Nova_Dictyo_Report2

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS INNOVA

ALEX WHITE

1. Introduction

Modeling Biological organisms in silico is paramount to understanding biological sys-tems. Without the computer, computational analysis would be near impossible; thereforeit is beneficial to create and test new tools that can be used to analyze systems. Withthis philosophy in mind this department partnered with Oberlin to test the new modelingsoftware NOVA on an actual problem in computational biology. The goal is to use NOVAto model Chemotaxis is Dictyostelium discoideum.The hope was to use NOVA to bringmultiple levels of chemotaxis analysis together and simulate them simultaneously.

1.1. Intro to Nova. Nova was created by Richard Salter at Oberlin College as, ”a newmodeling application that seamlessly integrates the system dynamics, spacial, and agentbased paradigms.” (Nova website) Nova combines paradigms like Venism and Agentsheetsinto a single programming language. Venism allows us to model dynamical systems withflow charts. This method is easy to learn quickly, easy to write and helps to visualize theset up. Nova takes a similar interface and style and uses said stye to create a cell grid andagents. Agentsheets simulates discrete objects called agents; these agents can interact withone another, change states.

The advantage of a program like Nova is dynamical systems can be embedded into theagents. This allows us to describe each individual object with sets of ODEs or PDE’s andhave them interact, often a difficult task. These agents move in a universe dubbed the cellgrid. Another advantage is this cell grid can also contain dynamic systems, thus allowingthe user to program diffusion or a similar process into the background. These propertiesnaturally lend themselves to modeling biological systems.

1.2. Quick Background to Chemotaxis. Dictyostelium lives most of its life as a singleamoeba,hunting bacteria. However, when food gets scarce, they enter a new phase in theirlife cycle. The amoebae will emit a cyclic adenosine monophosphate (cAMP) signal.[1]Other Dictyostelium amoebae pick up this cAMP signal, and begin to secrete their owncAMP signal.[1][6] Dictyostelium use the surrounding cAMP concentration gradient todetermine the location of other nearby Dictyostelium[4]. The Amoebae will move towardsone another aggregating together. Once a clump forms they can enter the multicellularreproductive stage of their life cycle. [1] They form a spore stalk and shoot a new generationof Dictyostelium amoebae off to new hunting grounds.

1

2 ALEX WHITE

Even though this is a straight-foward process there are many questions to ask about thisaggregation process. , ”How do internal enzymes regulate cAMP production”, ”Why iscAMP excreted in pulses?” ”How do Dictyostelium Amoebae know which way the cAMPgradient is pointing?” There have been many proposed answers to these questions, yetrarely are these questions asked simultaneously. In 1998 M.T. Laub and W. F. Loomisproposed a 7 state ODE as a simplified model of the internal enzymes and their effect oncAMP production. Their model explains the periodic pulses of cAMP secretions that havebeen observed in chemotaxing Dictyostelium media. [5]There have been numerous modelsput forth explaining how Dictyostelium discoideum senses cAMP gradient direction. So ofthe more influential model as described by Wouter-Jan Rappel et al. and expounded onby Joseph Kimmel et al. and Bo Hu et al. , is the vector sum model.[6][?][3] This Modelproposes that each receptor on Dictyostelium’s surface acts like a vector whose magnitudeis the average time the receptor spends bound.[?][3] A sum of these vectors then points inthe estimated direction of the cAMP gradient.[6]

1.3. Where Nova Comes in. Before we begin, take care to note some unfortunate ter-minology in agent modeling. Strictly speaking ”cells” refer to boxes within the gird. Theyare not biological cells, those are referred to as agents. Care will be taking to refer to boxesin grids as ”grid cells” and dictyostelium amoeba as Dictyostelium agents or simply agents.

To really test Novas capabilities, we decided to answer these questions together. Tomake full use of Nova’s novelty we combined Loomis and Laubs models of cAMP secretionand Rappel, Kimmel, and Hu’s gradient sensing models. To tie it together we placed themin a grid were cAMP can diffuse freely.

The Nova program has three main components, Diffusion, Synthesis, and Movement.Diffusion refers simply to allowing the cAMP concentration to diffuse on the cell grid.Synthesis refers to the agents ability to create cAMP, and Movement is the agent’s abilityto sense the gradient direction and move in said direction. Diffusion is modeled with finitedifference. As with any finite difference model care must be taken to insure numericalstability.

To model Synthesis we followed a paper written by M.T. Laub and W.F. Loomis. In theirpaper A molecular Network That Produces Spontaneous Oscillations in Excitable Cells ofDictyostelium. they lay out a 7 state ODE that describes cAMP production. It relieson a Chemical pathway were cAMP stimulates receptors, that stimulates Adenyl cyclaseand other enzymes which in turn produce cAMP. It also stimulates the productions ofinhibitors. This interaction produces waves when a single cell is considered.

Gradient sensing is based off papers by Rappel and Hu.[6][3]. This assumes a singlereceptor is a stochastic isolated system. If the bound time is recorded and a vector sumacross the entire surface of the cell is taking the resulting vector points in the direction thecell will move.

With this in mind we set out to tie all these components together in a single model.

