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Nguyen K, Rutter EM, Flores KB. Estimation of Parameter Distributions for Reaction-Diffusion Equations with Competition using Aggregate Spatiotemporal Data. Bull Math Biol 2023; 85:62. [PMID: 37268762 DOI: 10.1007/s11538-023-01162-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2023] [Accepted: 05/03/2023] [Indexed: 06/04/2023]
Abstract
Reaction-diffusion equations have been used to model a wide range of biological phenomenon related to population spread and proliferation from ecology to cancer. It is commonly assumed that individuals in a population have homogeneous diffusion and growth rates; however, this assumption can be inaccurate when the population is intrinsically divided into many distinct subpopulations that compete with each other. In previous work, the task of inferring the degree of phenotypic heterogeneity between subpopulations from total population density has been performed within a framework that combines parameter distribution estimation with reaction-diffusion models. Here, we extend this approach so that it is compatible with reaction-diffusion models that include competition between subpopulations. We use a reaction-diffusion model of glioblastoma multiforme, an aggressive type of brain cancer, to test our approach on simulated data that are similar to measurements that could be collected in practice. We use Prokhorov metric framework and convert the reaction-diffusion model to a random differential equation model to estimate joint distributions of diffusion and growth rates among heterogeneous subpopulations. We then compare the new random differential equation model performance against other partial differential equation models' performance. We find that the random differential equation is more capable at predicting the cell density compared to other models while being more time efficient. Finally, we use k-means clustering to predict the number of subpopulations based on the recovered distributions.
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Affiliation(s)
- Kyle Nguyen
- Biomathematics Graduate Program, North Carolina State University, Raleigh, NC, USA
- Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, USA
| | - Erica M Rutter
- Department of Applied Mathematics, University of California, Merced, Merced, CA, USA
| | - Kevin B Flores
- Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, USA.
- Department of Mathematics, North Carolina State University, Raleigh, NC, USA.
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2
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An Information-Theoretic Framework for Optimal Design: Analysis of Protocols for Estimating Soft Tissue Parameters in Biaxial Experiments. AXIOMS 2021. [DOI: 10.3390/axioms10020079] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework considers the information gain about the parameters from the experiment as the key criterion to be maximised, which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have a relatively low information content compared to high angles. The results also show that a smaller number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly D-optimal design.
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3
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Lambert B, Gavaghan DJ, Tavener SJ. A Monte Carlo method to estimate cell population heterogeneity from cell snapshot data. J Theor Biol 2020; 511:110541. [PMID: 33271182 DOI: 10.1016/j.jtbi.2020.110541] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Revised: 07/16/2020] [Accepted: 11/10/2020] [Indexed: 11/26/2022]
Abstract
Variation is characteristic of all living systems. Laboratory techniques such as flow cytometry can probe individual cells, and, after decades of experimentation, it is clear that even members of genetically identical cell populations can exhibit differences. To understand whether variation is biologically meaningful, it is essential to discern its source. Mathematical models of biological systems are tools that can be used to investigate causes of cell-to-cell variation. From mathematical analysis and simulation of these models, biological hypotheses can be posed and investigated, then parameter inference can determine which of these is compatible with experimental data. Data from laboratory experiments often consist of "snapshots" representing distributions of cellular properties at different points in time, rather than individual cell trajectories. These data are not straightforward to fit using hierarchical Bayesian methods, which require the number of cell population clusters to be chosen a priori. Nor are they amenable to standard nonlinear mixed effect methods, since a single observation per cell is typically too few to estimate parameter variability. Here, we introduce a computational sampling method named "Contour Monte Carlo" (CMC) for estimating mathematical model parameters from snapshot distributions, which is straightforward to implement and does not require that cells be assigned to predefined categories. The CMC algorithm fits to snapshot probability distributions rather than raw data, which means its computational burden does not, like existing approaches, increase with the number of cells observed. Our method is appropriate for underdetermined systems, where there are fewer distinct types of observations than parameters to be determined, and where observed variation is mostly due to variability in cellular processes rather than experimental measurement error. This may be the case for many systems due to continued improvements in resolution of laboratory techniques. In this paper, we apply our method to quantify cellular variation for three biological systems of interest and provide Julia code enabling others to use this method.
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Affiliation(s)
- Ben Lambert
- Department of Zoology, University of Oxford, Oxford, Oxfordshire, UK; MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London W2 1PG, UK.
| | - David J Gavaghan
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Simon J Tavener
- Department of Mathematics, Colorado State University, Fort Collins, Colorado, USA
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4
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Daly AC, Gavaghan D, Cooper J, Tavener S. Inference-based assessment of parameter identifiability in nonlinear biological models. J R Soc Interface 2019; 15:rsif.2018.0318. [PMID: 30021928 DOI: 10.1098/rsif.2018.0318] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2018] [Accepted: 06/21/2018] [Indexed: 11/12/2022] Open
Abstract
As systems approaches to the development of biological models become more mature, attention is increasingly focusing on the problem of inferring parameter values within those models from experimental data. However, particularly for nonlinear models, it is not obvious, either from inspection of the model or from the experimental data, that the inverse problem of parameter fitting will have a unique solution, or even a non-unique solution that constrains the parameters to lie within a plausible physiological range. Where parameters cannot be constrained they are termed 'unidentifiable'. We focus on gaining insight into the causes of unidentifiability using inference-based methods, and compare a recently developed measure-theoretic approach to inverse sensitivity analysis to the popular Markov chain Monte Carlo and approximate Bayesian computation techniques for Bayesian inference. All three approaches map the uncertainty in quantities of interest in the output space to the probability of sets of parameters in the input space. The geometry of these sets demonstrates how unidentifiability can be caused by parameter compensation and provides an intuitive approach to inference-based experimental design.
