Modeling Cellular Uptake, Proliferation, and Death Gillian Baker and Angel Carrillo The Ohio State University, California State University of Los Angeles Mentored by Dr. Zhenzhen Shi, Dr. Heman Shakeri, and Dr. Majid Jaberi-Douraki Kansas State University SUMaR Program The Institute of Computational Comparative Medicine NSF Grant # 1659123 May 29 - July 24, 2018 Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 1 / 35
Research Overview Literature Review 1 outlining past research on the interactions between nanoparticles and cells in the context of chemotherapeutic treatments Modeling Methods and Creation 2 Project Goals Mathematical Model Types Results Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 2 / 35
Project Goals Mathematical modeling of the binding process between a nanoparticle and a cell. Modeling of proliferation and necrotic development. Incorporate mathematical equations to agent-based model and predict cellular development under nanoparticle exposure. Compare model results to experimental results. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 3 / 35
The Hill Equation Used to find the fraction of cell receptor(s) bound to a ligand (notritious molecules) as a function of ligand concentration [ L ] n θ = [ K d ] n + [ L ] n θ : The fraction of the receptor protein bound to a ligand [ L ] : The free ligand concentration K d : The dissociation constant n : The Hill coefficient depicting cooperativity Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 4 / 35
The Hill Coefficient, n The Hill coefficient indicates positively cooperative binding: log(81) log( EC 90 / EC 10 ) = occupied receptors n H = total receptors where the constants EC 90 and EC 10 are elaborated upon in the forthcoming slide. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 5 / 35
Finding EC 90 and EC 10 Finding EC 90 and EC 10 the 90% and 10% maximal responses was determined by the number of nanoparticles in the substrate: � � √ H F EC F = × EC 50 100 − F EC 90 = 5 . 4 × 10 13 EC 10 = 6 . 7 × 10 11 where F is the response rate for which the equation is solved (90 or 10) and where H is the Hill slope of 1. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 6 / 35
Finding K d Based on the literature review Goutelle et al. where the reaction: L + M ⇋ LM has the equilibrium constant K d = [12 × 10 12 ][8 × 10 4 ] = 2 . 1 × 10 13 [ . 57(8 × 10 4 )] Where [ LM ] is the number of cells multiplied by the average percentage of the uptake of raw nanoparticles, both sizes of 5nm (72% after 24 hours) 40nm (42% after 24 hours). This was instituted to mimic the reaction process that [ LM ] implies. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 7 / 35
Finding the Hill Equation The final form of the Hill Equation: [ L ] n [ L ] K d +[ L ] n = (2 . 1 × 10 13 )+[ L ] where [ L ] is the number of nanoparticles. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 8 / 35
Definitions k + : transport rate constant per receptor ( cm 3 / s ) ( k + ) cell : transport rate constant per cell ( cm 3 / s ) K d : equilibrium disassociation constant (M) k f : association rate constant per receptor ( cm 3 / s ) ( k f ) cell : association rate constant per cell ( cm 3 / s ) k r : rate constant for disassociation of receptor/ligand complexes (1/min) k on : intrinsic association rate per constant per receptor ( cm 3 / s ) (or the rate at which ligands are bounded to a receptor) k off : intrinsic disassociation rate per constant per receptor ( cm 3 / s ) R : number of free receptors/sites (number/cell) D : diffusion coefficient ( cm 2 / s )(or the sum of the ligand diffusivities) L : ligand concentration (M) L o : bulk ligand concentration (M) a : radius of the cell ( µ m) s : encounter radius (nm) Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 9 / 35
Development of Solution-only Shoup and Szabo (1982) developed a method of incorporating diffusion models into receptor-ligand binding. Their partially diffusion-controlled reaction assumption implies that not all ligands bind to the receptors. Not all surfaces are uniformly reactive, and the authors focused on uniting equations from reaction kinetics to probability of ligand-receptor binding. k r Original: A + B AB ⇋ k f k + k 1 k + k 1 k r = k − 1 k A + B k − AB ... B k − 1 AB k f = ⇋ ⇋ k 1 + k − 1 k 1 + k Unlike the previous model, k f and k r are functions. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 10 / 35
