Mathematical Modeling Mathematical Modeling of Cellular Behaviorof Cellular Behavior

Ken DupontKen Dupont

Graduate StudentGraduate Student

(Bio) Mechanical Engineering(Bio) Mechanical Engineering

Math 8803 – Discrete Mathematical BiologyMath 8803 – Discrete Mathematical Biology

Introduction – Tissue EngineeringIntroduction – Tissue Engineering Tissue engineering (TE)Tissue engineering (TE) aims to create, restore, and/or aims to create, restore, and/or

enhance function of biological tissues through a combination enhance function of biological tissues through a combination of engineering and biochemical techniques of engineering and biochemical techniques

Bone TE Aim: Bone TE Aim: regrow bone that has been lost due to causes regrow bone that has been lost due to causes such as trauma, congenital defect, or removal due to excision such as trauma, congenital defect, or removal due to excision of tumors of tumors

The basic method of TE is to implant a construct consisting of The basic method of TE is to implant a construct consisting of scaffold +/- cells +/- growth factorsscaffold +/- cells +/- growth factors

PLDL Scaffold, 4 mm D x 8 mm L

(R Guldberg, GA Tech)

Human mesenchymal stem cells (green) on PLDL scaffold (black struts), 20X

(K Dupont, GA Tech)

Cells can either be seeded onto scaffolds Cells can either be seeded onto scaffolds ex vivoex vivo (outside the body) prior to implantation or can be (outside the body) prior to implantation or can be enticed to infiltrate the scaffold enticed to infiltrate the scaffold in vivo in vivo (within the (within the body) body)

Stem cells can both differentiate into other cells Stem cells can both differentiate into other cells and continue to proliferate (divide); mesenchymal and continue to proliferate (divide); mesenchymal stem cells are adult stem cells found in marrow stem cells are adult stem cells found in marrow cavities of long bones that can become muscle, cavities of long bones that can become muscle, cartilage, or bone cellscartilage, or bone cells

Introduction – Tissue Engineering - CellsIntroduction – Tissue Engineering - Cells

Introduction – Tissue Engineering – ModelingIntroduction – Tissue Engineering – Modeling

Mathematical/Computational modeling Mathematical/Computational modeling

of cell dynamics has the potential to be of cell dynamics has the potential to be

a very useful tool in TEa very useful tool in TE

Advanced knowledge of the behavior of the Advanced knowledge of the behavior of the

cells on constructs could help to optimize TE cells on constructs could help to optimize TE

construct design and limit the number of construct design and limit the number of

expensive and time-consuming empirical expensive and time-consuming empirical

experimentsexperiments

Introduction – Processes in TE ConstructsIntroduction – Processes in TE Constructs

Sengers has listed the many of the events happening at the Sengers has listed the many of the events happening at the cellular level in TE constructs:cellular level in TE constructs:

Proliferation – cells divide during mitosisProliferation – cells divide during mitosis Senescence/Death – cessation of division and later death Senescence/Death – cessation of division and later death Motility – cells adhere to and move throughout their Motility – cells adhere to and move throughout their

environment due to a variety of guiding signals (taxis)environment due to a variety of guiding signals (taxis) Differentiation – stem cells turn into other cell typesDifferentiation – stem cells turn into other cell types Nutrient transport/utilization – nutrient concentrations Nutrient transport/utilization – nutrient concentrations

higher outside of constructs than inside, and cellular higher outside of constructs than inside, and cellular demands may varydemands may vary

Matrix changes - cells produce extracellular matrix proteins Matrix changes - cells produce extracellular matrix proteins (i.e. collagen) and degradation of matrix may occur as well(i.e. collagen) and degradation of matrix may occur as well

Cell-cell interactions – Cells can communicate with each Cell-cell interactions – Cells can communicate with each other (such as during contact inhibition)other (such as during contact inhibition)

NOTE – All of the processes can vary with space and time NOTE – All of the processes can vary with space and time

Processes - Cell MotilityProcesses - Cell Motility

A moving cell – note the ovular nucleus (Dickinson)

Modeling – Cell Motility – Random Walk BackgroundModeling – Cell Motility – Random Walk Background

Cell motion can be modeled as a random walkCell motion can be modeled as a random walk

