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Modeling

We created an advanced multiscale model of the ComCDE quorum sensing system and biofilm formation via WIG synthesis in S. mutans. Employing the capabilities of the modelling environment Morpheus to simulate cell migration and adhesion as well as the the diffusion of extracellular small molecules and proteins, we modelled the logarithmic phase of bacterial growth in two dimensions. Our goal in modelling was to determine the most important factors in the bacterial biofilm initiation system in order to predict the effectiveness of various methods of inhibiting biofilm growth.


Symbol	Meaning	Value	Units


k_tx	Transcription Rate of a Gene		transcripts per gene per second

k_tf	Transcription Factor Activity of Phosphorylated ComE	0.15	transcripts per ComEP per gene per second

δ_mCSP	Decay Rate of CSP mRNA		transcripts per second

δ_mComD	Decay Rate of ComD mRNA		transcripts per second

δ_mComE	Decay Rate of ComE mRNA		transcripts per second

δ_mGTFC	Decay Rate of mGTFC mRNA		transcripts per second

pComC	Number of ComC genes	1	genes

pComD	Number of ComD genes	1	genes

pComE	Number of ComE genes	1	genes

pGTFC	Number of GTFC genes	1	genes

α_CSP	Translation Rate of CSP		proteins per second

α_ComD	Translation Rate of ComD		proteins per second

α_ComE	Translation Rate of ComE		proteins per second

α_GTFC	Translation Rate of GTFC		proteins per second

δ_CSP	Decay Rate of CSP	∣∣	proteins per second

δ_ComE	Decay Rate of ComE	∣∣	proteins per second

δ_ComD	Decay Rate of ComD	1.5 × 10^-5	proteins per second

δ_GTFC	Decay Rate of GTFC	∣∣	proteins per second

δ_CSPComDP	Decay Rate of CSPComDP	1.5 × 10^-5	complexes per second

δ_ComEP	Decay Rate of Phosphorylated ComE		proteins per second

k_b	Binding Activity of CSP to ComD		complexes per CSP per ComD per second

k_ub	Unbinding Activity of CSP:Phosphorylated ComD Complex		dissasociations per CSPComDP per second

k_k	Kinase Activity of Phosphorylated ComD		phosphorylations per CSPComDP per ComE per second

k_e	Enzyme Activity of GTFC		glucans formed per GTFC per Sugar per second

ϕ_CSP	Export Rate of CSP from a Cell		¹∕_{CSP × second}

ϕ_Glucan	Export Rate of Glucans from a Cell		¹∕_{Glucan × second}

k_ab	Binding Activity of ScFv to GTFC		Complexes per ScFv per GTFC per second

δ_GTFCScFv	Decay Rate of GTFC:ScFv Complex	∣∣	complexes per second

δ_{ScFv_cell}	Decay Rate of ScFv	∣∣	proteins per second

k_kc	Binding Activity of Kappa-Casein	1	¹∕_{KCasein × Glucan × second}

δ_{KCasein_cell}	Decay Rate of Kappa-Casein	∣∣	proteins per second

2.1 Cell Energy and Migration

P (σ ′ → σ ) = { 1-i(fΔ ΔHH+Y)+ Y > 0 (1) x x e---T---otherwise

∑ H = λV(νσ - Vt)2 σ>0

(2)

∑ H = [λV (νσ - Vt)2 + λP (ρσ - Pt)2] σ>0

(3)

∑ H = J[τ(σi),τ(σj)](1- δσiσj) i,j

(4)

δσiσj = {1,σi = σj;0,σi ⁄= σj}

(5)

Morpheus is based on the Cellular Potts Model, which defines cells as spaces in a two or three dimensional lattice (our model used a hexagonal lattice in 2d) and determines changes to cell shape and size by using the Hamiltonian (Equations 2-4). Equation 2 determines changes to the volume of a cell σ with current volume vσ in lattice sites and intended volume V _t where λ_V is a constant parameter of elasticity governing the extent to which the difference between the cell’s immediate and intended volume contributes to a rise in its free energy H. Equation 3 incorporates a similar system for cell perimeter into the equation.

In addition to modelling adjustments to cells’ shape and size as they migrate, the Cellular Potts Model also uses the Hamiltonian to model the interactions between cells. Equation 4 determines the interaction energies between different cell types where τ(σ_i,σ_j) represents the cell types of two cells σ_i and σ_j and J specifies said energies in matrix form. In order to prevent cells from returning values for interactions with themselves, the term known as the Kronecker Delta defined in Equation 5 is included. Each update to the configuration of cells in the lattice as a result of equations 2-4 only occurs with a certain likelihood governed by the Boltzmann probability in Equation 1. This equation states that the cell’s chance of changing state is 100% if its change in energy ΔH added to its resistance to change Y is favorable (ΔH+Y < 0) and decreases exponentially with a rate of 1T-

where T represents the amount of unfavorable updates to the cell lattice, defined as modifications to the cell’s perimeter or volume.

2.2 Diffusion and the Cell Lattice Space

-∂c -∂2c ∂T = D ∂X2

(6)

X- T- x = L ,t = τ

(7)

∂c τD-∂2c- ∂t = L2 ∂x2

(8)

c(x,t)  ci,j ≡ c(xi,tj),xi ≡ iδx,tj ≡ jδt

(9)

∂c-|xi,tj ci+1,j --ci--1,j= ci+1,j---ci-1,j- ∂x xi - xi-1 2δx

(10)

∂c ∂c ∂2c-|  -∂x |xi+-12 --∂x |xi--12= ci+1,j---2ci,j-+ci-1,j ∂x2 xi xi+12 - xi- 12 (δx)2

(11)

c = c + -δt--(c - 2c + c ) i,j+1 i,j (δx)2 i+1,j i,j i-1,j

(12)

( 2 2 ) ∂-c+ ∂-c  ci+1,j --2ci,j-+-ci-1,j-+ ck+1,j --2ck,j +-ck-1,j ∂x2 ∂y2 (δx)2 (δy)2

(13)

ci,k,j+1 = ci,j +-δt2(ci+1,j - 2ci,j +ci-1,j)+ --δt2(ck+1,j - 2ck,j + ck-1,j) (δx) (δy)

(14)

4 ∑∞ sin[(n-+-1)θ]sin[(m-+-1)θ]--sin-(nθ)sin[(m---1)θ] h2(2s) = 3 [(n+ 12m )2 + 3(12m)2]s m,n=-∞

(15)

Another important component of our model was treating sugar, CSP, and our potential outputs not simply as cell-associated values but as scalar fields representing the concentration of said molecules. Fortunately Morpheus also incorporates the ability to overlay fields like these on simulated cell populations and allows the field to be updated locally based on cell activity at a specific lattice site (cells can also report on fields or other cells surrounding them, a function which is explained in the next section). Diffusion Fields were evaluated using the Central Difference Method to solve the 2-D Diffusion Equation (Equations 6-14). Equation 6 is a differential equation modelling diffusion in one dimension where the concentration c is a function of the X-coordinate and time T. D signifies the diffusion coefficient and L represents the length between nodes in (X, T) space where the domain of solutions is 0≤X≤L. By making the change of variables in Equation 7, the Diffusion Equation can be rewritten in the form of Equation 8.

Next, a set of approximations are made in order to solve for the first and second derivatives of concentration with respect to x (Equation 9). The central difference method is a version of the finite difference method of approximations to solve the Diffusion Equation, which assumes that there is a minimum distance in both dimensions, δx and δt, for which the concentration changes. This creates a grid in (x, t) space as shown in Figure . The central difference method approximates the x-derivative of concentration based on concentration values on either side of a point (i, j), one for each of the smallest change in x from the starting point: (i+1, j) and (i-1, j) (Equation 10). This is more precise than the forwards or backwards difference methods which only take into account one direction of incrementation in x. Next, the second derivative of concentration with respect to x is approximated using the values of the first derivative on a range half as large as in the previous approximation (Equation 11).

Using these results, an approximation of the concentration at a point 1 time step in the future can be obtained based on the concentrations at the surrounding points at the initial time (Equation 12), which can be appended with terms based on the approximations for the first and second derivatives in Equations 10 and 11 in order to obtain a more accurate approximation. Of course, in our model this was done in two dimensions, but the steps for the second dimension y are the same as those above and can be combined into Equations 13 and 14 in order to obtain values for a 2-D diffusion model.