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 3

2. Methods

2.1. Diffusion. The Model means nothing if the signaling molecule cAMP cannot diffuse.We create a grid for the cAMP to diffuse on, and then proceed to write the code for a singlegrid cell. Nova will iterate each grid cell for us and create a cell grid. We define cAMP andhave cAMP flow from one grid cell to its neighboring grid cells, and simultaneously havecAMP flow from the neighbors to the grid cell. We multiply the differnce between to cellsby D/8 where here D = .5859[40µm]2[15sec]−1 is the diffusion constant of cAMP (notewe this comes from converting 2.5 ∗ 10−6cm2sec−1 into NOVA’s units[6]. We divide by 8because there are 8 neighbors(we handle cells on the edge differently.) . Nova will handlethe grid cell grid cell interactions for us. We now have a grid that allows cAMP to diffusearound.

Figure 1. cAMP diffusion on a Nova cell grid. The blue and green repre-sent higher concentrations while red and black represent lower concentra-tions

We must be cognizant of units; in this particular model 1µM is equivalent to 10,000 unitsin NOVA. We do this for two reasons. First Nova color codes each grid cell dependent onthe value the grid cell holds. At the start of writing the program this color code couldnot change. Thus this large number allows us to see changing concentration values on thecell grid. The ability to rescale the colors has been added, however, take care not to usetoo low of numbers where decimals will distinguish color. As of now, Nova will drop thedecimals before determining color.

However, just because each cell grid contains cAMP does not mean the grid containsenough information for a given agent( the biological cell) to determine which direction thegradient is pointing. Our grid cells are about 40µm by 40µm and a agent has a diameterof about 10µm. [6] A single grid cell does not contain the steepness or the gradient of thecAMP concentration. Also note that the grid cell contains the average concentration inthe grid cell. Agent’s are allowed to move freely and therefore they be located anywherein the grid cell.

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Therefore we must program the grid cells to figure out their local gradient. The easiestway to accomplish this is to use a vector sum. We consider each neighboring grid cell tohave a vector pointing in the corresponding direction of each neighboring grid cell. Weassume the cAMP concentration in that grid cell is the concentration in the center of thegrid cell. We multiply the vector by the concentration at the center of the grid cell. Thenwe take the sum of those 8 vectors.

Let XN = cN ~UN where N signifies to the north of our reference grid cell ~UN is thevectorpointing to the direction of the north (< 0, 1 >), and cN be the concentration in the gridcell to the north. Therefore XSE is the grid cell to the southeast, ~USE =<

√2,−√

2 >.

(1) Grad(~c) = (XN + XNE + ...+ XW + XNW )

Once we have this vector, we can find which direction the cAMP gradient is pointing,as well as its steepness. Within each grid cell we assume the gradient can be accuratelyrepresented by a linear approximation. Therefore we determine the concentration at anygiven point using this linear approximation. Given a given distance from the center of thegrid d and an displacement angle from the gradient direction θ, and our steepness is givenby ∆ and mean concentration given by c̄ we have:

(2) c = d∆ cos(θ) + c̄

where c is our concentration at any given point.We have now constructed the universe that our Dictyostelium agents will live in.

2.2. Gradient Sensing Motion. Next we must consider how will our dictyosteliumagents will move. First we create a submodel that will contain everything we need toknow about an individual dictyostelium agent. This submodel will call on its cell grid andgather all the information about its environment, Gradient direction θ, Gradient steep-ness ∆, and cAMP concentrationc. We then use a pre-made sub-model called Mover thathandles motion, and define a speed. From there we proceed to add the stochastic element.

Determining Direction in Dictyostelium discoideum can be simplified to drawing tworandom variables from two Gaussians. One variable is the x -component and the other they-component. We can use arctan to determine the noise the angle. However, we must takecare to set up these Gaussians correctly.

Let us start with a single receptor. We assume that each receptor is an isolated chemicalsystem, that is effected only by the concentration of the signaling molecule cyclic-AMP(cAMP). We assume that the receptor will not affect the concentration of the cAMP. (agross over simplification). The receptor switches to the bound state to the unbound state.The following equations govern the chemical system:

R+ L→ RLwith hazard k1[L](3)

RL→ R+ Lwith hazard k2(4)

k1 = 2.4 ∗ 10−4 converted from 24µM−1sec−1[6]

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 5

k2 = . converted from .7sec−1 [7]k1 has units of sec−1µ−1M and k2 has units of sec−1. [6]. [7]R is the receptor and L is the ligand (or signaling molecule cAMP) where the hazard

describes the rate at which the chemical equation occurs[4]. Because we are considering asingle receptor only one reaction happens at a time. We are concerned with how long anaverage receptor stays bound. The cell will use this information to determine the whichdirection has the highest concentration. The higher the concentration, the longer thereceptor is bound. To find the average bound time we consider a well mixed macroscopicsolution. We can set up the differential equations by looking at equations (3) and (4).[2]

dRL

dt= k1[R][L]− k2[RL](5)

R

dt= −k1[R][L] + k2[RL](6)

dL

dt= −k1[R][L] + k2[RL](7)

there is a few simplifacations that we can make. Because our Ligand concentrationnever changes when ever the receptors number changes, we can treat [L] as a ”constant”.Rather [L] is a measure of concentration so it will be convenient if we let c(θ) = [L]. In thefuture we will be considering concentration as a function of θ. This will make the followingequations make more sense Also note that because there is a total number of molecules,say N we can say [R]= N-[RL]. we know can reduce our system to the following:

(8)dRL

dt= k1c(θ)(N − [RL])− k2[RL]

solving for equilibrium yeilds

(9) [RL] =Nk1c(θ)

k1c(θ) + k2]