• Recall the Bridges of Konigsberg/random walks on graphs from Recall the Bridges of Konigsberg/random walks on graphs from classclass

Random walk (RW) - stochastic process made up of a Random walk (RW) - stochastic process made up of a sequence of discrete steps of certain length(s). A random sequence of discrete steps of certain length(s). A random variable can determine the step length and/or walk direction variable can determine the step length and/or walk direction

A more formal description of a random walk is as follows: “LA more formal description of a random walk is as follows: “Let et XX((tt) define a trajectory that begins at position ) define a trajectory that begins at position XX(0) = (0) = XX00. A . A random walk is modeled by the following expression: random walk is modeled by the following expression: XX((tt + τ) + τ) = = XX((tt) + Φ(τ) , where Φ is the random variable that describes ) + Φ(τ) , where Φ is the random variable that describes the probabilistic rule for taking a subsequent step and τ is the probabilistic rule for taking a subsequent step and τ is the time interval between steps” (Wikipedia) the time interval between steps” (Wikipedia)

A random walk is an example of a Markov chain, A random walk is an example of a Markov chain, which is a “collection of random variables {Xwhich is a “collection of random variables {Xtt} } (where the index t runs through 0,1,…..) having (where the index t runs through 0,1,…..) having the property that, given the present, the future is the property that, given the present, the future is conditionally independent of the past” (Weisstein):conditionally independent of the past” (Weisstein):

Modeling – Cell Motility – RW BackgroundModeling – Cell Motility – RW Background

Modeling – Cell Motility - 1D RWModeling – Cell Motility - 1D RW

Endothelial cell Endothelial cell taking a 1D random taking a 1D random walk (Jones)walk (Jones)

Paths taken for Paths taken for eight separate eight separate random walks in random walks in 1D originating at 1D originating at the origin and the origin and taking 100 steps taking 100 steps (Wikipedia)(Wikipedia)

Modeling – Cell Motility – RW LatticesModeling – Cell Motility – RW Lattices

The paths allowed during a The paths allowed during a random walk can be restricted to random walk can be restricted to the space of a point latticethe space of a point lattice

A lattice is a set of connected A lattice is a set of connected horizontal and vertical [for 2D+] horizontal and vertical [for 2D+] line segments, each passing line segments, each passing between adjacent lattice points between adjacent lattice points [which are regularly spaced] [which are regularly spaced]

A lattice path is therefore a A lattice path is therefore a sequence of points P0, P1, …Pn sequence of points P0, P1, …Pn with n > 0, such that each Pi is a with n > 0, such that each Pi is a lattice point and Pi +1 is obtained lattice point and Pi +1 is obtained by offsetting one unit east (or by offsetting one unit east (or west) or one unit north (or south) west) or one unit north (or south) (Weisstein) (Weisstein)

Path created during 2D Path created during 2D walk on a point lattice walk on a point lattice (lattice not shown) (lattice not shown) (Weisstein)(Weisstein)

Modeling – Cell Motility – RW LatticesModeling – Cell Motility – RW Lattices

Rat mesenchymal stem cells on a 2D cell culture dish with nuclei stained by Hoechst dye (K Dupont, GA Tech)

Point lattice unit cells Point lattice unit cells are generally in the shape are generally in the shape of squares, such that the of squares, such that the point lattices are point lattices are sometimes referred to as sometimes referred to as grids or meshesgrids or meshes

Square lattices helps to quare lattices helps to minimize memory use and minimize memory use and computation times computation times

Cells are far from squares or points, but their position in the mesh can be represented by the location of the cell’s nucleus

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

Endothelial cells forming Endothelial cells forming monolayer on blood vessel walls monolayer on blood vessel walls ≈ 2D surface≈ 2D surface

A moving cell will usually stop for A moving cell will usually stop for a period of time before continuing a period of time before continuing on its walk, or it may divide, on its walk, or it may divide, followed by walks of both followed by walks of both daughter cells daughter cells

As the number of cells fills up the As the number of cells fills up the

surface contact inhibition will surface contact inhibition will dominate the process and the dominate the process and the cells will no longer move or cells will no longer move or proliferate proliferate

Lee tracked individual EC motion Lee tracked individual EC motion experimentally in 2D - average experimentally in 2D - average cell speed, duration of time cell speed, duration of time remaining stationary, and remaining stationary, and average direction changes were average direction changes were determined for use as parameters determined for use as parameters in simulations in simulations