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Affiliation(s)
- Aidan C Daly
- Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
| | - David Gavaghan
- Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
| | - Jonathan Cooper
- Research IT Services, University College London, Gower Street, London WC1E 6BT, UK
| | - Simon Tavener
- Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
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5
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Banks HT, Hu S, Link K, Rosenberg ES, Mitsuma S, Rosario L. Modeling Immune Response to BK Virus Infection and Donor Kidney in Renal Transplant Recipients. INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 2016; 24:127-152. [PMID: 26925154 PMCID: PMC4767521 DOI: 10.1080/17415977.2015.1017484] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2014] [Accepted: 01/25/2015] [Indexed: 06/04/2023]
Abstract
In this paper we develop and validate with bootstrapping techniques a mechanistic mathematical model of immune response to both BK virus infection and a donor kidney based on known and hypothesized mechanisms in the literature. The model presented does not capture all the details of the immune response but possesses key features that describe a very complex immunological process. We then estimate model parameters using a least squares approach with a typical set of available clinical data. Sensitivity analysis combined with asymptotic theory is used to determine the number of parameters that can be reliably estimated given the limited number of observations.
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Affiliation(s)
- H T Banks
- Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212 USA
| | - Shuhua Hu
- Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212 USA
| | - Kathryn Link
- Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212 USA
| | - Eric S Rosenberg
- Partners Human Research Committee, Massachusetts General Hospital, Boston, MA 02114 USA
| | - Sheila Mitsuma
- Partners Human Research Committee, Massachusetts General Hospital, Boston, MA 02114 USA
| | - Lauren Rosario
- Partners Human Research Committee, Massachusetts General Hospital, Boston, MA 02114 USA
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6
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Banks HT, Rubio D, Saintier N, Troparevsky MI. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2015; 12:739-760. [PMID: 25974344 DOI: 10.3934/mbe.2015.12.739] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical D-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.
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Affiliation(s)
- H T Banks
- Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, United States.
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7
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Adoteye K, Banks H, Flores KB. Optimal Design of Non-equilibrium Experiments for Genetic Network Interrogation. APPLIED MATHEMATICS LETTERS 2015; 40:84-89. [PMID: 25558126 PMCID: PMC4281269 DOI: 10.1016/j.aml.2014.09.013] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2014] [Accepted: 09/21/2014] [Indexed: 06/04/2023]
Abstract
Many experimental systems in biology, especially synthetic gene networks, are amenable to perturbations that are controlled by the experimenter. We developed an optimal design algorithm that calculates optimal observation times in conjunction with optimal experimental perturbations in order to maximize the amount of information gained from longitudinal data derived from such experiments. We applied the algorithm to a validated model of a synthetic Brome Mosaic Virus (BMV) gene network and found that optimizing experimental perturbations may substantially decrease uncertainty in estimating BMV model parameters.
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Affiliation(s)
| | - H.T. Banks
- Department of Mathematics, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, United States
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8
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Katz Y, Nevo U. Quantification of pore size distribution using diffusion NMR: Experimental design and physical insights. J Chem Phys 2014; 140:164201. [DOI: 10.1063/1.4871193] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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9
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Banks HT, Rehm KL. Experimental Design for Vector Output Systems. INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 2014; 22:557-590. [PMID: 24563655 PMCID: PMC3929304 DOI: 10.1080/17415977.2013.797973] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
We formulate an optimal design problem for the selection of best states to observe and optimal sampling times for parameter estimation or inverse problems involving complex nonlinear dynamical systems. An iterative algorithm for implementation of the resulting methodology is proposed. Its use and efficacy is illustrated on two applied problems of practical interest: (i) dynamic models of HIV progression and (ii) modeling of the Calvin cycle in plant metabolism and growth.
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Affiliation(s)
- H T Banks
- Center for Research in Scientific Computation Center for Quantitative Sciences in Biomedicine N.C. State University Raleigh, NC
| | - K L Rehm
- Center for Research in Scientific Computation Center for Quantitative Sciences in Biomedicine N.C. State University Raleigh, NC
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10
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Banks HT, Robbins D, Sutton KL. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2013; 10:1301-1333. [PMID: 24245618 DOI: 10.3934/mbe.2013.10.1301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.
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Affiliation(s)
- H Thomas Banks
- Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, Raleigh, NC 27695-8212, United States.