Remember, however, that we are dealing with a single receptor. We want to relate themean number of bound receptors to determine the mean time bound. therefor we canset N = 1. Note however that we only record the time spent bound during a sampleperiod τ = 15sec. [2] This is akin to stating if half of the receptors are bound then onereceptor over a period of 15 seconds would spend half this time bound. This attempts tomathematically model the polarization stage of Dictyostelium chemotaxis. Polarization isthe period before Dictyostelium extends a pseudopod extension to move; In other wordsthe time the cell is spending to determine direction[2]. It is important to note we assumethat the Dictyostelium amoeba does not polarize during motion. Therefore we multiplyequation (9) by the sample time τ , which gives us

(10) µt =τk1c(θ)

k1c(θ) + k2]

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This is the average time that the receptor spends bound; however we must be careful notto forget that at this scale random variation plays a huge role. Therefor we must calculatethe variance. Because this is a telegraphic process we can assume that the variance is givenby:

σt = µt(1− µt)(11)

σt = τ2 k1k2c(θ)(k1c(θ) + k2)2

Now we have an expressions for the mean time a receptor is bound and the correspondingvariance. The next step is to use this information to determine the direction the cell willmove. First we assume the cell is a two dimensional object (a artifact of Nova). Wethen assume receptors are evenly distributed around the cell and that the cell is a perfectcircle. Therefore the ith receptor has coordinates (r cos(θi), r sin(θi)). Next we assumethe gradient is in the direction of θ = 0 and is increasing in that direction. (In nova thegradient will always be converted to this angle). Then we can described the concentrationby:

(12) c(θ) = c̄+ δ cos(θ)

c̄ is the concentration at the center of the cell and δ = r ∗∆ were r is the radius and ∆is the gradient of the concentration. If we consider each receptor to have a correspondingunit vector:

(13) ~U =(

cos(θi)sin(θi)

)We consider µt and σt as magnitudes of the unit vector. This describes how each

receptor contributes information to the entire cell. Because we assume that the time areceptor spends bound is represented as a gaussian we take the sum of the means andvariances. so we must calculate the following sums:

~µ =m∑i=0

τk1(c̄+ δ cos(θi))

k2 + k1(c̄+ δ cos(θi)r

(cos(θi)sin(θi)

)(14)

~σ =m∑i=0

τ2k2k1(c̄+ δ cos(θi)

(k2 + k1(c̄δ + cos(θi)))2r2

(cos2(θi)sin2(θi)

)

we can simplify by reducing the equations to

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 7

~µ = τKr

m∑i=0

(c̄+ δ cos(θi))1 +K(c̄+ δ cos(θi)

(cos(θi)sin(θi)

)(15)

~σ = τ2Kr2m∑i=0

(c̄+ δ cos(θi)(1 +K(c̄+ δ cos(θi)))2

(cos2(θi)sin2(θi)

)Where K = k1

k2. Assuming the number of receptors is large enough we can estimate the

sums with an integral. However, we must remember to multiply by the number of termsin our sum, this corresponds to the number of receptors. let the number of receptors bem(m is still being fined tuned).

~µ = mτKr

∫ 2π

0

(c̄+ δ cos(θi))1 +K(c̄+ δ cos(θi)

(cos(θi)sin(θi)

)dθ(16)

~σ = mτ2Kr2

∫ 2π

0

(c̄+ δ cos(θi)(1 +K(c̄+ δ cos(θi)))2

(cos2(θi)sin2(θi)

)dθ

We can solve these equations analytically.

~µ =τmr

δKπ

((1+c̄K)√

((1+c̄K)2−(Kδ)2− 1

0

)(17)

~σ =τ2r2m

δ2K2π

(1+c̄K)(2+5c̄K+4(c̄K)2+(c̄K)3−3(δK)2−(δK)2(c̄K)√

((1+c̄K)2−(Kδ)23 + (2 + c̄K)

−(2+3c̄K+(c̄K)2+(δK)2√((1+c̄K)2−(Kδ)2

+ (2 + c̄K)

See Appendix for the reasoning behind the anylitical solutions.Now we draw two random variables. One for the y-coordinate and the other for the

x-coordinate. We use are newly calculated ~µ and ~σ as are parameters for our gaussians.We then take the arctan of the two variables and we now have the direction the cell willmove. However we assume the distance the cell will move is one cell radius.

2.3. Synthesis. Dictyostelium agents are more then just objects that move around in aenvironment. They can directly alter their environment. So far we have done nothingdifficult for agent sheets to handle by its self. To truly test Nova’s capabilities we willinsert a 7-state ODE into every single Agent.

Every Dictyostelium amoeba has the capabillity to produce its own cAMP signal[5][4].Researchers have noticed that Dictyostelium Amoebae do not release cAMP steadily, butrather in consistant pulses[5]. These oscillating cAMP ”waves” occur with a period any-where from 5 to 10 minutes[5]. To go about modeling this phenomina, we used the Loomis-Laub model[5]. It contains six different chemical species (seven when we consider extra-cellular and intracellular cAMP separately). The Loomis-Laub model, discribes these 7molecules interactions with a set of ODE’s .