Confluent monolayer of ECs on tissue culture well (Lee)

Cell paths over 36 hours (Lee)

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

2D lattice of square computational sites 2D lattice of square computational sites • Each Each site ≈ size of a cell (28 micron sides) site ≈ size of a cell (28 micron sides) • each site has a finite # of possible states and 8 each site has a finite # of possible states and 8

nearest neighborsnearest neighbors• The size of the total grid was made to simulate the The size of the total grid was made to simulate the

size of one well of a 96-well size of one well of a 96-well in vitroin vitro cell culture plate cell culture plate with diameter of seven millimeterswith diameter of seven millimeters (Jones)

Lee used a 2D discrete cellular automaton model of the Lee used a 2D discrete cellular automaton model of the proliferation dynamics of populations of migrating cellsproliferation dynamics of populations of migrating cells

Assumed steady state nutrient concentrations and neglected cell Assumed steady state nutrient concentrations and neglected cell lossloss

These “discrete systems provide an alternative approach to These “discrete systems provide an alternative approach to continuous models that use ordinary and partial differential continuous models that use ordinary and partial differential equation to describe the dynamics of systems evolving in space equation to describe the dynamics of systems evolving in space and time” (Lee) and time” (Lee)

Discrete models can be used to describe movements of Discrete models can be used to describe movements of individual cells rather than looking at entire populations of cellsindividual cells rather than looking at entire populations of cells

At each time point a lattice site, automaton i, is in a certain state At each time point a lattice site, automaton i, is in a certain state xxi i (x(xi i = 0 means no cell present)= 0 means no cell present)

If a cell is present, xIf a cell is present, xii needs to specify if the cell is moving, the needs to specify if the cell is moving, the direction of locomotion, and the time remaining until a change of direction of locomotion, and the time remaining until a change of directiondirection

Time is viewed as discrete steps with uniform increments Δt Time is viewed as discrete steps with uniform increments Δt

The state xThe state xii of any automaton takes values from the set of 4-digit of any automaton takes values from the set of 4-digit integer numbers integer numbers klmnklmn• kk is the direction that the cell is moving in; is the direction that the cell is moving in; kk can take any can take any

value from the set {0,1,2,…8}, with 0 value from the set {0,1,2,…8}, with 0 no motion, 1 no motion, 1 motion east, 2 motion east, 2 motion northeast, etc.. motion northeast, etc..

• ll is the persistence counter that tells how much time is left is the persistence counter that tells how much time is left until the next change of direction (tuntil the next change of direction (tcc = = ll * Δt) * Δt)

• mnmn is the cell phase counter, which tells the amount of time is the cell phase counter, which tells the amount of time left until the next cell division (tleft until the next cell division (trr = (10 = (10mm + + nn) * Δt) ) * Δt)

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

Initial cell direction Initial cell direction kk assigned randomly assigned randomly

Experimental measurements of the cell trajectories were then Experimental measurements of the cell trajectories were then used to assign initial values of used to assign initial values of ll

• The value of the counter decreasing by one after each iteration, with The value of the counter decreasing by one after each iteration, with the cell direction changing when the counter reaches zero the cell direction changing when the counter reaches zero

• The experimental data showed that cells generally change directions in The experimental data showed that cells generally change directions in a gradual fashion, so transition probabilities of a cell making a large a gradual fashion, so transition probabilities of a cell making a large angle change in direction are small angle change in direction are small

mnmn is assigned to each cell, again using the distribution obtained is assigned to each cell, again using the distribution obtained from experimental observations of real cell cycles from experimental observations of real cell cycles

• 64% of cells divided after 12-18 h passed, 32% after 18-24 h passed, 64% of cells divided after 12-18 h passed, 32% after 18-24 h passed, and 4% after 24 -30 h passed and 4% after 24 -30 h passed

• mn mn also decreases by one with each iteration and the cell divides when also decreases by one with each iteration and the cell divides when it reaches zero it reaches zero

ll and and mnmn are reset after each direction change and division, are reset after each direction change and division, respectivelyrespectively

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

Example: Example:

• Assume a 2D square lattice with N x N sites, with time step Δt Assume a 2D square lattice with N x N sites, with time step Δt = 0.5 hours = 0.5 hours