In order to sense nearby concentrations or cells in Morpheus, each cell can take advantage of the NeighborhoodReporter function, which maps values within a specified node length from a cell to an average, variance, or sum specific to that cell. In our model, cells employed the sum function of the NeighborhoodReporter to interact with the various diffusion fields. Equation 15 is the equation for a lattice sum in a 2-dimensional hexagonal lattice, which Morpheus approximates to write field values to individual cells.

The first objective of accurately modelling a bacterial population was to model population growth over time. We chose to define our time step in the model as one second, and used the canonical doubling time of around 3800 seconds to determine a probability of cell division at each time step. Next, in order to reflect the constantly shifting environment of saliva on teeth, we programmed the cells with random motion defined within a realistic range for cell velocity in the salivary microbiome. This ensured that the cells would interact so that they would be affected by the modified adhesion associated with biofilm initiation.

3.1 Transcription and Translation of the Relevant Genes

dmCSP-- = ktx × pComC + ktf × ComEP × pComC - δmCSP × mCSP dt

(16)

dmComD---= ktx × pComD - δmComD × mComD dt

(17)

dmComE---= ktx × pComE - δmComE × mComE dt

(18)

dmGT--FC-= k × ComEP × pGTF C - δ × mGT FC dt tf mGTFC

(19)

dCSPin- = α × mCSP - δ × CSP dt CSP CSP in

(20)

dComD--= αComD ×mComD - δComD×ComD+kub ×CSP ComDP - kb×CSPout ×ComD dt

(21)

dComE--= αComE ×mComE - δComE ×ComE - kk×CSP ComDP ×ComE+ktf ×pComC ×ComEP +ktf×pGT F C×ComEP dt

(22)

dGT-FC- = αGTFC × mGT F C - δGTFC × GT FC dt

(23)

The biggest hurdle in creating an accurate model of biofilm formation in S. mutans was simulating the two-component signaling system known as ComCDE. We began by creating differential equations for the transcription and translation of the ComC, ComD, and ComE genes based on transcription rates, translation rates and decay rates for mRNAs and proteins from the literature. We also researched the genome size and operon structure of the bacteria in order to better reflect its production of the relevant proteins in our model. Equations 16-18 represent the transcription of mRNAs for the three genes we chose to study, and the first four terms in each of Equations 20-22 represent the translation of those mRNAs. All terms and constants are defined in the table at the beginning of this section. Now that we had the cells generating values for these variables at each time step, we could add terms to model the kinetics of the two-component sensing system itself.

3.2 Two-Component Sensing System: Ligand Binding, Response Regulator Phosphorylation and DNA Binding

dCSPout ---dt---= kub×CSP ComDP - kb×CSPout ×ComD+kk ×CSP ComDP ×ComE

(24)

dCSP-ComD-- dt = kb×CSPout×ComD - kub×CSP ComDP - kk×CSP ComDP ×ComE - δCSP ComDP ×CSP comDP

(25)

dComEP---= kk×CSP ComD ×ComEP - ktf×pComC ×ComEP - ktf×pGT FC ×ComEP - δComEP×ComEP dt

(26)

We used the Law of Mass Action to create differential equations reflecting the kinetics of ComD binding its ligand CSP, the autophosphorylation of ComD, the phosphorylation of ComE by ComD, and finally the DNA binding and transcription factor activity of phosphorylated ComE. For a full explanation of the way the two-component signalling system works in live bacteria, please refer to our project page, as this section assumes knowledge of the relevant components and their functions. The first step in the signalling system is the binding of CSP to ComD. In order to initiate this activity, we used the NeighborhoodReporter function to update the variable CSPout at each time step. ComD binds CSP at the rate kb, at which point the two are considered one complex, CSPComDP for autophosphorylated ComD:CSP, and the amount of CSP and ComD decrease accordingly (Equations 21 and 24). CSPComDP has an unbinding rate kub which decreases the amount of ComD:CSP complexes while increasing both CSPout and ComD by the same amount. In order to simplify the system of equations, the assumption was made that once CSPComDP phosphorylates its response regulator ComE, the complex dissociates and returns to separate ComD and CSPout. kk represents the rate of the complex phosphorylating ComE (Equation 25). Once ComE is phosphorylated, it is considered a different variable, ComEP for phosphorylated ComE. Phosphorylated ComE upregulates the ComC and Glycosyltransferase genes with the rate ktf in order to create a positive feedback loop within the population characteristic of quorum-sensing systems and produce the necessary enzymes for biofilm formation. Once again, the assumption was made that after ComEP binds DNA it is dephosphorylated by available phosphatases and returns to its initial state, hence the amount of ComEP decreases with the rate ktf for each gene it upregulates and ComE increases by the same amount (Equations 22 and 26).

3.3 Water-Insoluble Glucan Synthesis via Glucosyltransferase Activity

dGlucan ---dt---= ke × GT F C ×Sugarcell - ϕGlucan × Glucan

(27)

Biofilm formation in the model depends on two main factors: Sugar available to the cell and the amount of Glycosyltransferase enzymes a cell possesses (GTFC). mRNAs for GTFC (Equation 19) are only transcribed when ComEP binds the gene to upregulate it, and GTFC proteins are translated according to Equation 23. Sugar concentration (Sugar_field) is given an initial value as a diffusion field in the model, which was defined as a homogeneous diffusion field due to sugar’s high solubility in saliva. Sugar_cell is defined at each time step via the NeighborhoodReporter function based on the current concentration of the Sugar_cell.

The final differential equation based on the Law of Mass Action in the most basic form of the model, Equation 27, governs the production of water-insoluble glucans from available sugar. The enzymatic activity of GTFC is denoted ke, and the glucans become the basis of the extracellular biofilm by being incorporated into the extracellular field of the same name at the rate ϕ_Glucan. In order to simulate the transition of S. mutans from free-floating planktonic cells to biofilm-bound immobilized cells which grow in localized colonies, each cell was programmed to also use the NeighborhoodReporter to return a constantly updated value for how concentrated biofilm-forming glucans are around that cell. Once the local biofilm reaches a threshold value of 5 arbitrary units, cells are considered to be adhered and their movement ceases.

3.4 Extracellular Diffusion Fields

dCSP --dt-- = ϕCSP ×CSPin

(28)

dBiofilm ---------= ϕGlucan ×Glucan dt

(29)

dSugarfield ----dt----= - ke × GT FC × Sugarcell

(30)

Equations 28-30 are evaluated at each time step at each point in the lattice in order to affect the changes individual cells make to field values. Both the biofilm and CSP field equations increase at an export rate times the number of proteins a cell at a given point has, and the Sugar decreases as GTFC transforms it into glucans. Sugar undergoes homogeneous or well-mixed diffusion, CSP was given a diffusion constant of D = 1, and the Biofilm was given the diffusion constant D = 0.001, both in as employed in Equation 6 and its subsequent approximate solutions.

Our model proved quite successful in qualitatively reproducing multiple aspects of the formation of live S. mutans biofilms. The simulated cells form clear microcolonies in the locations where glucan concentration is the highest which expand from there. CSP increases everywhere throughout the simulation as the time steps modelled only encompass the phase of growth when that is the case.

4.1 Effect of Sugar Concentration on Biofilm Formation

The first parameter we varied was sugar by modifying the initial value of the diffusion field. After fifteen thousand time steps, there is a clear difference in biofilm density in plots of the cells themselves. Graphs of data collected from a representative cell in each condition show a clear distinction in glucan production between sugar concentrations. For the time period simulated, limiting sugar concentration was between 10 and 25 arbitrary units, at which point the cells were unable to generate enough glucans to adhere and create microcolonies in a biofilm. These results reflect our experiments with live S. mutans in which we varied sucrose concentration and measured its effect on biofilm growth through image analysis of of colony-forming units as well as crystal violet staining. As expected based on our differential equations, the concentration of sugar decreases exponentially over time, whereas the total amount of glucans produced increases exponentially over time. The rates of exponential growth and decay also depend on the sugar available, with the difference in glucans produced by the end of the simulation being around one order of magnitude less than the difference in sugar available in all cases. This data was useful in helping us match the arbitrary units in the model to real experiments.