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cAMP does not spontaneously oscillate however. When external cAMP (EcAMP) bindsto the receptor (CAR 1). There is a cascade of excitatory and inhibitory enzymes thatregulate cAMP synthesis. [5] When CAR1 is bound the MAP kinease ERK 2 s produced.The job of ERK 2 is to stimulate Adenylyl Cyclase (ACA).[5] ACA’s job is to turn AMPinto cAMP. The cAMP can either be excreted and become the signaling molecule to otherDictyostelium Amoebae. The job of internal cAMP (IcAMP) is to regulate the productionof itself. However it does not do this directly. IcAMP catalyzes the production of PKA.PKA inhibits the binding affenity of the CAR 1 receptor and ERK 2. This prevents EcAMPfrom stimulating CAR1. [5]

Figure 2. The above pathway shows the simplified chemical network thatregulates cAMP production and excretion. A cAMP pulse activates thereceptor CAR 1. CAR 1 excites ERK 2, which in turns stimulates ACA(adenyly cyclase). ACA produces cAMP, some which is excreted. The restof the cAMP stimulates PKA. PKA inhibits both CAR 1 and ERK 2 pro-duction. Also note REG A inhibits internal cAMP. Take care to note thatPDE (phosphodiesterase) is responsible for extracellular decay outside theameoba. This figure comes from M.T. Laub and W.F. Loomis’ 1998 paperA molecular Network That Produces Spontaneous Oscillations in ExcitableCells of Dictyostelium.[5]

Still there is one more player involved. If IcAMP did not degrade over time then theamoebae would eventually stop binding to the EcAMP. Dictyostelium produces the phos-phodiesterase REG A which degrades IcAMP. However REG A is inhibited by the presencesof ERK 2. [5]

With this in mind Loomis and Laub reduced the above chemical pathway to the set ofODE’s:

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 9

d[ACA]dt

= k1[CAR1]− k2[ACA][PKA]

d[PKA]dt

= k3[IcAMP ]− k4[PKA]

d[ERK2dt

= k5[CAR1]− k6[ERK2][PKA]

d[REGA]dt

= k7 − k8[REGA][ERK2]

d[IcAMP ]dt

= k9[ACA]− k10[REGA][IcAMP ]

d[EcAMP ]dt

= k11[ACA]− k12[EcAMP ]

d[CAR1]dt

= k13[EcAMP ]− k14[CAR1][PKA]

k1 2 min−1

k2 .9 min−1µM−1

k3 2.5 min−1

k4 1.5 min−1

k5 .6 min−1

k6 .8 min−1µM−1

k7 1 min−1µMk8 1.3 min−1µM−1

k9 .3 min−1

k10 .8 min−1µM−1

k11 .7 min−1

k12 4.9 min−1

k13 23 min−1

k14 4.5 min−1µM−1

Take care to note the above ODE’s are not the original set. Dr. Wouter-JanRappel hasimproved them, he changed the first equation from [ACA] to [ACA][PKA] as well as someof the constants. His set is more accurate and does not have a damping effect.[8]

These ODE’s have only been tested considering a single Dd. amoeba, and have ignoreddiffusion entirely.[5] They however do produce spontaneous ocilations from a variety ofstarting conditions. Using the 7 ODE’s has its advantages rather than using a sine wavewith a period of 5 minutes. Not only would a sine wave be a gross oversimplifcation, theODE’s requires no IF statement to initiate oscillations. If the cAMP is zero, then theoscillations do not occur. Furthermore it allows the Dd. Amoeba time to ”warm up” tocAMP production. When a cAMP signal reaches a Dd. amoeba there is no sudden releaseof cAMP, but rather a slow build up as internal chemicals are produced.

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We program this into NOVA using venism like conventions. We must take special carewith EcAMP (external cAMP) as it must be excreted out of the agent and the actual cAMPin the cell grid must enter the cell. We simply add an inflow to our cAMP variable in ourgrid and have it call the External cAMP, then we have a outflow model the degradation.Then the Agent uses the concentration of cAMP in the cell grid as its EcAMP value in theODEs.

3. Results and Discussion

”How does the Nova preform?” Considering all of the individual parts, fairly well. Be-cause Nova is still in Beta, it still has a few bugs, but beside for that it is remarkablyimpressive. As a quick recap, We have created a model of a living organism that: Movesaround in a universe that allows diffusion of the signaling molecule cAMP. Not only canthe amoebae determine which direction to move, it does so nosily. Even more impressive,each Dictyostelium agent produces its own signal, using a 7-state ODE as the rules tocAMP production. This program can run 250 agents on a standard p.c. laptop with ani5 processor. To put this in perspective, 250 independent 7-state ODE’s are interactingwith an environment with 1600 different grid cells. Each grid cell is running the diffusionof cAMP as well as calculating the local gradient. Most impressively it can run smoothly.With more computing power than a undergraduate’s school laptop, one could speed up thesimulation, and add more time steps or even add more cells. Still at the end of the day thedictyostelium agents find each-other and will form clumps.

Figure 3. From top left to bottom left, 2 iteration , 60 iterations, 120 iter-ations, 158 iterations, 244 iterations. These are the Dictyostelium Amoeba’sexcreting a cAMP signal (dark blue to white with white being the highestconcentration. purple signifies low level cAMP excretion, while green is highexcretion. notice the formation of clumps. (Note this is an example fromAugust 1, Improvements may be made in the future.)