• Choosing an arbitrary automaton site i gives a value of xChoosing an arbitrary automaton site i gives a value of x ii = = 3319 at t3319 at too

• This means that the site contains a cell moving north for three This means that the site contains a cell moving north for three more iterations (1.5 h) and that the cell will divide after 19 more iterations (1.5 h) and that the cell will divide after 19 iterations (9.5 h) iterations (9.5 h)

• At time tAt time too + Δt, the cell will have moved to site i + N, located + Δt, the cell will have moved to site i + N, located one site north of site i, and the value of xone site north of site i, and the value of x ii + N = 3218 + N = 3218

• The value of xThe value of xii will then be equal to zero unless another cell will then be equal to zero unless another cell moves into the sitemoves into the site

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

Each simulation run of the model starts by randomly distributing Each simulation run of the model starts by randomly distributing cells at varying densities throughout the 2D space cells at varying densities throughout the 2D space

An algorithm is then begun to increment cell activity at each site An algorithm is then begun to increment cell activity at each site with the motion of a cell stopping when it no longer has a free site with the motion of a cell stopping when it no longer has a free site in which to move in which to move

If a cell tries to move into an occupied site during one iteration, it If a cell tries to move into an occupied site during one iteration, it will stay in its current location until the next iteration will stay in its current location until the next iteration

If a cell divides during one iteration it will not move, and one If a cell divides during one iteration it will not move, and one daughter cell will remain in the current site and the other will be daughter cell will remain in the current site and the other will be randomly assigned to one of the neighbor sitesrandomly assigned to one of the neighbor sites

The rows and columns are scanned randomly for incrementation The rows and columns are scanned randomly for incrementation during each iteration to prevent artifacts due to scanning sites in during each iteration to prevent artifacts due to scanning sites in one repeated order one repeated order

CPU time per run lasts between 50-200 seconds on an IBMRS/6000 CPU time per run lasts between 50-200 seconds on an IBMRS/6000 POWERStation 350 computer, with time varying based on grid POWERStation 350 computer, with time varying based on grid size, initial density of cells, and spatial distribution of cells size, initial density of cells, and spatial distribution of cells

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

RESULTS:RESULTS:

• Confluence reached faster when (nonmotile) cells were Confluence reached faster when (nonmotile) cells were seeded at higher densities (left)seeded at higher densities (left)

• Increasing cell speed (S) decreases time to confluenceIncreasing cell speed (S) decreases time to confluence

Less of an effect for cells seeded at higher density (0.81%, right) Less of an effect for cells seeded at higher density (0.81%, right) than those seeded at lower density (0.081%, middle) than those seeded at lower density (0.081%, middle)

This behavior due to increased contact inhibition in the cells This behavior due to increased contact inhibition in the cells seeded at higher density seeded at higher density

(Lee)

Modeling – Cell Motility/Proliferation – 2D RW SimulationModeling – Cell Motility/Proliferation – 2D RW Simulation

RESULTS:RESULTS:

Lee’s model appears to Lee’s model appears to accurately predict 2D accurately predict 2D endothelial cell population endothelial cell population dynamics when compared dynamics when compared to actual experimental to actual experimental endothelial cell counts (n=3 endothelial cell counts (n=3 per time point) after per time point) after seeding at various initial seeding at various initial densitiesdensities

(Lee)

Modeling – Cell Motility/Proliferation – 3D RW SimulationModeling – Cell Motility/Proliferation – 3D RW Simulation

Cheng, from the same research group as Lee, investigated Cheng, from the same research group as Lee, investigated application of random walk model of cell motility in 3D application of random walk model of cell motility in 3D

Assumes highly porous scaffoldAssumes highly porous scaffold

• Allows unrestricted motionAllows unrestricted motion• A cell at one site can move to any of its 6 adjacent cubic faces A cell at one site can move to any of its 6 adjacent cubic faces

The algorithm for 3D motion is very similar to that of 2D The algorithm for 3D motion is very similar to that of 2D motion, again containing a migration index, cell division motion, again containing a migration index, cell division counter, direction persistence counter, waiting time, and counter, direction persistence counter, waiting time, and varying transition probabilities to determine the new varying transition probabilities to determine the new direction that a cell will move in after stopping, colliding, or direction that a cell will move in after stopping, colliding, or dividing dividing

