embryonic development is difficult without a mathematical model, but the inherent limitations of a model can have a profound impact on its utility. A central issue is the level of abstraction needed, and here we focus on the role of geometry in models and how the choice of the spatial dimension can influence the conclusions reached. A widely-studied system in which the proper choice of geometry is critical is embryonic development of Drosophila melanogaster, and we discuss recent work in which 3D embryo-scale modeling is used to identify key modes of transport, analyze gap gene expression, and test BMP-mediated positive feedback mechanisms.
doi:10.1016/j.gde.2012.09.007
PMCID: PMC3765006  PMID: 23107453
3.  Regulation of BMP activity and range in Drosophila wing development 
Current Opinion in Cell Biology  2011;24(2):158-165.
Bone morphogenetic protein (BMP) signaling controls development and maintenance of many tissues. Genetic and quantitative approaches in Drosophila reveal that ligand isoforms show distinct function in wing development. Spatiotemporal control of BMP patterning depends on a network of extracellular proteins Pent, Ltl and Dally that regulate BMP signaling strength and morphogen range. BMP-mediated feedback regulation of Pent, Ltl, and Dally expression provides a system where cells actively respond to, and modify, the extracellular morphogen landscape to form a gradient that exhibits remarkable properties, including proportional scaling of BMP patterning with tissue size and the modulation of uniform tissue growth. This system provides valuable insights into mechanisms that mitigate the influence of variability to regulate cell-cell interactions and maintain organ function.
doi:10.1016/j.ceb.2011.11.004
PMCID: PMC3320673  PMID: 22152945
4.  Efficient calculation of steady state probability distribution for stochastic biochemical reaction network 
BMC Genomics  2012;13(Suppl 6):S10.
The Steady State (SS) probability distribution is an important quantity needed to characterize the steady state behavior of many stochastic biochemical networks. In this paper, we propose an efficient and accurate approach to calculating an approximate SS probability distribution from solution of the Chemical Master Equation (CME) under the assumption of the existence of a unique deterministic SS of the system. To find the approximate solution to the CME, a truncated state-space representation is used to reduce the state-space of the system and translate it to a finite dimension. The subsequent ill-posed eigenvalue problem of a linear system for the finite state-space can be converted to a well-posed system of linear equations and solved. The proposed strategy yields efficient and accurate estimation of noise in stochastic biochemical systems. To demonstrate the approach, we applied the method to characterize the noise behavior of a set of biochemical networks of ligand-receptor interactions for Bone Morphogenetic Protein (BMP) signaling. We found that recruitment of type II receptors during the receptor oligomerization by itself doesn't not tend to lower noise in receptor signaling, but regulation by a secreted co-factor may provide a substantial improvement in signaling relative to noise. The steady state probability approximation method shortened the time necessary to calculate the probability distributions compared to earlier approaches, such as Gillespie's Stochastic Simulation Algorithm (SSA) while maintaining high accuracy.
doi:10.1186/1471-2164-13-S6-S10
PMCID: PMC3481438  PMID: 23134718
5.  Analysis of Gap Gene Regulation in a 3D Organism-Scale Model of the Drosophila melanogaster Embryo 
PLoS ONE  2011;6(11):e26797.
The axial bodyplan of Drosophila melanogaster is determined during a process called morphogenesis. Shortly after fertilization, maternal bicoid mRNA is translated into Bicoid (Bcd). This protein establishes a spatially graded morphogen distribution along the anterior-posterior (AP) axis of the embryo. Bcd initiates AP axis determination by triggering expression of gap genes that subsequently regulate each other's expression to form a precisely controlled spatial distribution of gene products. Reaction-diffusion models of gap gene expression on a 1D domain have previously been used to infer complex genetic regulatory network (GRN) interactions by optimizing model parameters with respect to 1D gap gene expression data. Here we construct a finite element reaction-diffusion model with a realistic 3D geometry fit to full 3D gap gene expression data. Though gap gene products exhibit dorsal-ventral asymmetries, we discover that previously inferred gap GRNs yield qualitatively correct AP distributions on the 3D domain only when DV-symmetric initial conditions are employed. Model patterning loses qualitative agreement with experimental data when we incorporate a realistic DV-asymmetric distribution of Bcd. Further, we find that geometry alone is insufficient to account for DV-asymmetries in the final gap gene distribution. Additional GRN optimization confirms that the 3D model remains sensitive to GRN parameter perturbations. Finally, we find that incorporation of 3D data in simulation and optimization does not constrain the search space or improve optimization results.
doi:10.1371/journal.pone.0026797
PMCID: PMC3217930  PMID: 22110594
6.  Analysis of dynamic morphogen scale invariance 
During the development of some tissues, fields of multipotent cells differentiate into distinct cell types in response to the local concentration of a signalling factor called a morphogen. Typically, individual organisms within a population differ in size, but their body plans appear to be scaled versions of a common template. Similarly, closely related species may differ by three or more orders of magnitude in size, yet common structures between species scale to have similar proportions. In standard reaction–diffusion equations, the morphogen range has a length scale that depends on a balance between kinetic and transport processes and not on the length or size of the field of cells being patterned. However, as shown here for a class of morphogen-patterning systems, a number of conditions lead to scale invariance of the morphogen distribution at equilibrium and during the transient approach to equilibrium. Equilibrium scale invariance requires conservation of the total binding site number and total input flux. Dynamic scale invariance additionally requires sufficient binding to slow the diffusion of ligand. The equations derived herein can be extended to the study of other perturbations to gain further insight into the processes regulating the robustness and scaling of morphogen-mediated pattern formation.
doi:10.1098/rsif.2009.0015
PMCID: PMC2817156  PMID: 19324667
scale invariance; bicoid; bone morphogenetic protein; systems biology; development; morphogen
7.  Organism-scale modeling of early Drosophila patterning via Bone Morphogenetic Proteins 
Developmental cell  2010;18(2):260-274.
Advances in image acquisition and informatics technology have led to organism-scale spatio-temporal atlases of gene expression and protein distributions. To maximize the utility of this information for the study of developmental processes, a new generation of mathematical models is needed for discovery and hypothesis testing. Here we develop a data-driven, geometrically-accurate, model of early Drosophila embryonic bone morphogenetic protein (BMP)-mediated patterning. We tested nine different mechanisms for signal transduction with feedback, eight combinations of geometry and gene expression prepatterns, and two scale-invariance mechanisms for their ability to reproduce proper BMP signaling output in wild-type and mutant embryos. We found that a model based on positive feedback of a secreted BMP binding protein, coupled with the experimentally-measured embryo geometry, provides the best agreement with population-mean image data. Our results demonstrate that using bioimages to build and optimize a 3D model provides significant new insights into mechanisms that guide tissue patterning.
doi:10.1016/j.devcel.2010.01.006
PMCID: PMC2848394  PMID: 20159596

Results 1-7 (7)