As for the accuracy of the model, Qualitatively it looks semi-realistic. As for Quanti-tatively it does not fair too well. This is not a failure on the part of Nova however. Thesimulation is still being fine tuned, and is a slow process. The main problem is keepingtract of all of the constants involved. For example the Loomis-Laub model was much moresensitive to the parameters than the paper leads on. (Not the intital condisions, but ratherthe 1998 paper by Loomis and Laub had incorrect equations and constants.)[8] If we’renot careful the Loomis-Laub model becomes a damped oscillating system. The Stochastic

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 11

component is extremely sensitive to its constants, and if the constants are off the agentscan be come perfectly accurate, or become completely lost.

One of the biggest problems to quantitive accuracy is the Adhesion. When the agentsclump together and the cAMP production is waning, the local external cAMP concentra-tion will drop. This causes the concentration around the clump to be higher than theconcentration in the clump. The agents will then follow the cAMP wave outward and splitapart. This spontaneous explosion of aggregates doesn’t occur in nature, or at least in wildtypes. An effective model of adhesion is probably necessary to fix the exploding clumps.As of now, Adhesion is being simulated by a ad-hoc quick fix. If three agents get within acetian distance then their speed divided by the number of agents in a given distance. It isnot perfect but it works for now.

Figure 4. Pre and Post Explosion. The Amoeba’s will clump togetherthen once the cAMP excretion reaches a temporary low the Amoebae willfollow the cAMP wave away. With the adhesion quick fix this explosiondoes not occur.

There is a current bug within the simulation (not nova), were the conditions c̄ > δ fail.This occurs when the gradient is too steep. Thus if the cell is sufficiently close to the bordersome of it surface lies in negative cAMP molarity. This will cause the program to crash.This is usually caused by an early clump formation. This clump produces a huge spikein the cAMP concentration which has the tendency to cause the gradient sensing moduleto return a negitave number. This causes varience to return an imaginary number, andthus it fails. The crash has a characteristic blacking of the screen starting in the upperleft hand corner. The blackness will spread across the screen. (This occurs rarely in thecurrent version, but abnormal conditions may cause it to occur.)

Another bug(again not in Nova) is more of a nuisance. The if statement controlling thecolor of the agents will fail occasionally. To prevent a crash, Nova will revert the agentshape to a circle, and randomize the color.

However as the program is debugged and expand upon, many of these small quantizedinaccuracies will be fixed.

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3.1. Emergent Properties. From the basic properties above we can observe complexbehavior in the our simulation. The most apparent is the clumping of the Dictyosteliumagents. At the current iteration program it takes about 400 iterations u 100 Nova secondsor 25 minutes. On previous iterations of the program we were able to achieve streaming(the property for the Dictyostelium Amoebae to form branch like rivers of amoebae flowingtowards an aggregate center.) However with the current constants, the cells are either tooslow or to noisy to form streams.

Another interesting property comes from the phase of the dictyostelium agents cAMPexcretions. Nova has been programmed to change the Agent color based of how muchcAMP the agent is releasing. This allows us to see where in a given oscillation. In thecurrent model the amoebae’s excreting the most cAMP will move outward in a wave likemotion, almost like watching a colorful ripples on the agents.

Also the stochastic component follows the predicted behavior of Dictyostelium Amoebaeaccurately. At low and high concentrations of cAMP the amoebae are very noisy, howeveraround 25nM of cAMP there is a minimum of noise.[6][3] Biologically this happens for tworeasons; in low concentrations there is not enough cAMP signal to determine the directionfrom which the cAMP signal originates from. [6][3]. Counterintuitively, Too much cAMPcan cause Dictyostelium to revert to a random walk. This is because over-saturation,if the receptors are constantly bound, then no information can be extracted from thecAMP concentration gradient.[6][3] At low concentrations the accuracy of the Dictyosteliumamoebae is a random walk, as the concentration increases so does the accuracy, until acritical point and it begins to decrease again.

Figure 5. 10−10µM , 10−1µM and 1010µM . At low concentrations the ac-curacy of the Dictyostelium amoebae is a random walk, as the concentrationincreases so does the accuracy, until a critical point and it begins to decreaseagain.

3.2. Bugs. Because Nova is in Beta, there are bugs present. One of the more bothersomebugs is agent-grid cell interaction. For example, if two agents are in the came grid cell,then only one agent can interact with the grid. This is not much of an design flaw, butrather an small oversight.

Also if an agent makes a calculation that returns an error, then the agent relocatesitself to the upper left corner and sits. This is not so much of a bug, but rather a strange

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 13

behavior. Usually this helps with debugging, however, sometimes the error may cause theprogram to crash and the black wave will travel from upper left corner to lower right corner.

Also there is no ability to have agent-agent interaction, it is possible through the cellgrid, but rather difficult to remove the middle man. Where this becomes apparent is tryingto implement adhesion. It is not difficult to make agents aware of other agents location,but rather the agents can’t be grouped together, or affect each others internal constantsbased off their own internal constants.

The Grid cannot handle negative numbers and decimals. It is capable of storing thenegative numbers, however it can not assign colors to grid cells with a negative numbers.The same with decimals; when assigning colors it drops the decimals then assigns the color.If negative it defaults to black.

3.3. Where To next. The next phase of this project (aside from fine tuning it) is to testif the model can predict biological phenomena. the most logical starting point would beto compare the model to actual Dictyostelium discoideum chemotaxis. From there we canmodify the program to model mutants.

It is also possible to compare the NOVA model to Concentration clamp experiments.These experiments are the Dd. Amoebae that are placed in linear gradients. We cancompare experimental chemotaxis Indices to predicted chemotaxis indices. Also with ananalytical expression for the noisy we can graph a plot of chemotaxis indices and compareit to experimental data.

