Modeling Cellular Uptake, Proliferation, and Death Gillian Baker and - - PowerPoint PPT Presentation

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

1

Literature Review

• utlining past research on the interactions between nanoparticles and

cells in the context of chemotherapeutic treatments

2

Modeling Methods and Creation Project Goals Mathematical Model Types Results

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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.

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The Hill Equation

Used to find the fraction of cell receptor(s) bound to a ligand (notritious molecules) as a function of ligand concentration θ = [L]n [Kd]n + [L]n θ : The fraction of the receptor protein bound to a ligand [L] : The free ligand concentration Kd : The dissociation constant n : The Hill coefficient depicting cooperativity

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The Hill Coefficient, n

The Hill coefficient indicates positively cooperative binding: nH =

log(81) log(EC90/EC10) = occupied receptors total receptors

where the constants EC90 and EC10 are elaborated upon in the forthcoming slide.

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Finding EC90 and EC10

Finding EC90 and EC10 the 90% and 10% maximal responses was determined by the number of nanoparticles in the substrate: ECF =

• F

100−F

• √

H

× EC50 EC90 = 5.4 × 1013 EC10 = 6.7 × 1011 where F is the response rate for which the equation is solved (90 or 10) and where H is the Hill slope of 1.

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Finding Kd

Based on the literature review Goutelle et al. where the reaction: L + M ⇋ LM has the equilibrium constant Kd = [12×1012][8×104]

[.57(8×104)]

= 2.1 × 1013 Where [LM] is the number of cells multiplied by the average percentage

• f 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.

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Finding the Hill Equation

The final form of the Hill Equation:

[L]n Kd+[L]n = [L] (2.1×1013)+[L]

where [L] is the number of nanoparticles.

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Definitions

k+: transport rate constant per receptor (cm3/s) (k+)cell: transport rate constant per cell (cm3/s) Kd: equilibrium disassociation constant (M) kf : association rate constant per receptor (cm3/s) (kf )cell: association rate constant per cell (cm3/s) kr: rate constant for disassociation of receptor/ligand complexes (1/min) kon: intrinsic association rate per constant per receptor (cm3/s) (or the rate at which ligands are bounded to a receptor) koff : intrinsic disassociation rate per constant per receptor (cm3/s) R: number of free receptors/sites (number/cell) D: diffusion coefficient (cm2/s)(or the sum of the ligand diffusivities) L: ligand concentration (M) Lo: bulk ligand concentration (M) a: radius of the cell (µm) s: encounter radius (nm)

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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. Original: A + B

kr

⇋

kf

AB A + B

k+

⇋

k− AB...B k1

⇋

k−1 AB

kf =

k+k1 k1+k−1

kr = k−1k

k1+k

Unlike the previous model, kf and kr are functions.

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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

• n the cell itself.

Comparing the values of kf to those of solution-only interactions: The rate constant for ligand binding for the whole cell is: (kf )cell =

(k+)cellR(kon) (k+)cell+R(kon)

Per receptor, divided by the number of receptors: (kf )cell =

(k+)cell(kon) (k+)cell+(kon)

and (kr)cell =

(k+)cell(koff ) (k+)cell+(koff )

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Finding k+, kon

Assembling the portions of γ included the equations: (k+)cell = 4πDa where D is the diffusion coefficient experimentally measured to be within the range (1.0 × 10−5 − 1.0 × 10−7cm2/s) and a is the radius of the cell, around 10µm kon: the experimentally measured association rate constant ranging from (1.0 × 10−10 − 1.0 × 10−13cm2/s) (kon)cell= Rkon, where R is the number of free receptor site

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The New Capture Probability

Revised from:

kon kon+k+ for a solution-only interaction to become:

γcell =

Rkon (k+)cell+Rkon

which depicts the capture probability for the whole cell. Where: R is the number of free receptor sites kon is the rate at which ligands are bound to a receptor k+ is the transport rate constant per receptor

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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: γcell

• [L]n

Kd+[L]n

• =

Rkon (k+)cell+Rkon

• [L]n

Kd+[L]n

• Gillian Baker and Angel Carrillo (KSU)

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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: V (t) = V0e

α β (1−e−βt)

where Vmax = V0e( α

β )

V(t): the volume of the tumor at time t V0: the initial volume of the tumor,treated as one cell and equal to V(0) α, β: fitting parameters for the graphed data

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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: V (t) = V0e

α β − V0e α β (1−βt) + V0 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:

dV (t) dt

= αV0e

α β (1−βt)

to a discrete equation:

αkV0e−αT0 t

where t = kT0 and V0 = 1, representing a ”tumor” of initially one cell. T0 is the discrete time interval (1,2,3...,24) in hours and k=(1,2,3,...)