4.2 Inhibition of GTFC Activity by Single-Chain Variable Fragments

4.2.1 Additional Cellular Equations Governing ScFv Activity

dGT-F-CScF-v = kab × ScFvcell × GT FC - δGTFCScFv × GT FCScF v dt

(31)

dScF-vcell= - k ×ScF v × GT FC - δ × ScF v dt ab cell ScFvcell cell

(32)

4.2.2 Additional Field Equations Governing ScFv Activity

dScF-vfield-= (- k × ScF v ×GT F C - δ × ScFv ) ÷100 dt ab cell ScFvcell cell

(33)

4.2.3 Modified Equations Integating ScFv Activity

dGT FC --dt---= αGT FC × mGT F C - δGT FC ×GT F C - kab × ScFvcell × GT FC

(34)

The primary purpose of our model was to preemptively compare the effectiveness of our potential outputs by modelling their inhibition of different parts of the pathway and its effects on overall biofilm growth. We began by implementing the above differential equations (Equations 31-33), and modifying Equation 23 into equation 34. The amount of ScFvs affecting the cell is determined via the NeighborhoodReporter function, and the ScFvs bind GTFC at the rate kab. Once the ScFv binds a GTFC, it forms a GTFC:ScFv complex represented in Equation 31, decreasing both the cell’s GTFC and ScFv count accordingly (Equations 32 and 34). The complex degrades quickly based on a rate from the literature, and once it has degraded both the ScFv and GTFC bound have been eliminated from the simulation.

The effects of the ScFv on biofilm formation are immediately obvious from plots of the cells at the end of the simulation, with only 0.075 arbitrary units of concentration to start with being enough to prevent the bacteria from forming adherent microcolonies throughout the time modelled. The critical point at which the amount of GTFC: ScFv complexes goes from increasing to decreasing determines how soon the cells are able to begin initiating biofilm formation, assuming they are still in the logarithmic growth phase where CSP production is increasing. Since GTFC production is decreased, everything downstream of it is also affected: cells consume less sugar, produce fewer glucans, and create less of a biofilm field in the presence of ScFvs, as shown in the following figures. In equation 33, the decrease of the ScFv field is divided by 100 due to the membrane resolution being set to 100 for the simulation.

4.3 Inhibition of Glucan Activity by Kappa-Casein

4.3.1 Additional Cellular Equations Governing K-Casein Activity

dKCaseincell -----dt-----= - kkc × KCaseincell × Glucan - δKCaseincell × KCaseincell

(35)

4.3.2 Additional Fiels Equations Governing K-Casein Activity

dKCaseinfield -----dt------= (- kkc×KCaseincell×Glucan- δKCaseincell×KCaseincell)÷100

(36)

4.3.3 Modified Equations Integating K-Casein Activity

dGlucan-= ke×GT F C×Sugarcell- ϕGlucan×Glucan - kkc×KCaseincell×Glucan dt

(37)

Next, we sought to create a similar set of equations to model the effect of Kappa-Casein on biofilm formation. Equation 35 governs the effect of K-Casein on an individual cell. Cells use the NeighborhoodReporter to determine the amount of K-Casein molecules affecting it, and those molecules bind glucans at a rate of kkc. Rather than creating a new variable for the complex of K-Casein and a glucan, both are removed from the simulation immediately. Due to being outside the cell, the binding rate of K-Casein was set to 1. Equation 37 is a modification of equation 27 integrating the decrease in glucans.

It became clear from varying the initial K-Casein concentration that the equations were successful in modelling the decrease in biofilm formation due to a decrease in effective adherent glucans surrounding the cells. However, despite the K-Casein being given a dramatically higher binding rate than the ScFvs, a much higher concentration of K-Casein was required to limit the formation of a biofilm during the time simulated. In fact, ten times as high a concentration of K-Casein was required to achieve the same effects as ScFvs. The K-Casein also affected fewer aspects of the overall system of equations because its influence was only effective on one of the most downstream components of the simulation.

The most obvious takeaway from the results of our computational model was that ScFvs were more effective than K-Casein at inhibiting biofilm formation, or, more broadly, inhibiting GTFC is a better method for limiting the virulence of S. mutans than reducing the glucans themselves. This result had a profound effect on our experimental design and our project as a whole. While Kappa-Casein can be readily purchased and implemented in live experiments, the ScFvs we planned to express and implement were not available for purchase anywhere, would be very expensive to get synthesized, and would require additional training and lab techniques to isolate from mammalian cells expressing the protein itself. However, thanks to data from the model, we were able to confirm the superiority of the ScFvs for our purposes. Based on these results, we decided to move further ahead with characterization experiments for GTFC-inhibiting ScFvs.