4. Discussion

What has been accomplished with this Model; we have not explained anything novelbiological about Dictyostelium, yet! What has been accomplished with this model is agreat starting point for combining multiple phases of Dictyostelium chemotaxis. Nova hasset the groundwork for future work in biological modeling.

Despite the Quasi accuracy of the Nova Model there are endless directions to proceed.With fine tuning the Nova model can accurately model Dictyostelium aggregation , butwhat are the advantages to Nova over models by Laub and Loomis, Kessler and Levine,Hofferm, Sherratt and Maini, and Dallon, Dolton and Malani. Nova can combine morethan just one aspect of Dictyostelium Chemotaxis. We have shown that Nova is capable ofsimulating populations of Dictyostelium Amoebae each interacting with its environment.Nova has allowed us to combine models of Dictyostelium usually considered separate fromone another. We took the stochastic gradient sensing models of Rappel, Hu et al. andcombined it with a model of cAMP production put forth by Loomis and Laub. Thereis nothing stopping an ambitious programmer from added other components to the Novamodel. Perhaps introducing more Enzymes to the Loomis-Laub Model, or maybe adhesiondynamics.

However this goes beyond Dictyostelium discoideum research; we have show the utilityof Nova. The possibilities Nova brings to the table in mathematical biology our endless.Biological modelers has been torn trying to decide between dynamics and discrete forms ofmodeling. When choosing an appropriate model for Biological phenomena we the Biologists

14 ALEX WHITE

have had to pick between the agent based and the dynamic routes, unable to choose bothwithout resorting to copious amounts of programming. With Nova we can seamlesslyintegrate ODE’s PDE’s even SDE’s into our = agent based models. We can now modelmotion and communication between whole populations without having to simplify themechanics behind a cell’s behavior. We now can see how our ODE’s interact when we scalethe simulation up to a full population.

Advancing technology at the interface of agent and dynamic modeling is key if biologicalmodeling wants to advance. Differential equations can no longer describe Biological phe-nomena alone. Agent Models, don’t have the sophistication to capture an entire chemicalsystem. With programs in Nova, we can bridge this gap, unite the two, and even unravelnew mathematics.

5. Appendix

Below describes how to integrate ”Rational Polynomials of Cosines” and hopes to fills inthe giant leap between the integrals for mean and variance and there analytical solutions.we leave out the solutions for the y-coordninate. The mean y-coordinate comes from adirectly from a u−substitution with u = sin(θ) and goes directly to zero. We leave outthe variance for y-coordinate because it is the same method as x-coordinate variance.(Just let sin2(θ) = 1 − cos2(θ)). The goal here is to show how to solve to go aboutintegrating Polynomials of Cosines, not to show all the work that solved the mean andvarience intergrals.

First lets begin by defining what a ”Rational Polynomial of Cosine” is. if P1(χ) is somepolynomial say Aχ3 +Bχ2 +Cχ+D and χ = cos(θ) then a rational polynomial of cosinesis P1(χ)

P2(χ) . It is possible to solve any rational polynomial of cosines by following the stepsoutlined below:

(1) Synthetic division(2) Partial fractions(3) Integrate using Quotient Integration if necessary (this step can be delayed until

after 6)(4) Substitute cos(θ) = cos2(θ/2)− sin2(θ/2) and 1 = cos2(θ/2) + sin2(θ/2)(5) Multiply top and bottom by sec2(θ/2). (Note if step 3 was delayed we need to use

more sec2 n instead.)(6) Use u-substitution with u = tan(θ/2) and du = sec2(θ/2)(7) Integrate the remaining terms like an rational polynomial

Step 6 is a special case of integration of parts that is required for RPoC’s in the formA

(B+C cos(θ))n where n is an integer greater than 2. This is because partial fractions cannotsimplify these farther (unless complex partials are considered).

As an example let us work with the equation for the x-coordinate mean. so we have tointegrate

(18)∫ 2π

0

cos θ(c̄+ δ cos(θi))1 +K(c̄+ δ cos(θi)

dθ

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 15

We want to reduce this integral into a rational polynomial in terms of cos(θ). In otherwords, we want to make χ = cos(θ).we must stress here that χ is not a substitution,butrather we want to preform synthetic devision and partial fractions of the integrand. Wewill substitute cos(θ) back in for χ before integration is preformed.

so convert the intergrand into

(19)cos θ(c̄+ δ cos(θi))1 +K(c̄+ δ cos(θi)

=c̄χ+ δχ2

1 +Kc̄+Kδχ

preform synthetic devision and reduces to

(20)χ

K− 1K2δ

+1K2 δ + c̄

Kδ

1 +Kc̄+Kδχ

in this example we do not need to use partial fractions on the denominator, howeverwith variance we will need to use partial fractions to reduce the denominator further.

now we can substiute the cos(θ) back in for χ. We know have

(21)∫ 2π

0

cos(θ)K

− 1K2δ

+1K2 δ + c̄

Kδ

1 +Kc̄+Kδ cos(θ)dθ

the first two terms are trivial, however the last term requires a creative substution. Tosimplify let

A =1K2

δ +c̄

Kδ(22)

B = 1 +Kc̄(23)

C = Kδ(24)

This simplifies the problem to

(25)∫ 2π

0

A

B + C cos(θ)dθ

next we use the double angle formula to convert C cos(θ) = C cos2(θ/2) − C sin2(θ/2)and the Pythogertian identity B = B cos2(θ/2) +B sin2(θ/2) into the denominator to get