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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

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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)T0) = αkV0e−αTt+1

Tt+1

− αkV0e−αTt

Tt

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.

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Spherical Coordinates and Parameters for Cell Death

Death affects the overall volume of the cell, which is displayed in spherical coordinates as a function of the radius:

1 r2 ∂ ∂r (r2 dσ dr ) − ΓH(r − rnec) = 0

Where: r is the radius of the cell σ is the distribution of the nutrient rnec is the necrotic radius of the cell Γ is the rate at which proliferating cells consume nutrients

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Heaviside and Proliferation

Since the necrotic core has no proliferating cells, there is no nutrient consumption in the region, and H(r − rnec) is the Heaviside function: H(x)=

• 1

x > 0 x ≤ 0 representing the differences in nutrient uptake. Proliferation is bounded between: rnec < r < R

• r the necrotic radius and the outer radius of the cell.

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Critical Value of R*

If σ(0, t) = σnec necrosis begins when the tumor’s radius surpasses a value R∗ which is defined as: R∗ =

• 6(σ∞−σnec)

Γ

Where: σ∞ is the constant nutrient concentration on exterior portion of the tumor Since σ∞ is continuous on the tumor boundary, it ensures the continuity of σ and ∂σ

∂r

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Change in Necrotic Radius

The change in the radius of the necrotic region is:

dR dt = s 15R∗ (3σ∞ − 5

σ + 2σnec) Where: σ and s are positive constants σ∞ is the constant nutrient concentration on the exterior of the tumor σnec is the nutrient distribution in the necrotic region

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Probability of the Necrotic Area Changing

dR dt = s 15R∗ (3σ∞ − 5

σ + 2σnec)

R

′kon

(k+)cell+R′kon

Where R

′ is the number of surface receptors and is changed from the

• riginal ”capture probability” equation:

γ =

Rkon (k+)cell+Rkon

The change in the area of the necrotic radius multiplied by the probability that the cell and ligand will successfully bind (thus inciting necrosis in a toxic-coated ligand) yields the rate at which individual cells will become necrotic.

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NetLogo Model: Cell Binding Rate

The cells were given individual attributes in the form of equations representing a Hill binding rate θ =

[L]n [Kd]n+[L]n , the probability of

successfully forming bonds with toxic ligands

Rkon (k+)cell+Rkon , and the

number of surrounding nanoparticles. Ranges for constants such as D and kon were retrieved from literature and included in the interface. If a range was included, the program was designed to assign the cells a random value within the span of available values.

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NetLogo Model: Cell Proliferation Rate

Cellular proliferation rates were meant to mimic actions of the cancer cells that were not killed by the chemotherapy, as around 28% of the cells are unaffected. Proliferation commands were set to be a global variable, and cells were commanded to act according to: V ((k + 1)T0) = αkV0e(−αTt+1)

Tt+1

− αkV0e−αTt

Tt

Similar to the case of cell binding rates, any fitting parameters were set in the interface and randomly chosen between a range of appropriate values.

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NetLogo Model: Cell Necrosis

The cell death rates of:

s∗ 15R∗ (3σ∞ − 5

σ + 2σnec)

R

′kon

(k+)cell+R′kon

were assigned to 72% of the cells allotted to die. The change in the necrotic portion of the tumors radius is a function

• f its necrotic core and its uptake of outside nutrients multiplied by

the chance that a bioincompatible ligand will bind with each cell. The second portion of the equation links the binding to the cell death rates.

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NetLogo Model Output

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Results

The NetLogo program was found to accurately predict the positive correlation between the binding and death rates of the MCF-7 cells. As more of the toxic ligands bound to receptors, the higher their death rates became. Cellular proliferation increased as the number of cells increased, even though the Gompertz model was refined to accept decreasing values

• f the cell population.

However, when evaluated on the individual cellular level, the binding rate based on the Hill equation, the probability of a successful binding, and the probability of necrosis developing were miniscule, ranging from 10−5 to 10−14.

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Acknowledgements

We would like to thank the National Science Foundation and Kansas State University for making this research possible. We would also like to thank

• ur mentors Dr. Shi, Dr. Jaberi, and Dr. Shakeri and the SUMaR

program coordinators, Dr. Korten and Dr. Yetter for an excellent experience and a great summer!