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-	<h1 class="subtitle text-center">Model: Background</h1>
-<ul>
-<li>
-<p>Simple growth models of bacterial populations are based on two interdependent factors: the concentration of bacteria (<em>X</em>) and concentration of a growth limiting substrate (<em>S</em>) (Monod, 1949)(Figure 1A). Bacteria-phage model can be considered as an extended version of a simple model that includes a third dynamic factor: the phage (<em>P</em>)(Figure 1B). The first model was presented as a set of ordinary differential equations (ODDs) by Campbell (1961). In this model, the change rate of the bacterial population is dependent on the concentration of bacteria and substrate, as well as the rate of phage-induced lysis, which in turn is also dependent on the bacterial concentration (Figure 1B).</p>
-<div class="div-fig" style="width: 800px;">
-		<img src="https://static.igem.org/mediawiki/2017/1/11/T--Edinburgh_OG--model_bg_Picture1.png">
-<p><strong>Figure 1.</strong> Schematic diagram showing interactions in (A) a two-factor model containing bacteria and substrate, and (B) an extended three-factor model also containing a lytic phage as proposed by Campbell (1961). Taken from Krysiak-Baltyn <em>et al.</em> (2016).</p>
-</div>
-<h2>Lytic and lysogenic cycles of phage replication</h2>
-<p>In order to model phage-bacteria interactions it is important to understand a step-by-step process of phage infection. Figure 2 shows a lytic life cycle of a lytic phage in a simplified five-step process: from diffusion and attachment to the bacterial cell membrane to cell lysis. The lysogenic life cycle of a temperate phage starts in a similar way to the lytic phage, however following the infection the phage genome either integrates itself into the host genome (&lambda; phage) or forms a circularized plasmid-like structure (P1 phage)(Howard-Varona <em>et al.</em>, 2017). The phage genome that has been integrated into the bacterial chromosome is also known as a prophage. Once integrated, temperate phage replicates together with the bacterial genome upon bacterial cell division. However, either spontaneously at low frequency (10-8&ndash;10-5) or due to unfavourable stress conditions, such as exposure to UV light, antibiotics, hydrogen peroxide and changes in temperature, nutrients or pH, can lead to a phage induction. Once induced, lytic-lysogenic switch gets activated leading to expression of viral genes, a switch to lytic life cycle and cell lysis.</p>
-<div class="div-fig" style="width: 800px;">
-		<img src="https://static.igem.org/mediawiki/2017/c/cf/T--Edinburgh_OG--model_bg_Picture2.png">
-<p><strong>Figure 2.</strong> A simplified diagram of lytic and lysogenic life cycles of the phage. (1) diffusion of the phage particle to the bacterial cell; (2) phage attachment to the receptors along the bacterial membrane; (3) injection of phage genome into the cell; (3a) replication via cell division during lysogenic cycle; (3b) induction of lytic cycle; (4) replication of phage genome and assembly of capsid proteins; (5) cell lysis followed by the release of phage particles outside the cell. The burst size (<em>b</em>) shows the number of released phage particles. Adapted from Krysiak-Baltyn <em>et al. </em>(2016).</p>
-</div>
-<div class="div-fig" style="width: 800px;">
-<table>
-<tbody>
-<tr>
-<td>
-<p><em>b</em></p>
-</td>
-<td>
-<p>Burst size</p>
-</td>
-<td>
-<p><em>P</em></p>
-</td>
-<td>
-<p>Concentration of phage</p>
-</td>
-</tr>
-<tr>
-<td>
-<p><em>C</em></p>
-</td>
-<td>
-<p>Carrying capacity</p>
-</td>
-<td>
-<p><em>q</em></p>
-</td>
-<td>
-<p>Induction rate</p>
-</td>
-</tr>
-<tr>
-<td>
-<p><em>D</em></p>
-</td>
-<td>
-<p>Dilution rate </p>
-</td>
-<td>
-<p><em>S</em></p>
-</td>
-<td>
-<p>Concentration of substrate </p>
-</td>
-</tr>
-<tr>
-<td>
-<p><em>d</em></p>
-</td>
-<td>
-<p>Rate of decay of cells or phages </p>
-</td>
-<td>
-<p><em>T</em></p>
-</td>
-<td>
-<p>Latency time </p>
-</td>
-</tr>
-<tr>
-<td>
-<p><em>K</em><sub><em>i</em></sub></p>
-</td>
-<td>
-<p>Adsorption rate of phages to bacteria</p>
-</td>
-<td>
-<p><em>X</em><sub><em>S</em></sub></p>
-</td>
-<td>
-<p>Concentration of susceptible bacteria (here antibiotic resistant bacteria)</p>
-</td>
-</tr>
-<tr>
-<td>
-<p><em>K</em><sub><em>m</em></sub></p>
-</td>
-<td>
-<p>Half-saturation constant </p>
-</td>
-<td>
-<p><em>X</em><sub><em>I</em></sub></p>
-</td>
-<td>
-<p>Concentration of infected bacteria </p>
-</td>
-</tr>
-<tr>
-<td>
-<p><em>&mu;</em><sub><em>max</em></sub></p>
-</td>
-<td>
-<p>Maximum specific growth rate</p>
-</td>
-<td>
-<p><em>Y</em></p>
-</td>
-<td>
-<p>Bacterial yield, or bacterial cells per unit substrate</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p><strong>Table 1.</strong> Parameters used in models of populations of phage and bacteria.</p>
-</div>
-<h2>Basic model</h2>
-<p>Modelling of the processes described in Figure 2. involves developing and solving expressions for the concentrations of phages, infected and non-infected bacteria as well as the substrate. The key parameters involved into equations are listed in Table 1. Following the approach documented in Campbell (1961), the rate of infection is proportional to the concentration of bacteria, lytic phage and the absorption rate (<em>K</em><em>i</em>). The latter is the rate at which phage particles attach to the bacterial surface and it is dependent on (1) the concentrations of both bacteria and phage, (2) the time required for diffusion and attachment of the phage, and (3) the number of phage particles, which upon attachment to the bacterial cell, do not lead to infection.</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/3/32/T--Edinburgh_OG--model_bg_eq1.png">
-</div>
-<p>Oppositely, the growth rate of phage population decreases due to attachment of phage particles onto bacterial cells:</p>
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-		<img src="https://static.igem.org/mediawiki/2017/c/c4/T--Edinburgh_OG--model_bg_eq2.png">
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-<p>And increases due to cell lysis and release of newly produced phage particles:</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/f/f9/T--Edinburgh_OG--model_bg_eq3.png">
-</div>
-<p>The <em>latency time </em>(<em>T</em>) is caused by the time delay between the infection and cell lysis (steps 3 and 5 in Figure 2).</p>
-<p>The growth of bacterial population was represented as a logistic kinetic expression that is independent from the concentration of the growth limiting substrate, but dependent on the total <em>carrying capacity</em> (<em>C</em>) <em>i.e. </em>the maximum bacterial concentration after which bacteria cease cell division. Thus, the growth rate of bacteria slows down once it approaches the carrying capacity value.</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/f/f9/T--Edinburgh_OG--model_bg_eq4.png">
-</div>
-<p>Campbell&rsquo;s model was describing continuous culture in a chemostat, which is a bioreactor with a continuous inflow of fresh medium and outflow of liquid culture with phage particles and bacteria. Therefore, a washout rate of both phages and bacteria was also taken into account as simple first-order kinetics, where <em>D</em> is the <em>dilution rate</em>:</p>
-<div class="div-eq">
-		<img style="height: 130px;" src="https://static.igem.org/mediawiki/2017/0/0e/T--Edinburgh_OG--model_bg_eq5.png">
-</div>
-<p>In order to simulate real life environment, bacterial (<em>d</em><em>X</em>) and phage decay (<em>d</em><em>P</em>) rates were introduced. It has been shown that UV light can have a significant negative impact on phage infectivity by altering its DNA, which is does not necessarily destroy the virion particles (Suttle and Chen, 1992; Rastogi <em>et al.</em>, 2010). In nature phage decay is also associated with absorption by heat-labile particles or direct consumption by flagellates (Suttle and Chen, 1992; Deng <em>et al.</em>, 2014). Bacterial cells are more robust compared to the phages, but are subject to endogenous decay, which is caused by oxidation of internal cell components for production of energy and life maintenance whilst being limited by the availability of the growth substrate (Droste, 1998).</p>
-<div class="div-eq">
-		<img style="height: 130px;" src="https://static.igem.org/mediawiki/2017/f/f6/T--Edinburgh_OG--model_bg_eq6.png">
-</div>
-<p>Arguably, one of the most important aspects of mathematical modelling is the initial set of assumptions that are used to build a model. The very first model of phage-bacterial interactions published by Campbell (1961) was based on the following assumptions:</p>
-<ol>
-<li>1. A single bacterial cell can be infected only by one phage.</li>
-<li>2. Both rates of phage attachment and infection are dependent on bacterial and phage concentrations.</li>
-<li>3. The burst size does not change and is the same for each infection.</li>
-<li>4. The latency time does not change and is the same for each infection.</li>
-<li>5. The bacterial growth rate is expressed as a logistic function.</li>
-</ol>
-<p>Based on these assumptions, he came up with the following set of delayed differential equations (DDEs):</p>