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11
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Banks HT, Rehm KL. Experimental Design for Distributed Parameter Vector Systems. APPLIED MATHEMATICS LETTERS 2013; 26:10-14. [PMID: 23049163 PMCID: PMC3462031 DOI: 10.1016/j.aml.2012.08.003] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
We formulate an optimal design problem for the selection of best states to observe and optimal sampling times and locations for parameter estimation or inverse problems involving complex nonlinear nonlinear partial differential systems. An iterative algorithm for implementation of the resulting methodology is proposed.
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Affiliation(s)
- H T Banks
- Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, N.C. State University, Raleigh, NC
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12
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Banks HT, Kapraun DF, Thompson WC, Peligero C, Argilaguet J, Meyerhans A. A novel statistical analysis and interpretation of flow cytometry data. JOURNAL OF BIOLOGICAL DYNAMICS 2013; 7:96-132. [PMID: 23826744 PMCID: PMC3753657 DOI: 10.1080/17513758.2013.812753] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
A recently developed class of models incorporating the cyton model of population generation structure into a conservation-based model of intracellular label dynamics is reviewed. Statistical aspects of the data collection process are quantified and incorporated into a parameter estimation scheme. This scheme is then applied to experimental data for PHA-stimulated CD4+T and CD8+T cells collected from two healthy donors. This novel mathematical and statistical framework is shown to form the basis for accurate, meaningful analysis of cellular behaviour for a population of cells labelled with the dye carboxyfluorescein succinimidyl ester and stimulated to divide.
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Affiliation(s)
- H T Banks
- Center for Research in Scientific Computation and Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8212, USA.
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13
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Banks HT, Hu S, Jang T, Kwon HD. Modelling and optimal control of immune response of renal transplant recipients. JOURNAL OF BIOLOGICAL DYNAMICS 2012; 6:539-67. [PMID: 22873605 PMCID: PMC3691280 DOI: 10.1080/17513758.2012.655328] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
We consider the increasingly important and highly complex immunological control problem: control of the dynamics of immunosuppression for organ transplant recipients. The goal in this problem is to maintain the delicate balance between over-suppression (where opportunistic latent viruses threaten the patient) and under-suppression (where rejection of the transplanted organ is probable). First, a mathematical model is formulated to describe the immune response to both viral infection and introduction of a donor kidney in a renal transplant recipient. Some numerical results are given to qualitatively validate and demonstrate that this initial model exhibits appropriate characteristics of primary infection and reactivation for immunosuppressed transplant recipients. In addition, we develop a computational framework for designing adaptive optimal treatment regimes with partial observations and low-frequency sampling, where the state estimates are obtained by solving a second deterministic optimal tracking problem. Numerical results are given to illustrate the feasibility of this method in obtaining optimal treatment regimes with a balance between under-suppression and over-suppression of the immune system.
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Affiliation(s)
- H T Banks
- Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8212, USA.
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14
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Wendelsdorf K, Dean G, Hu S, Nordone S, Banks HT. Host immune responses that promote initial HIV spread. J Theor Biol 2011; 289:17-35. [PMID: 21871901 DOI: 10.1016/j.jtbi.2011.08.012] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2010] [Revised: 07/05/2011] [Accepted: 08/11/2011] [Indexed: 11/29/2022]
Abstract
The host inflammatory response to HIV invasion is a necessary component of the innate antiviral activity that vaccines and early interventions seek to exploit/enhance. However, the response is dependent on CD4+ T-helper cell 1 (Th1) recruitment and activation. It is this very recruitment of HIV-susceptible target cells that is associated with the initial viral proliferation. Hence, global enhancement of the inflammatory response by T-cells and dendritic cells will likely feed viral propagation. Mucosal entry sites contain inherent pathways, in the form of natural regulatory T-cells (nTreg), that globally dampen the inflammatory response. We created a model of this inflammatory response to virus as well as inherent nTreg-mediated regulation of Th1 recruitment and activation. With simulations using this model we sought to address the net effect of nTreg activation and its specific functions as well as identify mechanisms of the natural inflammatory response that are best targeted to inhibit viral spread without compromising initial antiviral activity. Simulation results provide multiple insights that are relevant to developing intervention strategies that seek to exploit natural immune processes: (i) induction of the regulatory response through nTreg activation expedites viral proliferation due to viral production by nTreg itself and not to reduced Natural Killer (NK) cell activity; (ii) at the same time, induction of the inflammation response through either DC activation or Th1 activation expedites viral proliferation; (iii) within the inflammatory pathway, the NK response is an effective controller of viral proliferation while DC-mediated stimulation of T-cells is a significant driver of viral proliferation; and (iv) nTreg-mediated DC deactivation plays a significant role in slowing viral proliferation by inhibiting T-cell stimulation, making this function an aide to the antiviral immune response.
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Affiliation(s)
- K Wendelsdorf
- Virginia Bioinformatics Institute, Virginia Polytechnic Institute and University, Washington Street, MC 0477, Blacksburg, VA 24061, USA
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