MULTI-SCALE INVERSE MODELING IN BIOLOGICAL MASS TRANSPORT PROCESSES

A state-of-the-art inverse modeling strategy was developed, analyzed, and applied in two different biological mass transport processes. The strategy was developed in the framework of the nonlinear optimization problem in which model parameters were estimated by minimizing an appropriate objective function which represents the discrepancy between the observed and predicted responses of the biological systems. The forward problems were solved numerically using the mass conservative Galerkin based linear finite element and finite difference methods. Before incorporating in the framework of the inverse code, the numerical simulators were validated with either analytical or reference solutions. In the inverse code, the Osborne- Moré extended version of the Levenberg- Marquardt algorithm was used to determine the search direction. The Jacobian matrix was constructed using partial derivatives of the state variables with respect to model parameters by one and two-sided finite difference approximations. A mixed termination criterion was used to end the optimization. The strategy was applied to parameter identification problem in Fluorescence Recovery after Photobleaching (FRAP) protocol to estimate the optimized values of the mass transport and binding rate parameters for GFP-tagged glucocorticoid receptor. Results indicate that the protocol provides enough information to uniquely estimate one parameter. It also provides enough information to uniquely estimate the individual values of the binding rate coefficients given the value of the molecular diffusion coefficient is known. However, the protocol provides insufficient information for unique simultaneous estimation of three parameters (diffusion coefficient and binding rate parameters) owing to the high intercorrelation between the molecular diffusion coefficient and pseudo-association rate parameter. Attempts to estimate macromolecule mass transport and binding rate parameters simultaneously from FRAP data result in misleading conclusions regarding concentrations of free macromolecule and bound complex inside the cell, average binding time per vacant site, average time for diffusion of macromolecules from one site to the next, and slow or rapid mobility of biomolecules in cells. To obtain unique values for molecular diffusion coefficient and binding rate parameters of biomolecule, two FRAP experiments should be conducted on the same class of macromolecule and cell. One experiment should be used to measure the molecular diffusion coefficient independently of binding in an effective diffusion regime and the other should be conducted in a reaction dominant or reaction-diffusion regime to quantify the binding rate parameters. The inverse modeling strategy was also successfully used to identify hydraulic parameters for both single and multi-objective optimization problems in homogeneous and heterogeneous variably saturated soils. Incorporating both soil water content information and soil water pressure head data in the framework of the multi-objective parameter optimization, produced excellent result for both soil water content and pressure head profiles

A finite element model for protein transport in vivo

<p>Abstract</p> <p>Background</p> <p>Biological mass transport processes determine the behavior and function of cells, regulate interactions between synthetic agents and recipient targets, and are key elements in the design and use of biosensors. Accurately predicting the outcomes of such processes is crucial to both enhancing our understanding of how these systems function, enabling the design of effective strategies to control their function, and verifying that engineered solutions perform according to plan.</p> <p>Methods</p> <p>A Galerkin-based finite element model was developed and implemented to solve a system of two coupled partial differential equations governing biomolecule transport and reaction in live cells. The simulator was coupled, in the framework of an inverse modeling strategy, with an optimization algorithm and an experimental time series, obtained by the Fluorescence Recovery after Photobleaching (FRAP) technique, to estimate biomolecule mass transport and reaction rate parameters. In the inverse algorithm, an adaptive method was implemented to calculate sensitivity matrix. A multi-criteria termination rule was developed to stop the inverse code at the solution. The applicability of the model was illustrated by simulating the mobility and binding of GFP-tagged glucocorticoid receptor in the nucleoplasm of mouse adenocarcinoma.</p> <p>Results</p> <p>The numerical simulator shows excellent agreement with the analytic solutions and experimental FRAP data. Detailed residual analysis indicates that residuals have zero mean and constant variance and are normally distributed and uncorrelated. Therefore, the necessary and sufficient criteria for least square parameter optimization, which was used in this study, were met.</p> <p>Conclusion</p> <p>The developed strategy is an efficient approach to extract as much physiochemical information from the FRAP protocol as possible. Well-posedness analysis of the inverse problem, however, indicates that the FRAP protocol provides insufficient information for unique simultaneous estimation of diffusion coefficient and binding rate parameters. Care should be exercised in drawing inferences, from FRAP data, regarding concentrations of free and bound proteins, average binding and diffusion times, and protein mobility unless they are confirmed by long-range Markov Chain-Monte Carlo (MCMC) methods and experimental observations.</p