-<div class="div-eq">
-		<img style="height: 130px;" src="https://static.igem.org/mediawiki/2017/6/68/T--Edinburgh_OG--model_bg_eq7.png">
-</div>
-<p>Notably, most of these assumptions are nowadays considered false by the community working on bacteria-phage interaction. Firstly, there are multiple binding sites to which phage particles can attach to on the bacterial membrane. Thus, many of them can be occupied at the same time, leading to multiple simultaneous infections (Figure 3). Secondly, the burst size is affected by a number of variables, including the cell size of the bacterium, its metabolic activity and particular phage-bacteria relationship (Parada, Herndl and Weinbauer, 2006). Similarly, the latency time depends on the cell density, selecting for a shorter latency time upon high cell density and opting for longer latency upon low density (Wang, Dykhuizen and Slobodkin, 1996). Lastly, the bacterial growth is naturally dependent on the availability of resources to feed on, therefore the logistic equation for the bacterial growth has been replaced for the Monod equation, which takes availability of the substrate into account.</p>
-<div class="div-fig" style="width: 350px;">
-		<img src="https://static.igem.org/mediawiki/2017/c/cd/T--Edinburgh_OG--model_bg_Picture3.png">
-<p><strong>Figure 3.</strong> Multiple phages attached to the bacterial membrane via phage binding sites. Taken from Smith and Trevino (2009).</p>
-</div>
-<p>To tackle some of these limitations, Levin <em>et al. </em>(1977) proposed a new generalized model, that could include any number of phages and bacterial species, as well as substrates. The change rate of the concentration of substrate depends on its consumption by both non-infected and infected bacteria, multiplied by the inverse value of the bacterial yield (<em>Y</em>), and concentration of inflow and outflow medium:</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/7/7f/T--Edinburgh_OG--model_bg_eq8.png">
-</div>
-<p>The growth of non-infected bacteria is described by the Monod equation:</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/4/42/T--Edinburgh_OG--model_bg_eq9.png">
-</div>
-<p>They also included a separate equation to track the change in population of infected bacteria as well as its removal from the chemostat, which had a negative impact on the number of phage particles generated from lysis and has not been included into Campbell&rsquo;s (1961) model.</p>
-<div class="div-eq">
-		<img style="height: 130px;" src="https://static.igem.org/mediawiki/2017/8/83/T--Edinburgh_OG--model_bg_eq10.png">
-</div>
-<p>Although Levin&rsquo;s model predicted the concentrations at steady-state within an order of magnitude from experimental values, it also predicted a total extinction of either phage or bacteria, or both, whereas this was not observed experimentally. Ultimately, the model was labelled not complex enough to accurately describe the interactions of phage and bacteria (Levin, Stewart and Chao, 1977). Since then, the model has been extended by others to include variable absorption rate and burst size (Smith and Thieme, 2012), multiple binding sites (Smith and Trevino, 2009), bacterial resistance to phages (Lenski and Levin, 1985; Cairns <em>et al.</em>, 2009) and spatial heterogeneity (Jones and Smith, 2011). Detailed description of these extensions is beyond the scope of this study.</p>
-<p>In 2007 Qiu augmented Levin&rsquo;s model by adding lysogenic cycle to it in order to study the dynamic mechanisms of the lytic-lysogeny decision. In contrast to Levin, <em>lysogenic bacteria</em>, that is bacteria infected with temperate phages, were able to grow by consuming substrate. He also introduced an <em>induction rate</em> (<em>q</em>) &ndash; that is the rate at which lysogenic bacteria enter lytic cycle either spontaneously or due to environmental factors discussed previously. The concentration of free phage particles (<em>P</em><em>T</em>) only increased due to induction of lysogenic bacteria (<em>X</em><em>L</em>).</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/9/96/T--Edinburgh_OG--model_bg_eq11.png">
-</div>
-<p>It is worth noting that Qiu&rsquo;s model did not account for time latency between infection and lysis, thus presenting a system of ODEs rather than DDEs (Qiu, 2007).</p>
-<div class="div-eq">
-		<img src="https://static.igem.org/mediawiki/2017/6/6f/T--Edinburgh_OG--model_bg_eq12.png">
-</div>
-<div class="btn-row">
-		<div><a class="btn-result" style="width:90%;" data-wow-delay=".9s" href="https://2017.igem.org/Team:Edinburgh_OG/Model" style="visibility: visible; animation-delay: 0.9s;">Model Homepage</a></div>
-		<div style="clear:both;"><a class="btn-result btn-active" data-wow-delay=".9s" href="https://2017.igem.org/Team:Edinburgh_OG/Model:Background" style="visibility: visible; animation-delay: 0.9s;">Background</a>
-			<a class="btn-result" data-wow-delay=".9s" href="https://2017.igem.org/Team:Edinburgh_OG/Model:Methodology" style="visibility: visible; animation-delay: 0.9s;">Methodology</a></div>
-		<div style="clear:both;" class="btn-row"><a class="btn-result" data-wow-delay=".9s" href="https://2017.igem.org/Team:Edinburgh_OG/Model:Results" style="visibility: visible; animation-delay: 0.9s;">Results</a>
-			<a class="btn-result" data-wow-delay=".9s" href="https://2017.igem.org/Team:Edinburgh_OG/Model:References" style="visibility: visible; animation-delay: 0.9s;">References</a></div>
-	</div>
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+.w3-animate-top{position:relative;animation:animatetop 0.4s}@keyframes animatetop{from{top:-300px;opacity:0} to{top:0;opacity:1}}
+.w3-animate-left{position:relative;animation:animateleft 0.4s}@keyframes animateleft{from{left:-300px;opacity:0} to{left:0;opacity:1}}
+.w3-animate-right{position:relative;animation:animateright 0.4s}@keyframes animateright{from{right:-300px;opacity:0} to{right:0;opacity:1}}
+.w3-animate-bottom{position:relative;animation:animatebottom 0.4s}@keyframes animatebottom{from{bottom:-300px;opacity:0} to{bottom:0;opacity:1}}
+.w3-animate-zoom {animation:animatezoom 0.6s}@keyframes animatezoom{from{transform:scale(0)} to{transform:scale(1)}}
+.w3-animate-input{transition:width 0.4s ease-in-out}.w3-animate-input:focus{width:100%!important}
+.w3-opacity,.w3-hover-opacity:hover{opacity:0.60}.w3-opacity-off,.w3-hover-opacity-off:hover{opacity:1}
+.w3-opacity-max{opacity:0.25}.w3-opacity-min{opacity:0.75}
+.w3-greyscale-max,.w3-grayscale-max,.w3-hover-greyscale:hover,.w3-hover-grayscale:hover{filter:grayscale(100%)}
+.w3-greyscale,.w3-grayscale{filter:grayscale(75%)}.w3-greyscale-min,.w3-grayscale-min{filter:grayscale(50%)}
+.w3-sepia{filter:sepia(75%)}.w3-sepia-max,.w3-hover-sepia:hover{filter:sepia(100%)}.w3-sepia-min{filter:sepia(50%)}
+.w3-tiny{font-size:10px!important}.w3-small{font-size:12px!important}.w3-medium{font-size:15px!important}.w3-large{font-size:18px!important}
+.w3-xlarge{font-size:24px!important}.w3-xxlarge{font-size:36px!important}.w3-xxxlarge{font-size:48px!important}.w3-jumbo{font-size:64px!important}
+.w3-left-align{text-align:left!important}.w3-right-align{text-align:right!important}.w3-justify{text-align:justify!important}.w3-center{text-align:center!important}
+.w3-border-0{border:0!important}.w3-border{border:1px solid #ccc!important}
+.w3-border-top{border-top:1px solid #ccc!important}.w3-border-bottom{border-bottom:1px solid #ccc!important}
+.w3-border-left{border-left:1px solid #ccc!important}.w3-border-right{border-right:1px solid #ccc!important}
+.w3-topbar{border-top:6px solid #ccc!important}.w3-bottombar{border-bottom:6px solid #ccc!important}
+.w3-leftbar{border-left:6px solid #ccc!important}.w3-rightbar{border-right:6px solid #ccc!important}
+.w3-section,.w3-code{margin-top:16px!important;margin-bottom:16px!important}
+.w3-margin{margin:16px!important}.w3-margin-top{margin-top:16px!important}.w3-margin-bottom{margin-bottom:16px!important}
+.w3-margin-left{margin-left:16px!important}.w3-margin-right{margin-right:16px!important}
+.w3-padding-small{padding:4px 8px!important}.w3-padding{padding:8px 16px!important}.w3-padding-large{padding:12px 24px!important}
+.w3-padding-16{padding-top:16px!important;padding-bottom:16px!important}.w3-padding-24{padding-top:24px!important;padding-bottom:24px!important}
+.w3-padding-32{padding-top:32px!important;padding-bottom:32px!important}.w3-padding-48{padding-top:48px!important;padding-bottom:48px!important}
+.w3-padding-64{padding-top:64px!important;padding-bottom:64px!important}
+.w3-left{float:left!important}.w3-right{float:right!important}
+.w3-button:hover{color:#000!important;background-color:#ccc!important}
+.w3-transparent,.w3-hover-none:hover{background-color:transparent!important}
+.w3-hover-none:hover{box-shadow:none!important}
+/* Colors */
+.w3-amber,.w3-hover-amber:hover{color:#000!important;background-color:#ffc107!important}
+.w3-aqua,.w3-hover-aqua:hover{color:#000!important;background-color:#00ffff!important}
+.w3-blue,.w3-hover-blue:hover{color:#fff!important;background-color:#2196F3!important}
+.w3-light-blue,.w3-hover-light-blue:hover{color:#000!important;background-color:#87CEEB!important}