2D Cell Membrane Lauffenburger et al. (1993) expanded upon Shoup and Szabo (1982) for a spherical cell with radius a ; the model ignores the diffusion of the receptors on the cell itself. Comparing the values of k f to those of solution-only interactions: The rate constant for ligand binding for the whole cell is: ( k + ) cell R ( k on ) ( k f ) cell = ( k + ) cell + R ( k on ) Per receptor, divided by the number of receptors: ( k + ) cell ( k on ) ( k + ) cell ( k off ) ( k f ) cell = and ( k r ) cell = ( k + ) cell +( k on ) ( k + ) cell +( k off ) Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 11 / 35
Finding k + , k on Assembling the portions of γ included the equations: ( k + ) cell = 4 π Da where D is the diffusion coefficient experimentally measured to be within the range (1 . 0 × 10 − 5 − 1 . 0 × 10 − 7 cm 2 / s ) and a is the radius of the cell, around 10 µ m k on : the experimentally measured association rate constant ranging from (1 . 0 × 10 − 10 − 1 . 0 × 10 − 13 cm 2 / s ) ( k on ) cell = Rk on , where R is the number of free receptor site Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 12 / 35
The New Capture Probability k on Revised from: k on + k + for a solution-only interaction to become: Rk on γ cell = ( k + ) cell + Rk on which depicts the capture probability for the whole cell. Where: R is the number of free receptor sites k on is the rate at which ligands are bound to a receptor k + is the transport rate constant per receptor Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 13 / 35
Binding Rate To find the rate at which cells bind to a nanoparticle, the probability of a receptor forming a complex with a ligand ( γ ) was multiplied by the Hill equation to form: � � � � [ L ] n [ L ] n Rk on γ cell = K d +[ L ] n ( k + ) cell + Rk on K d +[ L ] n Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 14 / 35
The Gompertz Equation Created in 1825, it was originally meant to explain human mortality curves. It generalizes the logistics model with a sigmodial curve is asymmetrical and has one inflection point defined as: α β (1 − e − β t ) V ( t ) = V 0 e where V max = V 0 e ( α β ) V(t): the volume of the tumor at time t V 0 : the initial volume of the tumor,treated as one cell and equal to V(0) α, β : fitting parameters for the graphed data Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 15 / 35
Revised Gompertz The data from Pitchaimani et al.(2017) for cellular viability is decreasing, since cells were dying throughout the study when treated with nanoparticles. A new Gompertz model designed for negative growth was derived implemented: β (1 − β t ) + V 0 α α β − V 0 e V ( t ) = V 0 e Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 16 / 35
Discretizing the Gompertz From Xiao-Gang and Hu Ri-Cha, the Gompertz equation was transformed from its derivative: α dV ( t ) β (1 − β t ) = α V 0 e dt to a discrete equation: α kV 0 e − α T 0 t where t = kT 0 and V 0 = 1, representing a ”tumor” of initially one cell. T 0 is the discrete time interval (1,2,3...,24) in hours and k=(1,2,3,...) Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 17 / 35
Rate for Tumor Growth and Cell Proliferation An intuitive rate for cell growth is: number of cells at time B − number of cells at time A B − A where B is the t + 1 timestep and A is the t timestep for all t ∈ Z Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 18 / 35
Combining the Two Substituting the discretized Gompertz equation at t + 1 time in for the number of the cells at time B and the discretized Gompertz equation at t time in for the number of cells at time A yields: V (( k + 1) T 0 ) = α kV 0 e − α Tt +1 − α kV 0 e − α Tt T t +1 T t Inputting distinct values of time, such as hours, for t and allowing α and β to be parameters allows for the prediction of the MCF − 7 cells’ proliferation rates. Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 19 / 35
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