Year of Publication


Degree Name

Doctor of Philosophy (PhD)

Document Type





Computer Science

First Advisor

Dr. Jun Zhang


This thesis demonstrates the use of numerical simulation in predicting the behavior of proteins in a flow environment.

A novel convection-diffusion-reaction computational model is first introduced to simulate fibroblast growth factor (FGF-2) binding to its receptor (FGFR) on cell surfaces and regulated by heparan sulfate proteoglycan (HSPG) under flow in a bioreactor. The model includes three parts: (1) the flow of medium using incompressible Navier-Stokes equations; (2) the mass transport of FGF-2 using convection-diffusion equations; and (3) the cell surface binding using chemical kinetics. The model consists of a set of coupled nonlinear partial differential equations (PDEs) for flow and mass transport, and a set of coupled nonlinear ordinary differential equations (ODEs) for binding kinetics. To handle pulsatile flow, several assumptions are made including neglecting the entrance effects and an approximate analytical solution for axial velocity within the fibers is obtained. To solve the time-dependent mass transport PDEs, the second order implicit Euler method by finite volume discretization is used. The binding kinetics ODEs are stiff and solved by an ODE solver (CVODE) using Newton’s backward differencing formula. To obtain a reasonable accuracy of the biochemical reactions on cell surfaces, a uniform mesh is used. This basic model can be used to simulate any growth factor-receptor binding on cell surfaces on the wall of fibers in a bioreactor, simply by replacing binding kinetics ODEs.

Circulation is an important delivery method for natural and synthetic molecules, but microenvironment interactions, regulated by endothelial cells and critical to the molecule’s fate, are difficult to interpret using traditional approaches. Growth factor capture under flow is analyzed and predicted using computer modeling mentioned above and a three-dimensional experimental approach that includes pertinent circulation characteristics such as pulsatile flow, competing binding interactions, and limited bioavailability. An understanding of the controlling features of this process is desired. The experimental module consists of a bioreactor with synthetic endotheliallined hollow fibers under flow. The physical design of the system is incorporated into the model parameters. FGF-2 is used for both the experiments and simulations. The computational model is based on the flow and reactions within a single hollow fiber and is scaled linearly by the total number of fibers for comparison with experimental results. The model predicts, and experiments confirm, that removal of heparan sulfate (HS) from the system will result in a dramatic loss of binding by heparin-binding proteins, but not by proteins that do not bind heparin. The model further predicts a significant loss of bound protein at flow rates only slightly higher than average capillary flow rates, corroborated experimentally, suggesting that the probability of capture in a single pass at high flow rates is extremely low. Several other key parameters are investigated with the coupling between receptors and proteoglycans shown to have a critical impact on successful capture. The combined system offers opportunities to examine circulation capture in a straightforward quantitative manner that should prove advantageous for biological or drug delivery investigations.

For some complicated binding systems, where there are more growth factors or proteins with competing binding among them moving through hollow fibers of a bioreactor coupled with biochemical reactions on cell surfaces on the wall of fibers, a complex model is deduced from the basic model mentioned above. The fluid flow is also modeled by incompressible Navier-Stokes equations as mentioned in the basic model, the biochemical reactions in the fluid and on the cell surfaces are modeled by two distinctive sets of coupled nonlinear ordinary differential equations, and the mass transports of different growth factors or complexes are modeled separately by different sets of coupled nonlinear partial differential equations. To solve this computationally intensive system, parallel algorithms are devised, in which all the numerical computations are solved in parallel, including the discretization of mass transport equations and the linear system solver Stone’s Implicit Procedure (SIP). A parallel SIP solver is designed, in which pipeline technique is used for LU factorization and an overlapped Jacobi iteration technique is chosen for forward and backward substitutions. For solving binding equations ODEs in the fluid and on cell surfaces, a parallel scheme combined with a sequential CVODE solver is used. The simulation results are obtained to demonstrate the computational efficiency of the algorithms and further experiments need to be conducted to verify the predictions.