Identification of biomolecule mass transport and binding rate parameters in living cells by inverse modeling

BACKGROUND: Quantification of in-vivo biomolecule mass transport and reaction rate parameters from experimental data obtained by Fluorescence Recovery after Photobleaching (FRAP) is becoming more important. METHODS AND RESULTS: The Osborne-Moré extended version of the Levenberg-Marquardt optimization algorithm was coupled with the experimental data obtained by the Fluorescence Recovery after Photobleaching (FRAP) protocol, and the numerical solution of a set of two partial differential equations governing macromolecule mass transport and reaction in living cells, to inversely estimate optimized values of the molecular diffusion coefficient and binding rate parameters of GFP-tagged glucocorticoid receptor. The results indicate that the FRAP protocol provides enough information to estimate one parameter uniquely using a nonlinear optimization technique. Coupling FRAP experimental data with the inverse modeling strategy, one can also uniquely estimate the individual values of the binding rate coefficients if the molecular diffusion coefficient is known. One can also simultaneously estimate the dissociation rate parameter and molecular diffusion coefficient given the pseudo-association rate parameter is known. However, the protocol provides insufficient information for unique simultaneous estimation of three parameters (diffusion coefficient and binding rate parameters) owing to the high intercorrelation between the molecular diffusion coefficient and pseudo-association rate parameter. Attempts to estimate macromolecule mass transport and binding rate parameters simultaneously from FRAP data result in misleading conclusions regarding concentrations of free macromolecule and bound complex inside the cell, average binding time per vacant site, average time for diffusion of macromolecules from one site to the next, and slow or rapid mobility of biomolecules in cells. CONCLUSION: To obtain unique values for molecular diffusion coefficient and binding rate parameters from FRAP data, we propose conducting two FRAP experiments on the same class of macromolecule and cell. One experiment should be used to measure the molecular diffusion coefficient independently of binding in an effective diffusion regime and the other should be conducted in a reaction dominant or reaction-diffusion regime to quantify binding rate parameters. The method described in this paper is likely to be widely used to estimate in-vivo biomolecule mass transport and binding rate parameters

Histograms of residuals for normalized average fluorescent intensity using one-site-mobile-immobile model

<p><b>Copyright information:</b></p><p>Taken from "A finite element model for protein transport in vivo"</p><p>http://www.biomedical-engineering-online.com/content/6/1/24</p><p>BioMedical Engineering OnLine 2007;6():24-24.</p><p>Published online 28 Jun 2007</p><p>PMCID:PMC1940256.</p><p></p

Spatial and temporal distributions of GFP-GR inside bleach spot after photo-chemical bleaching during time course of a FRAP experiment

<p><b>Copyright information:</b></p><p>Taken from "A finite element model for protein transport in vivo"</p><p>http://www.biomedical-engineering-online.com/content/6/1/24</p><p>BioMedical Engineering OnLine 2007;6():24-24.</p><p>Published online 28 Jun 2007</p><p>PMCID:PMC1940256.</p><p></p> The Figure shows comparison of the analytic solution (solid lines) and numerical model (dots) at times of 0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, and 2 seconds for free GFP-GR (a), bound complex (b), and total GFP-GR (c). Validation of the numerical model (dots) with the analytic solution (solid lines) of [25] is depicted in (d). The graph presents average normalized fluorescent intensity, obtained by equation (16), inside the bleach spot