(26)∫ 2π

0

A

(B + C) cos2(θ/2) + (B − C) sin2(θ/2)dθ

multiply top and bottom by sec2(θ/2)

(27)∫ 2π

0

A sec2(θ/2)(B + C) + (B − C) tan2(θ/2)

dθ

note now u = tan(θ/2) and du = 12 sec2(θ/2) and we get

(28)∫ 2π

0

2A(B + C) + (B − C)u2

note how we now have the much more manigable form of the traditional derivitive oftan−1 but what if C > B. If we look at what C and B equaled it would mean Kδ > 1+Kc̄

16 ALEX WHITE

if c̄ < δ then the cell would have a negitive concentration along part of the cells surface.This is impossible so we can impose c̄ > δ allowing us to use ArcTan instead of ArcTanh.

so solving the intergral we get∫ 2π

0

2A(B−C)

(B+C)B−C + u2

du(29) ∫du

a2 + u2=

1a

tan−1(u

a)

∴2A

2(B − C)

√B − C tan−1(

√B−C tan(θ/2)√

B+C)

√B + C

then resubstiute in A,B, andC to get

(30)1K2

(−θδ

+2(1 +Kc̄) tan−1

[((1+barcK)2−Kδ)Tan[ θ2 ]√

(1+c̄K)2K2δ2

]δ√

(1 +Kc̄)2 − k2δ2+KSin[θ])

note that on our bounds 0− 2π sin(θ) = 0 and θδ = −2π

δ however

(31) F (θ) =2(1 +Kc̄) tan−1

[((1+barcK)2−Kδ)Tan[ θ2 ]√

(1+c̄K)2K2δ2

]δ√

(1 +Kc̄)2 − k2δ2

(call it F (θ) for connivence).that at 0 and 2π there tangent is both zero;however it does not mean that F (2π) = 0

because tan(π) = 0 (remember the half angle). The ArcTangent will actually produce πbecause one remember the dicontinuity at π. One must solve the integral from 0− π andπ − 2π and add them together, this gives the effect of reducing the arctan into pi to giveu our final answer of:

(32) 2π((1 + c̄K)√

((1 + c̄K)2 − (Kδ)2− 1)

We use a similar process for the variance. Here lets use the x-coordinate variance forthis example so we have:

(33)∫ 2π

0

cos2(θ)(c̄+ δ cos(θ))(1 +Kc̄+Kδ cos(θ))2

dθ

like before we let χ = cos(θ) not as a subsitution but to facilitaty synthetic division.Note C̄ = 1 +Kc̄

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 17

δχ3 + c̄χ2

δ2χ2 + 2δC̄χ+ C̄2

χ

δK2+Kc̄− 2C̄δ2K3

+(K3C̄2 − 2K2c̄C̄)δχ−Kc̄C̄2 + 2C̄3

K2δ2(δ2χ2 + 2δC̄χ+ C̄2)(34)

once again the first two terms are trivial to integrate, however the last term needs to bebroken up via partial fractions. We get:

(35) − (2Kc̄− 3C̄)C̄K3δ2(C̄ +Kδχ)

+C̄2(Kc̄− C̄)

K3δ2(C̄ +Kδχ)2

when we resubsititute χ = cos(θ) into are integral we get:

(36)∫ 2π

0

cos(θ)K2δ

+Kc̄− 2C̄K3δ2

+− (K2c̄− 3C̄)C̄K3δ2(C̄ +Kδ cos(θ))

+C̄2(Kc̄− C̄)

K3δ2(C̄ +Kδ cos(θ))2dθ

As before the first two terms are trivial to integrate and we have a formula for the thirdterm, however we have to work a little more with the last term. As before lets make fewersymbols, let:

A =C̄2(c̄− C̄)K3δ2

(37)

B = C̄

C = Kδ

giving us:

(38)∫ 2π

0

A

(B + C cos(θ))2dθ

* we can proceed down a different path from here (see 5.1)*Next we introduce Quotient integration. Let us consider the quotient rule for derivitives.

If we haved

dθ

f(θ)g(θ)

=g(θ)f ′(θ)− f(θ)g(θ)

g(θ)2(39)

So if we consider an integral similar in structure, lets let g(θ) = B + C cos(θ) andf(θ) = C sin(θ) then

d

dθ

C sin(θ)(B + C cos(θ))

=(B + C cos(θ))(C cos(θ))− (C sin(θ))(−C sin(θ))

(B + C cos(θ))2(40)

=BC cos(θ) + C2

(B + C cos(θ))2

18 ALEX WHITE

However if manipulate equation (38) into the following form we get:

A

C2 −B2

∫ 2π

0

A

(B + C cos(θ))2∗ C

2 −B2

A+

B

B + C cos(θ)− B

B + C cos(θ)dθ(41)

A

C2 −B2

∫ 2π

0

BC cos(θ) + C2

(B + C cos(θ))2− B

B + C cos(θ)dθ

note however that the term BB+C cos(θ) is in the same form as the integral from equation

(25) therefore we can integrate it using the method outline above. Lets focus on the firstterm however. We have

(42)A

C2 −B2

∫ 2π

0

BC cos(θ) + C2

(B + C cos(θ))2=

A

C2 −B2

C sin(θ)(B + C cos(θ))