+.w3-brown,.w3-hover-brown:hover{color:#fff!important;background-color:#795548!important}
+.w3-cyan,.w3-hover-cyan:hover{color:#000!important;background-color:#00bcd4!important}
+.w3-blue-grey,.w3-hover-blue-grey:hover,.w3-blue-gray,.w3-hover-blue-gray:hover{color:#fff!important;background-color:#607d8b!important}
+.w3-green,.w3-hover-green:hover{color:#fff!important;background-color:#4CAF50!important}
+.w3-light-green,.w3-hover-light-green:hover{color:#000!important;background-color:#8bc34a!important}
+.w3-indigo,.w3-hover-indigo:hover{color:#fff!important;background-color:#3f51b5!important}
+.w3-khaki,.w3-hover-khaki:hover{color:#000!important;background-color:#f0e68c!important}
+.w3-lime,.w3-hover-lime:hover{color:#000!important;background-color:#cddc39!important}
+.w3-orange,.w3-hover-orange:hover{color:#000!important;background-color:#ff9800!important}
+.w3-deep-orange,.w3-hover-deep-orange:hover{color:#fff!important;background-color:#ff5722!important}
+.w3-pink,.w3-hover-pink:hover{color:#fff!important;background-color:#e91e63!important}
+.w3-purple,.w3-hover-purple:hover{color:#fff!important;background-color:#9c27b0!important}
+.w3-deep-purple,.w3-hover-deep-purple:hover{color:#fff!important;background-color:#673ab7!important}
+.w3-red,.w3-hover-red:hover{color:#fff!important;background-color:#f44336!important}
+.w3-sand,.w3-hover-sand:hover{color:#000!important;background-color:#fdf5e6!important}
+.w3-teal,.w3-hover-teal:hover{color:#fff!important;background-color:#009688!important}
+.w3-yellow,.w3-hover-yellow:hover{color:#000!important;background-color:#ffeb3b!important}
+.w3-white,.w3-hover-white:hover{color:#000!important;background-color:#fff!important}
+.w3-black,.w3-hover-black:hover{color:#fff!important;background-color:#000!important}
+.w3-grey,.w3-hover-grey:hover,.w3-gray,.w3-hover-gray:hover{color:#000!important;background-color:#9e9e9e!important}
+.w3-light-grey,.w3-hover-light-grey:hover,.w3-light-gray,.w3-hover-light-gray:hover{color:#000!important;background-color:#f1f1f1!important}
+.w3-dark-grey,.w3-hover-dark-grey:hover,.w3-dark-gray,.w3-hover-dark-gray:hover{color:#fff!important;background-color:#616161!important}
+.w3-pale-red,.w3-hover-pale-red:hover{color:#000!important;background-color:#ffdddd!important}
+.w3-pale-green,.w3-hover-pale-green:hover{color:#000!important;background-color:#ddffdd!important}
+.w3-pale-yellow,.w3-hover-pale-yellow:hover{color:#000!important;background-color:#ffffcc!important}
+.w3-pale-blue,.w3-hover-pale-blue:hover{color:#000!important;background-color:#ddffff!important}
+.w3-text-amber,.w3-hover-text-amber:hover{color:#ffc107!important}
+.w3-text-aqua,.w3-hover-text-aqua:hover{color:#00ffff!important}
+.w3-text-blue,.w3-hover-text-blue:hover{color:#2196F3!important}
+.w3-text-light-blue,.w3-hover-text-light-blue:hover{color:#87CEEB!important}
+.w3-text-brown,.w3-hover-text-brown:hover{color:#795548!important}
+.w3-text-cyan,.w3-hover-text-cyan:hover{color:#00bcd4!important}
+.w3-text-blue-grey,.w3-hover-text-blue-grey:hover,.w3-text-blue-gray,.w3-hover-text-blue-gray:hover{color:#607d8b!important}
+.w3-text-green,.w3-hover-text-green:hover{color:#4CAF50!important}
+.w3-text-light-green,.w3-hover-text-light-green:hover{color:#8bc34a!important}
+.w3-text-indigo,.w3-hover-text-indigo:hover{color:#3f51b5!important}
+.w3-text-khaki,.w3-hover-text-khaki:hover{color:#b4aa50!important}
+.w3-text-lime,.w3-hover-text-lime:hover{color:#cddc39!important}
+.w3-text-orange,.w3-hover-text-orange:hover{color:#ff9800!important}
+.w3-text-deep-orange,.w3-hover-text-deep-orange:hover{color:#ff5722!important}
+.w3-text-pink,.w3-hover-text-pink:hover{color:#e91e63!important}
+.w3-text-purple,.w3-hover-text-purple:hover{color:#9c27b0!important}
+.w3-text-deep-purple,.w3-hover-text-deep-purple:hover{color:#673ab7!important}
+.w3-text-red,.w3-hover-text-red:hover{color:#f44336!important}
+.w3-text-sand,.w3-hover-text-sand:hover{color:#fdf5e6!important}
+.w3-text-teal,.w3-hover-text-teal:hover{color:#009688!important}
+.w3-text-yellow,.w3-hover-text-yellow:hover{color:#d2be0e!important}
+.w3-text-white,.w3-hover-text-white:hover{color:#fff!important}
+.w3-text-black,.w3-hover-text-black:hover{color:#000!important}
+.w3-text-grey,.w3-hover-text-grey:hover,.w3-text-gray,.w3-hover-text-gray:hover{color:#757575!important}
+.w3-text-light-grey,.w3-hover-text-light-grey:hover,.w3-text-light-gray,.w3-hover-text-light-gray:hover{color:#f1f1f1!important}
+.w3-text-dark-grey,.w3-hover-text-dark-grey:hover,.w3-text-dark-gray,.w3-hover-text-dark-gray:hover{color:#3a3a3a!important}
+.w3-border-amber,.w3-hover-border-amber:hover{border-color:#ffc107!important}
+.w3-border-aqua,.w3-hover-border-aqua:hover{border-color:#00ffff!important}
+.w3-border-blue,.w3-hover-border-blue:hover{border-color:#2196F3!important}
+.w3-border-light-blue,.w3-hover-border-light-blue:hover{border-color:#87CEEB!important}
+.w3-border-brown,.w3-hover-border-brown:hover{border-color:#795548!important}
+.w3-border-cyan,.w3-hover-border-cyan:hover{border-color:#00bcd4!important}
+.w3-border-blue-grey,.w3-hover-border-blue-grey:hover,.w3-border-blue-gray,.w3-hover-border-blue-gray:hover{border-color:#607d8b!important}
+.w3-border-green,.w3-hover-border-green:hover{border-color:#4CAF50!important}
+.w3-border-light-green,.w3-hover-border-light-green:hover{border-color:#8bc34a!important}
+.w3-border-indigo,.w3-hover-border-indigo:hover{border-color:#3f51b5!important}
+.w3-border-khaki,.w3-hover-border-khaki:hover{border-color:#f0e68c!important}
+.w3-border-lime,.w3-hover-border-lime:hover{border-color:#cddc39!important}
+.w3-border-orange,.w3-hover-border-orange:hover{border-color:#ff9800!important}
+.w3-border-deep-orange,.w3-hover-border-deep-orange:hover{border-color:#ff5722!important}
+.w3-border-pink,.w3-hover-border-pink:hover{border-color:#e91e63!important}
+.w3-border-purple,.w3-hover-border-purple:hover{border-color:#9c27b0!important}
+.w3-border-deep-purple,.w3-hover-border-deep-purple:hover{border-color:#673ab7!important}
+.w3-border-red,.w3-hover-border-red:hover{border-color:#f44336!important}
+.w3-border-sand,.w3-hover-border-sand:hover{border-color:#fdf5e6!important}
+.w3-border-teal,.w3-hover-border-teal:hover{border-color:#009688!important}
+.w3-border-yellow,.w3-hover-border-yellow:hover{border-color:#ffeb3b!important}
+.w3-border-white,.w3-hover-border-white:hover{border-color:#fff!important}
+.w3-border-black,.w3-hover-border-black:hover{border-color:#000!important}
+.w3-border-grey,.w3-hover-border-grey:hover,.w3-border-gray,.w3-hover-border-gray:hover{border-color:#9e9e9e!important}
+.w3-border-light-grey,.w3-hover-border-light-grey:hover,.w3-border-light-gray,.w3-hover-border-light-gray:hover{border-color:#f1f1f1!important}
+.w3-border-dark-grey,.w3-hover-border-dark-grey:hover,.w3-border-dark-gray,.w3-hover-border-dark-gray:hover{border-color:#616161!important}
+.w3-border-pale-red,.w3-hover-border-pale-red:hover{border-color:#ffe7e7!important}.w3-border-pale-green,.w3-hover-border-pale-green:hover{border-color:#e7ffe7!important}
+.w3-border-pale-yellow,.w3-hover-border-pale-yellow:hover{border-color:#ffffcc!important}.w3-border-pale-blue,.w3-hover-border-pale-blue:hover{border-color:#e7ffff!important}
+          #contentSub, #footer-box, #catlinks, #search-controls, #p-logo, #sideMenu, #menubar, .logo_2017, .printfooter, .firstHeading,.visualClear {
+               display: none;
+          } /*-- hides default wiki settings --*/
+          #top-section { /*-- styling for default menu bar (edit, page, history, etc.) --*/
+               border: 0 none;
+               height: 14px;
+               z-index: 100;
+               top: 0;
+               position: fixed;
+               width: 975px;
+               left: 50%;
+               margin-left: -487px;
+          }
+          #top_title, #top {
+               display: none;
+               z-index: -1000;
+          }
+          .logo_2018 {
+               display: none;
+               z-index: -1000;
+          }
+          #globalWrapper, #content { /*-- changes default wiki settings --*/
+               width: 100%;
+               height: 100%;
+               border: 0px;
+               background-color: black;
+               margin: 0px;
+               padding: 0px;
+          }
+          html, body, .wrapper { /*-- changes default wiki settings --*/
+               width: 100%;
+               height: 100%;
+               background-color: black;
+          }
+.menu {
+        list-style-type: none;
+        margin: 0;
+        padding: 0;
+        /*overflow: hidden;*/
+        background-color: #333;
+        position: fixed;
+        width: 100%;
+        -webkit-box-shadow: 0 8px 6px -6px #2c2c2c;
+        -moz-box-shadow: 0 8px 6px -6px #2c2c2c;
+        box-shadow: 0 8px 6px -6px #2c2c2c;
+        top: 1.1vw;
+        z-index: 20;
+    }
+    .sub {
+        font-family: 'Dosis', sans-serif;;
+        font-size: 1.5vw;
+        float: left;
+    }
+    .sub a {
+        display: block;
+        color: white;
+        text-align: center;
+        padding: 15px 30px;
+        text-decoration: none;
+    }
+    .sub a:hover {
+        background-color: #410081;
+    }
+    .activemenu {
+        background-color: #9055ff;