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References I

Byrne, H M, and M A. J. Chaplain.“Necrosis and Apoptosis: Distinct Cell Loss Mechanisms in a Mathematical Model of Avascular Tumour Growth.” Computational and Mathematical Methods in Medicine. 1.3 (1998): 223-235. Print. Chesla, SE, P Selvaraj, and C Zhu. ”Measuring Two-Dimensional Receptor-Ligand Binding Kinetics by Micropipette.” Biophysical Journal. 75.3 (1998): 1553-72. Print. Gesztelyi, Rudolf, Judit Zsuga, Adam Kemeny-Beke, Balazs Varga, Bela Juhasz, and Arpad Tosaki. “The Hill Equation and the Origin of Quantitative Pharmacology.” Archive for History of Exact Sciences. 66.4 (2012): 427-438. Print.

Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 31 / 35

References II

Goutelle, Sylvain, Michel Maurin, Florent Rougier, Xavier Barbaut, Laurent Bourguignon, Michel Ducher, and Pascal Maire. “Review Article: the Hill Equation: a Review of Its Capabilities in Pharmacological Modelling.” Fundamental and Clinical Pharmacology. 22.6 (2008): 633-648. Print. Khosravi-Far, Roya. Programmed Cell Death: Part A. San Diego, Calif: Academic Press, 2008. Internet resource. Lauffenburger, Douglas A, and Jennifer J. Linderman. Receptors: Models for Binding, Trafficking, and Signaling. New York: Oxford University Press, 1993. Print. Marasini, Ramesh, Arunkumar Pitchaimani, Tuyen D. T. Nguyen, Jeffrey Comer, and Santosh Aryal. “Influence of Polyethylene Glycol Passivation

• n the Surface Plasmon Resonance Induced Photothermal Properties of

Gold Nanorods.” Nanoscale. (2018). Print.

Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 32 / 35

References III

Murphy, Hope, Hana Jaafari, and Hana M. Dobrovolny. “Differences in Predictions of ODE Models of Tumor Growth: A Cautionary Example” BMC Cancer 16 (2016): 163. PMC. Web. 1 July 2018. Nguyen, TDT, A Pitchaimani, C Ferrel, R Thakkar, and S Aryal. “Nano-confinement-driven Enhanced Magnetic Relaxivity of Spions for Targeted Tumor Bioimaging.” Nanoscale. 10.1 (2017): 284-294. Print. Olea, N, M Villalobos, MI Nuez, J Elvira, de A. J. M. Ruiz, and V

• Pedraza. “Evaluation of the Growth Rate of Mcf-7 Breast Cancer

Multicellular Spheroids Using Three Mathematical Models.” Cell

• Proliferation. 27.4 (1994): 213-23. Print.

Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 33 / 35

References IV

Pitchaimani, Arunkumar, Tuyen D. T. Nguyen, Mukund Koirala, Yuntao Zhang, and Santosh Aryal. “Impact of Cell Adhesion and Migration on Nanoparticle Uptake and Cellular Toxicity.”Toxicology in Vitro. 43 (2017): 29-39. Print. Pitchaimani, Arunkumar, Tuyen D. T. Nguyen, and Santosh Aryal. “Natural Killer Cell Membrane Infused Biomimetic Liposomes for Targeted Tumor Therapy.” Biomaterials. 160 (2018): 124-137. Print. Schutt, C, S Ibsen, E Zahavy, S Aryal, S Kuo, S Esener, M Berns, and S

• Esener. “Drug Delivery Nanoparticles with Locally Tunable Toxicity Made

Entirely from a Light-Activatable Prodrug of Doxorubicin.” Pharmaceutical Research. 34.10 (2017): 2025-2035. Print.

Gillian Baker and Angel Carrillo (KSU) Biomathematics Project May 29 - July 24, 2018 34 / 35

References V

Shoup, D, and A Szabo. “Role of Diffusion in Ligand Binding to Macromolecules and Cell-Bound Receptors.” Biophysical Journal. 40.1 (1982): 33-9. Print. Simbawa, Eman. “Mechanistic Model for Cancer Growth and Response to Chemotherapy.” Computational & Mathematical Methods in Medicine. (2017). Print. Xiao-Gang, Ruan, and Hu Ri-Cha. “Implementing Gompertz Model with a One-Dimensional Cellular Automaton.” 2 (2002): 1116. Print. Zinovyev, A, S Fourquet, L Tournier, L Calzone, and E Barillot. “Cell Death and Life in Cancer: Mathematical Modeling of Cell Fate Decisions.” Advances in Experimental Medicine and Biology. 736 (2012): 261-74. Print.

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