= 0

because sin(0)and sin(2π) = 0so we have left

− A

C2 −B2

B

B + C cos(θ)=

C̄3(Kc̄− C̄)K3δ2(C̄ +Kδ cos(θ))(C̄2 −K2δ2)

(43)

∴∫ 2π

0

cos(θ)K2δ

+Kc̄− 2C̄K3δ2

+− (K2c̄− 3C̄)C̄K3δ2(C̄ +Kδ cos(θ))

− C̄3(Kc̄− C̄)K3δ2(C̄ +Kδ cos(θ))(C̄2 −K2δ2)

dθ∫ 2π

0

cos(θ)K2δ

+Kc̄− 2C̄K3δ2

+−(K2c̄− 3C̄)(C̄2 −K2δ2)C̄ − C̄3(Kc̄− C̄)K3δ2(C̄ +Kδ cos(θ))(C̄2 −K2δ2)

dθ

now lets integrate; once agian the first two terms are trival. For the third term we do asimilar style to the mean, we let:

A = (K2c̄− 3C̄)(C̄2 −K2δ2)C̄ − C̄3(Kc̄− C̄)(44)

B = K3δ2C̄(C̄2 −K2δ2)

C = K4δ3(C̄2 −K2δ2)

MODELING DICTYOSTELIUM DISCOIDEUM CHEMTOAXIS IN NOVA 19

we then have

∫ 2π

0

A

B + C cos(θ)dθ(45) ∫ 2π

0

A sec2(θ/2)sec2(θ/2) ∗ [(B + C) cos2(θ/2) + (B − C) sin2(θ/2)]

dθ

let u = tan θ/2 and du =sec2(θ/2)

2∫ 2π

0

2A(B + C) + (B − C)u2

du∫ 2π

0

2AB−C

B+CB−C + u2

du

2AB − C

√B − CB + C

tan−1(

√B − CB + C

tan(θ/2))

remember from the arguments from before tan−1 = π so we have

π2A

B − C

√B − CB + C

(46)

π2A√

B2 − C2

2π(K2c̄− 3C̄)(C̄2 −K2δ2)C̄ − C̄3(Kc̄− C̄)

K3δ2(C̄2 −K2δ2)√

(C̄2 −K2δ2)remember C̄ = 1 +Kc̄

2π−(3−Kc̄)(1 +Kc̄)((1 +Kc̄)2 −K2δ2) + (1 +Kc̄)3

K3δ2√

((1 +Kc̄)2 −K2δ2)3

2π(1 + c̄K)(2 + 5c̄K + 4(c̄K)2 + (c̄K)3 − 3(δK)2 − (δK)2(c̄K)

K3δ2√

((1 +Kc̄)2 −K2δ2)3

add the other two terms on

2π(1 + c̄K)(2 + 5c̄K + 4(c̄K)2 + (c̄K)3 − 3(δK)2 − (δK)2(c̄K)

K3δ2√

((1 +Kc̄)2 −K2δ2)3+ 2π

(2 + c̄K)K3δ2

+ 0

We can use a similar method to solve the variance for the y coordinate.remember C cos(θ) = C cos2(θ/2)− C sin2(θ/2) and B = B cos2(θ/2) +B sin2(θ/2)

20 ALEX WHITE

5.1. Alternate Method. A alternate way to proceed is to do substituteC cos(θ) = C cos2(θ/2)−C sin2(θ/2) and B = B cos2(θ/2) +B sin2(θ/2) in before Quotient Integration.

∫ 2π

0

A sec4(θ/2)((B + C) + (B − C) tan2(θ/2))2

(47)

∫ 2π

0

A sec2(θ/2)(1 + tan2(θ/2))((B + C) + (B − C) tan2(θ/2))2

u subsitiute∫A+Au2

((B + C) + (B − C)u2)2du∫

−2AC(B − C)((B + C) + (B − C)u2)2

+A

(B − C)((B + C) + (B − C)u2)du

rearrange to prepare for Quotient integration( note using complex partials here is another option)

A

B − C

∫ 2C(B−C)B+C u2

((B + C) + (B − C)u2)2+

1− 2CB+C

(B + C) + (B − C)u2du

letf(u) = u, f ′(u) = 1, g(u) = (B + C) + (B − C)u2, g′(u) = 2(B − C)u2

∴−AC

B2 − C2

∫−2(B − C)u2

((B + C) + (B − C)u2)2=−AC

B2 − C2(∫f(u)g′(u)g(u)2

du =f(u)g(u)

−∫f ′(u)g(u)

du)

−ACB2 − C2

u

(B + C) + (B − C)u2− −ACB2 − C2

∫1

(B + C)− (B + C)u2du

One can proceed from this point.

References

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cell biology.Volume 20.[5] Michael T. Laub and William F. Loomis. 1998. A Molecular Network That Produces Spontaneous

Oscillations in Excitable Cells of Dictyostelium. Molecular Biology of the cell. Volume 9.[6] Wouter-Jan Rappel, Peter Thoams, Herbert Levine, and William F. Loomis. 2002. emphEstablishing

Direction during Chemotaxis in Eukaryotic Cells. Biophysical Journal. Volume 83.[7] Peter j.M. Van Hasstert. 1984 The modulation of cell surface cAMP receptor from Dictyostelium dis-

coideum by ammonium sulfate. Bichimica et Biophysica Acta. Volume 85[8] biology.ucsd.edu/labs/loomis/network/laubloomis.html