+    }
+    .dropdown {
+        position: relative;
+        display: inline-block;
+    }
+    .dropdown-content {
+        font-family: 'Raleway', sans-serif;;
+        font-size: 1.2vw;
+        display: none;
+        position: absolute;
+        top: 4.2vw;
+        background-color: #2c2c2c;
+        min-width: 160px;
+        box-shadow: 0px 8px 16px 0px rgba(0,0,0,0.2);
+        z-index: 1;
+    }
+    .dropdown-content a {
+        color: white;
+        padding: 12px 16px;
+        text-decoration: none;
+        display: block;
+    }
+    .dropdown-content a:hover {background-color: #474747;}
+    .dropdown:hover .dropdown-content {display: block;}
+body{
+        margin:0;
+    }
+     </style>
+     <style>
+          .MIT-content {
+               font-family: "Times New Roman", Times, serif;
+               padding: 14px 16px;
+          }
+          #project-overview {
+               text-indent: 50px;
+          }
+.cmr-5{font-size:50%;}
+        .cmr-7{font-family: 'Raleway', sans-serif;;font-size:70%;}
+        .cmmi-5{font-family: 'Raleway', sans-serif;;font-size:50%;font-style: italic;}
+        .cmmi-7{font-family: 'Raleway', sans-serif;;font-size:70%;font-style: italic;}
+        .cmmi-10{font-family: 'Raleway', sans-serif;;font-style: italic;}
+        .cmsy-5{font-size:50%;}
+        .cmsy-7{font-size:70%;}
+        .ecrm-1728{font-size:170%;}
+        .ecti-1728{font-size:170%; font-style: italic;}
+        .ecti-1728{ font-style: italic; }
+        .ecti-1728{ font-style: italic;}
+        .ecti-1728{ font-style: italic;}
+        .ecti-1728{ font-style: italic;}
+        .ecti-1728{ font-style: italic;}
+        .ecrm-1200{font-size:120%;}
+        .ectt-1000{ font-family: 'Raleway', sans-serif;;}
+        .ectt-1000{ font-family: 'Raleway', sans-serif;;}
+        .ectt-1000{ font-family: 'Raleway', sans-serif;;}
+        .ectt-1000{ font-family: 'Raleway', sans-serif;;}
+        .ectt-1000{ font-family: 'Raleway', sans-serif;;}
+        .ectt-1000{ font-family: 'Raleway', sans-serif;;}
+        .ecti-1000{ font-family: 'Raleway', sans-serif;;font-style: italic;}
+        .ecti-1000{ font-family: 'Raleway', sans-serif;;font-style: italic;}
+        .ecti-1000{ font-family: 'Raleway', sans-serif;;font-style: italic;}
+        .ecti-1000{ font-family: 'Raleway', sans-serif;;font-style: italic;}
+        .ecti-1000{ font-family: 'Raleway', sans-serif;;font-style: italic;}
+        .ecti-1000{ font-family: 'Raleway', sans-serif;;font-style: italic;}
+        p.noindent { font-family: 'Raleway', sans-serif;;text-indent: 0em }
+        td p.noindent { font-family: 'Raleway', sans-serif;;text-indent: 0em; margin-top:0em; }
+        p.nopar { font-family: 'Raleway', sans-serif;;text-indent: 0em; }
+        p.indent{ font-family: 'Raleway', sans-serif;;text-indent: 1.5em }
+        @media print {div.crosslinks {visibility:hidden;}}
+        a img { border-top: 0; border-left: 0; border-right: 0; }
+        center { margin-top:1em; margin-bottom:1em; }
+        td center { margin-top:0em; margin-bottom:0em; }
+        .Canvas { position:relative; }
+        img.math{vertical-align:middle;}
+        li p.indent { font-family: 'Raleway', sans-serif;;text-indent: 0em }
+        li p:first-child{ font-family: 'Raleway', sans-serif;;margin-top:0em; }
+        li p:last-child, li div:last-child {font-family: 'Raleway', sans-serif;; margin-bottom:0.5em; }
+        li p~ul:last-child, li p~ol:last-child{ font-family: 'Raleway', sans-serif;;margin-bottom:0.5em; }
+        .enumerate1 {list-style-type:decimal;}
+        .enumerate2 {list-style-type:lower-alpha;}
+        .enumerate3 {list-style-type:lower-roman;}
+        .enumerate4 {list-style-type:upper-alpha;}
+        div.newtheorem { margin-bottom: 2em; margin-top: 2em;}
+        .obeylines-h,.obeylines-v {white-space: nowrap; }
+        div.obeylines-v p { margin-top:0; margin-bottom:0; }
+        .overline{ text-decoration:overline; }
+        .overline img{ border-top: 1px solid black; }
+        td.displaylines {text-align:center; white-space:nowrap;}
+        .centerline {text-align:center;}
+        .rightline {text-align:right;}
+        div.verbatim {font-family: monospace; white-space: nowrap; text-align:left; clear:both; }
+        .fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }
+        div.fbox {display:table}
+        div.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }
+        div.minipage{width:100%;}
+        div.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}
+        div.center div {text-align: left;}
+        div.flushright, div.flushright div.flushright {text-align: right;}
+        div.flushright div {text-align: left;}
+        div.flushleft {text-align: left;}
+        .underline{ text-decoration:underline; }
+        .underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }
+        .framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }
+        .framebox-c {text-align:center;}
+        .framebox-l {text-align:left;}
+        .framebox-r {text-align:right;}
+        span.thank-mark{ vertical-align: super }
+        span.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }
+        div.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }
+        table.tabular td p{margin-top:0em;}
+        table.tabular {margin-left: auto; margin-right: auto;}
+        td p:first-child{ font-family: 'Raleway', sans-serif;;margin-top:0em; }
+        td p:last-child{ font-family: 'Raleway', sans-serif;;margin-bottom:0em; }
+        div.td00{ margin-left:0pt; margin-right:0pt; }
+        div.td01{ margin-left:0pt; margin-right:5pt; }
+        div.td10{ margin-left:5pt; margin-right:0pt; }
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+        .abstract p {margin-left:5%; margin-right:5%;}
+        div.abstract {width:100%;}
+        .equation td{text-align:center; }
+        .equation-star td{text-align:center; }
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+        td.gather {text-align:center; }
+        table.gather {width:100%;}
+        div.gather-star {text-align:center;}
+        .figure img.graphics {margin-left:10%;}
+        #TBL-2 colgroup{border-left: 1px solid black;border-right:1px solid black;}
+        #TBL-2{border-collapse:collapse;}
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+        #TBL-2 colgroup{border-left: 1px solid black;border-right:1px solid black;}
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+        #TBL-2 colgroup{border-left: 1px solid black;border-right:1px solid black;}
+        #TBL-2{border-collapse:collapse;}
+        #TBL-2 colgroup{border-left: 1px solid black;border-right:1px solid black;}
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+            font-family: 'Raleway', sans-serif;;
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+        span{
+            font-family: 'Raleway', sans-serif;;
+            font-size: 1.3vw;
+        }
+        .panel{
+            font-family: 'Raleway', sans-serif;;
+            font-size: 1.3vw;
+        }
+        body{
+        margin-top: 4.2vw;
+    }
+.accordion {
+        font-family: 'Dosis', sans-serif;;
+        font-size: 1.5vw;
+        background-color: #2c2c2c;
+        color: #e8e8e8;
+        cursor: pointer;
+        padding: 18px;
+        width: 100%;
+        border: none;
+        text-align: left;
+        outline: none;
+        transition: 0.4s;
+    }
+    .active, .accordion:hover {
+        background-color: #202020;
+    }
+    .accordion:after {
+        content: '\002B';
+        color: #777;
+        font-weight: bold;
+        float: right;
+        margin-left: 5px;
+    }
+    .active:after {
+        content: "\2212";
+    }
+    .panel {
+        padding: 0 18px;
+        background-color: white;
+        max-height: 0;
+        overflow: hidden;
+        transition: max-height 0.2s ease-out;
+    }
+        .h3{
+            color: white;
+            font-family: 'Dosis', sans-serif;;
+            font-size: 8vw;
+position: absolute;
+            left: 2vw;
+top: 3vw;
+            z-index: 5;
+        }
+     </style>
+</head>
+<body >
+ <div class="menu">
+        <div class="sub"><a href="https://2018.igem.org/Team:MIT">Home</a></div>
+        <div class="dropdown">
+            <div class="sub"><a href="https://2018.igem.org/Team:MIT/Team">Team</a></div>
+            <div class="dropdown-content">
+                <a href="https://2018.igem.org/Team:MIT/Team">Team Members</a>
+                <a href="https://2018.igem.org/Team:MIT/Collaborations">Collaborations</a>
+            </div>
+        </div>
+        <div class="dropdown">
+            <div class="sub"><a href="https://2018.igem.org/Team:MIT/Description">Project</a></div>
+            <div class="dropdown-content">
+                <a href="https://2018.igem.org/Team:MIT/Design">Design</a>
+                <a href="https://2018.igem.org/Team:MIT/Results">Results</a>
+                <a href="https://2018.igem.org/Team:MIT/InterLab">InterLab</a>
+                <a href="https://2018.igem.org/Team:MIT/Notebook">Notebook</a>
+                <a href="https://2018.igem.org/Team:MIT/Experiments">Protocols</a>
