MBI Publications
MBI Publications for HyeWon Kang (9)

H. Kang, A. Friedman and P. NanaSinkam
A mathematical model for miR9, let7, and EMT in lung cancer(Submitted)Abstract

H. Kang, T. Kurtz and L. Popovic
Central limit theorems and diffusion approximations for multiscale Markov chain models(In Preparation)Abstract

J. Hu, H. Kang and H. Othmer
Stochastic analysis of reactiondiffusion processesBulletin of Mathematical Biology (Accepted)Abstract
Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatiallydiscretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reactiondiffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems. 
L. Zheng, H. Othmer and H. Kang
The effect of the signaling scheme on the robustness of pattern formation in developmentInterface FocusVol. 2 No. 4 (2012) pp. 465486 (To Appear)Abstract

H. Kang
A multiscale approximation in a heat shock response model in E.coliBMC Systems BiologyVol. 6 No. 1 (2012) pp. 143Abstract
Background: A heat shock response model of Escherichia coli developed by Srivastava, Peterson, and Bentley (2001) has multiscale nature due to its species numbers and reaction rate constants varying over wide ranges. Applying the method of separation of timescales and model reduction for stochastic reaction networks extended by Kang and Kurtz (2012), we approximate the chemical network in the heat shock response model.
Results: Scaling the species numbers and the rate constants by powers of the scaling parameter, we embed the model into a oneparameter family of models, each of which is a continuoustime Markov chain. Choosing an appropriate set of scaling exponents for the species numbers and for the rate constants satisfying balance conditions, the behavior of the full network in the time scales of interest is approximated by limiting models in three time scales. Due to the subset of species whose numbers are either approximated as constants or are averaged in terms of other species numbers, the limiting models are located on lower dimensional spaces than the full model and have a simpler structure than the full model does.
Conclusions: The goal of this paper is to illustrate how to apply the multiscale approximation method to the biological model with significant complexity. We applied the method to the heat shock response model involving 9 species and 18 reactions and derived simplified models in three time scales which capture the dynamics of the full model. Convergence of the scaled species numbers to their limit is obtained and errors between the scaled species numbers and their limit are estimated using the central limit theorem. 
H. Kang, L. Zheng and H. Othmer
A new method for choosing the computational cell in stochastic reactiondiffusion systemsJournal of Mathematical BiologyVol. 65 No. Series 67 (2012) pp. 10171099Abstract
How to choose the computational compartment or cell size for the stochastic simulation of a reactionâ€“diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finitedimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reactionâ€“diffusion system. We show that for a broad class of firstorder reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably renormalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatiallyuniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and nondiffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anteriorposterior pattern formation in Drosophila, and with several other examples. 
H. Kang, M. Crawford, M. Fabbri, G. Nuovo, M. Garofalo, P. NanaSinkam and A. Friedman
A mathematical model for microRNA in lung cancerPLoS OneVol. 8 No. 1 (2013)Abstract
Lung cancer is the leading cause of cancerrelated deaths worldwide. Lack of early detection and limited options for targeted therapies are both contributing factors to the dismal statistics observed in lung cancer. Thus, advances in both of these areas are likely to lead to improved outcomes. MicroRNAs (miRs or miRNAs) represent a class of noncoding RNAs that have the capacity for gene regulation and may serve as both diagnostic and prognostic biomarkers in lung cancer. Abnormal expression patterns for several miRNAs have been identified in lung cancers. Specifically, let7 and miR9 are deregulated in both lung cancers and other solid malignancies. In this paper, we construct a mathematical model that integrates let7 and miR9 expression into a signaling pathway to generate an in silico model for the process of epithelial mesenchymal transition (EMT). Simulations of the model demonstrate that EGFR and Ras mutations in nonsmall cell lung cancers (NSCLC), which lead to the process of EMT, result in miR9 upregulation and let7 suppression, and this process is somewhat robust against random input into miR9 and more strongly robust against random input into let7. We elected to validate our model in vitro by testing the effects of EGFR inhibition on downstream MYC, miR9 and let7a expression. Interestingly, in an EGFR mutated lung cancer cell line, treatment with an EGFR inhibitor (Gefitinib) resulted in a concentration specific reduction in cMYC and miR9 expression while not changing let7a expression. Our mathematical model explains the signaling link among EGFR, MYC, and miR9, but not let7. However, very little is presently known about factors that regulate let7. It is quite possible that when such regulating factors become known and integrated into our model, they will further support our mathematical model. 
H. Kang and T. Kurtz
Separation of timescales and model reduction for stochastic reaction networksAnnals of Applied ProbabilityVol. 23 No. 1 (2013) pp. 529583Abstract
A stochastic model for a chemical reaction network is embedded in a oneparameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have different timescales, the subnetworks can be approximated separately, providing insight into the behavior of the full network through the analysis of these lowerdimensional approximations. 
H. Kang, M. Crawford, M. Fabbri, G. Nuovo, M. Garofalo and A. Friedman
A mathematical model for microRNA in lung cancer.PloS oneVol. 8 No. 1 (2013) pp. e53663Abstract
Lung cancer is the leading cause of cancerrelated deaths worldwide. Lack of early detection and limited options for targeted therapies are both contributing factors to the dismal statistics observed in lung cancer. Thus, advances in both of these areas are likely to lead to improved outcomes. MicroRNAs (miRs or miRNAs) represent a class of noncoding RNAs that have the capacity for gene regulation and may serve as both diagnostic and prognostic biomarkers in lung cancer. Abnormal expression patterns for several miRNAs have been identified in lung cancers. Specifically, let7 and miR9 are deregulated in both lung cancers and other solid malignancies. In this paper, we construct a mathematical model that integrates let7 and miR9 expression into a signaling pathway to generate an in silico model for the process of epithelial mesenchymal transition (EMT). Simulations of the model demonstrate that EGFR and Ras mutations in nonsmall cell lung cancers (NSCLC), which lead to the process of EMT, result in miR9 upregulation and let7 suppression, and this process is somewhat robust against random input into miR9 and more strongly robust against random input into let7. We elected to validate our model in vitro by testing the effects of EGFR inhibition on downstream MYC, miR9 and let7a expression. Interestingly, in an EGFR mutated lung cancer cell line, treatment with an EGFR inhibitor (Gefitinib) resulted in a concentration specific reduction in cMYC and miR9 expression while not changing let7a expression. Our mathematical model explains the signaling link among EGFR, MYC, and miR9, but not let7. However, very little is presently known about factors that regulate let7. It is quite possible that when such regulating factors become known and integrated into our model, they will further support our mathematical model.