One additional feature of the model is that it incorporates a One additional feature of the model is that it incorporates a waiting time that a cell will remain stationary after colliding waiting time that a cell will remain stationary after colliding with another cell, which accounts for the tendency of cells with another cell, which accounts for the tendency of cells to form clusters in 3Dto form clusters in 3D

Modeling – Cell Motility/Proliferation – 3D RW SimulationModeling – Cell Motility/Proliferation – 3D RW Simulation

Cell seeding in two modes is Cell seeding in two modes is considered:considered:

• Uniform cell seeding Uniform cell seeding throughout the 3D space throughout the 3D space

• ““Wound healing” seeding, Wound healing” seeding, with cells seeded along a with cells seeded along a edges of a cylindrical edges of a cylindrical “wound” portion of the “wound” portion of the entire 3D grid entire 3D grid

The simulation runs until the The simulation runs until the cell volume fraction, κ(t), cell volume fraction, κ(t), increases to the point that all increases to the point that all available sites are occupied available sites are occupied by cellsby cells

(Cheng)

Modeling – Cell Motility/Proliferation – 3D RW SimulationModeling – Cell Motility/Proliferation – 3D RW Simulation

A)

B) RESULTS

Uniform Seeding

“Wound” Seeding(Cheng)

Modeling – Cell Motility/Proliferation – 3D RW SimulationModeling – Cell Motility/Proliferation – 3D RW Simulation

Cheng’s model also allowed Cheng’s model also allowed study of the effects of study of the effects of chemotaxis on the amount chemotaxis on the amount of time to reach confluencyof time to reach confluency

Chemotaxis causes cells toChemotaxis causes cells to

• A) Migrate preferentially in A) Migrate preferentially in one direction over all one direction over all others (creating a others (creating a biased/reinforced random biased/reinforced random walk) (top figure)walk) (top figure)

• B) Proliferate B) Proliferate anisotropically (bottom anisotropically (bottom figure – note that only four figure – note that only four nearest neighbors are used nearest neighbors are used in this figure)in this figure)

P1 > P2 > P3 (Perez)

(Jones)

Modeling – Cell Motility/Proliferation – 3D RW SimulationModeling – Cell Motility/Proliferation – 3D RW Simulation

With chemotaxis, the time to With chemotaxis, the time to confluence drastically increased, confluence drastically increased, because most of the cells bunched because most of the cells bunched up near the end of the grid near the up near the end of the grid near the “attractant” and became contact “attractant” and became contact inhibitedinhibited

CHEMOTAXIS RESULTS

(Cheng)

ConclusionConclusion

The list of individual phenomena occurring during tissue repair is a long The list of individual phenomena occurring during tissue repair is a long one even without considering the specific spatial and temporal interactions one even without considering the specific spatial and temporal interactions between them between them

Currently, no model can completely describe the tissue growth process, Currently, no model can completely describe the tissue growth process, because there are still too many unknowns regarding the process itself because there are still too many unknowns regarding the process itself

Application of discrete models of cell behavior and treatment of cells as Application of discrete models of cell behavior and treatment of cells as individual stochastic objects can be advantageous compared to continuous individual stochastic objects can be advantageous compared to continuous models because the complex behavior of cells can be broken down into models because the complex behavior of cells can be broken down into constituent elementsconstituent elements

• In the words of Jones: “by modeling crucial steps as discrete processes, In the words of Jones: “by modeling crucial steps as discrete processes, it is then possible to develop individual areas independently of the rest it is then possible to develop individual areas independently of the rest of the model” of the model”

Caution must be used in applying models to living systems because Caution must be used in applying models to living systems because “theoretical understanding is required as a check on the great risk of error “theoretical understanding is required as a check on the great risk of error in software and to bridge the enormous gap between computational results in software and to bridge the enormous gap between computational results and insight or understanding” (Cohen) and insight or understanding” (Cohen)

Until more of the basic biology is known, as well as the math to represent Until more of the basic biology is known, as well as the math to represent that biology, models will serve as fair predictors for simplified cases of cell that biology, models will serve as fair predictors for simplified cases of cell dynamics and tissue growth dynamics and tissue growth

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