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1.  Multiple Model-Informed Open-Loop Control of Uncertain Intracellular Signaling Dynamics 
PLoS Computational Biology  2014;10(4):e1003546.
Computational approaches to tune the activation of intracellular signal transduction pathways both predictably and selectively will enable researchers to explore and interrogate cell biology with unprecedented precision. Techniques to control complex nonlinear systems typically involve the application of control theory to a descriptive mathematical model. For cellular processes, however, measurement assays tend to be too time consuming for real-time feedback control and models offer rough approximations of the biological reality, thus limiting their utility when considered in isolation. We overcome these problems by combining nonlinear model predictive control with a novel adaptive weighting algorithm that blends predictions from multiple models to derive a compromise open-loop control sequence. The proposed strategy uses weight maps to inform the controller of the tendency for models to differ in their ability to accurately reproduce the system dynamics under different experimental perturbations (i.e. control inputs). These maps, which characterize the changing model likelihoods over the admissible control input space, are constructed using preexisting experimental data and used to produce a model-based open-loop control framework. In effect, the proposed method designs a sequence of control inputs that force the signaling dynamics along a predefined temporal response without measurement feedback while mitigating the effects of model uncertainty. We demonstrate this technique on the well-known Erk/MAPK signaling pathway in T cells. In silico assessment demonstrates that this approach successfully reduces target tracking error by 52% or better when compared with single model-based controllers and non-adaptive multiple model-based controllers. In vitro implementation of the proposed approach in Jurkat cells confirms a 63% reduction in tracking error when compared with the best of the single-model controllers. This study provides an experimentally-corroborated control methodology that utilizes the knowledge encoded within multiple mathematical models of intracellular signaling to design control inputs that effectively direct cell behavior in open-loop.
Author Summary
Most cell behavior arises as a response to external forces. Signals from the extracellular environment are passed to the cell's nucleus through a complex network of interacting proteins. Perturbing these pathways can change the strength or outcome of the signals, which could be used to treat or prevent a pathological response. While manipulating these networks can be achieved using a variety of methods, the ability to do so predictably over time would provide an unprecedented level of control over cell behavior and could lead to new therapeutic design and research tools in medicine and systems biology. Hence, we propose a practical computational framework to aid in the design of experimental perturbations to force cell signaling dynamics to follow a predefined response. Our approach represents a novel merger of model-based control and information theory to blend the predictions from multiple mathematical models into a meaningful compromise solution. We verify through simulation and experimentation that this solution produces excellent agreement between the cell readouts and several predefined trajectories, even in the presence of significant modeling uncertainty and without measurement feedback. By combining elements of information and control theory, our approach will help advance the best practices in model-based control applications for medicine.
PMCID: PMC3983080  PMID: 24722333
2.  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.
PMCID: PMC3952817  PMID: 24626201
3.  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.
PMCID: PMC3481438  PMID: 23134718
4.  Modeling Mitochondrial Bioenergetics with Integrated Volume Dynamics 
PLoS Computational Biology  2010;6(1):e1000632.
Mathematical models of mitochondrial bioenergetics provide powerful analytical tools to help interpret experimental data and facilitate experimental design for elucidating the supporting biochemical and physical processes. As a next step towards constructing a complete physiologically faithful mitochondrial bioenergetics model, a mathematical model was developed targeting the cardiac mitochondrial bioenergetic based upon previous efforts, and corroborated using both transient and steady state data. The model consists of several modified rate functions of mitochondrial bioenergetics, integrated calcium dynamics and a detailed description of the K+-cycle and its effect on mitochondrial bioenergetics and matrix volume regulation. Model simulations were used to fit 42 adjustable parameters to four independent experimental data sets consisting of 32 data curves. During the model development, a certain network topology had to be in place and some assumptions about uncertain or unobserved experimental factors and conditions were explicitly constrained in order to faithfully reproduce all the data sets. These realizations are discussed, and their necessity helps contribute to the collective understanding of the mitochondrial bioenergetics.
Author Summary
Mathematically modeling biological systems challenges our current understanding of the physical and biochemical events contributing to the observed dynamics. It requires careful consideration of hypothesized mechanisms, model development assumptions and details regarding the experimental conditions. We have adopted a modeling approach to translate these factors that explicitly considers the thermodynamic constraints, biochemical states and reaction mechanisms during model development. Such models have numerous constant parameters that must be determined. Integrating thermodynamics and detailed mechanistic representation of the principal phenomena help constrain these parameter values; therefore, only a handful of the total number of model parameters (∼10%) must be adjusted during parameter estimation through model simulations. Additionally, all models must undergo some form of corroboration prior to application. In practice, this corroboration should challenge all possible dynamics of the model, but it is recognized that in this data rich world, we are surprisingly data poor. Eventually such developed and corroborated models are capable of supporting current hypotheses, guiding experimental designs and contributing to the overall knowledge base of biological processes.
PMCID: PMC2793388  PMID: 20052270
5.  Identification of IKr Kinetics and Drug Binding in Native Myocytes 
Annals of Biomedical Engineering  2009;37(7):1294-1309.
Determining the effect of a compound on IKr is a standard screen for drug safety. Often the effect is described using a single IC50 value, which is unable to capture complex effects of a drug. Using verapamil as an example, we present a method for using recordings from native myocytes at several drug doses along with qualitative features of IKr from published studies of HERG current to estimate parameters in a mathematical model of the drug effect on IKr. IKr was recorded from canine left ventricular myocytes using ruptured patch techniques. A voltage command protocol was used to record tail currents at voltages from −70 to −20 mV, following activating pulses over a wide range of voltages and pulse durations. Model equations were taken from a published IKr Markov model and the drug was modeled as binding to the open state. Parameters were estimated using a combined global and local optimization algorithm based on collected data with two additional constraints on IKrI–V relation and IKr inactivation. The method produced models that quantitatively reproduce both the control IKr kinetics and dose dependent changes in the current. In addition, the model exhibited use and rate dependence. The results suggest that: (1) the technique proposed here has the practical potential to develop data-driven models that quantitatively reproduce channel behavior in native myocytes; (2) the method can capture important drug effects that cannot be reproduced by the IC50 method. Although the method was developed for IKr, the same strategy can be applied to other ion channels, once appropriate channel-specific voltage protocols and qualitative features are identified.
Electronic supplementary material
The online version of this article (doi:10.1007/s10439-009-9690-5) contains supplementary material, which is available to authorized users.
PMCID: PMC2690829  PMID: 19353268
Mathematical modeling; Drug–ion current interaction; Parameter estimation; Global optimization; Verapamil; Cardiac electrophysiology

Results 1-5 (5)