+                <a href ="https://2018.igem.org/Team:MIT/Attributions"> Attributions </a>
+            </div>
+        </div>
+        <div class="dropdown">
+            <div class="sub"><a href="https://2018.igem.org/Team:MIT/Parts">Parts</a></div>
+            <div class="dropdown-content">
+                <a href="https://2018.igem.org/Team:MIT/Basic_Part">Basic Parts</a>
+                <a href="https://2018.igem.org/Team:MIT/Composite_Part">Composite Parts</a>
+            </div>
+        </div>
+        <div class="sub"><a href="https://2018.igem.org/Team:MIT/Safety">Safety</a></div>
+        <div class="dropdown">
+            <div class="sub"><a href="https://2018.igem.org/Team:MIT/Human_Practices">Human Practices</a></div>
+            <div class="dropdown-content">
+                <a href="https://2018.igem.org/Team:MIT/Human_Practices">Integrated Human Practices</a>
+                <a href="https://2018.igem.org/Team:MIT/Public_Engagement">Public Engagement</a>
+            </div>
+        </div>
+        <div class="sub"><a class="activemenu" href="https://2018.igem.org/Team:MIT/Model">Model</a></div>
+    </div>
+<div class="h3">Modeling</div>
+    <img src="https://static.igem.org/mediawiki/2018/b/b7/T--MIT--MITmodel.gif" style="width: 50vw; position: relative; left: 25vw; top: 10vw;"></img>
+    <p style="position: relative; top: 11vw; font-family: 'Raleway', sans-serif;; font-size: 1.3vw; color: white; margin-left: 2vw;
+margin-right: 2vw; margin-bottom: 3vw;">
+        We created an advanced multiscale model of the ComCDE quorum sensing system and biofilm formation via WIG synthesis in
+        S. mutans. Employing the capabilities of the modelling environment Morpheus to simulate cell migration and adhesion as
+        well as the the diffusion of extracellular small molecules and proteins, we modelled the logarithmic phase of bacterial
+        growth in two dimensions. Our goal in modelling was to determine the most important factors in the bacterial biofilm
+        initiation system in order to predict the effectiveness of various methods of inhibiting biofilm growth.
+    <p> </p>
+    </p>
+<div style="position: absolute; top: 65vw">
+    <button class="accordion">1. List of Constants and Variables Used in the Model</button>
+    <div class="panel">
+        <div class="tabular"> <table id="TBL-2" class="tabular"
+                                     cellspacing="0" cellpadding="0"
+        ><colgroup id="TBL-2-1g"><col
+                id="TBL-2-1"></colgroup><colgroup id="TBL-2-2g"><col
+                id="TBL-2-2"></colgroup><colgroup id="TBL-2-3g"><col
+                id="TBL-2-3"></colgroup><colgroup id="TBL-2-4g"><col
+                id="TBL-2-4"></colgroup><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-1-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-1-1"
+                                                                    class="td11">  Symbol   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-1-2"
+                                                                                                      class="td11">                     Meaning                                  </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-1-3"
+                                                                                                                                                                                           class="td11">  Value   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-1-4"
+                                                                                                                                                                                                                            class="td11">                     Units                                  </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-2-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-2-1"
+                                                                    class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">tx</span></sub>      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-2-2"
+                                                              class="td11">             Transcription Rate of a Gene                    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-2-3"
+                                                                                                                                                  class="td11">    <img
+                src="https://static.igem.org/mediawiki/2018/2/2a/T--MIT--MITMorpheqs0x.png" alt="13"  class="frac" align="middle">       </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-2-4"
+                                                                                            class="td11">           transcripts per gene per second                 </td></tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-3-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-3-1"
+                                                                    class="td11"> <span
+                class="cmmi-10">k</span><sub>
+<span
+        class="cmmi-7">tf</span></sub>      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-3-2"
+                                                      class="td11">  Transcription Factor Activity of Phosphorylated ComE   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-3-3"
+                                                                                                                                      class="td11">   0.15    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-3-4"
+                                                                                                                                                                        class="td11">     transcripts per ComEP per gene per second        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-4-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-4-1"
+                                                                    class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">mCSP</span></sub>    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-4-2"
+                                                              class="td11">              Decay Rate of CSP mRNA                      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-4-3"
+                                                                                                                                               class="td11">   <img
+                src="https://static.igem.org/mediawiki/2018/4/45/T--MIT--MITMorpheqs1x.png" alt="1--
+"  class="frac" align="middle">      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-4-4"
+                                                  class="td11">              transcripts per second                       </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-5-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-5-1"
+                                                                    class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">mComD</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-5-2"
+                                                              class="td11">             Decay Rate of ComD mRNA                    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-5-3"
+                                                                                                                                             class="td11">   <img
+                src="https://static.igem.org/mediawiki/2018/c/c1/T--MIT--MITMorpheqs2x.png" alt="1--
+"  class="frac" align="middle">      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-5-4"
+                                                  class="td11">              transcripts per second                       </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-6-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-6-1"
+                                                                    class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">mComE</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-6-2"
+                                                              class="td11">             Decay Rate of ComE mRNA                    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-6-3"
+                                                                                                                                             class="td11">   <img
+                src="https://static.igem.org/mediawiki/2018/1/19/T--MIT--MITMorpheqs3x.png" alt="1--
+"  class="frac" align="middle">      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-6-4"
+                                                  class="td11">              transcripts per second                       </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-7-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-7-1"
+                                                                    class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">mGTFC</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-7-2"
+                                                              class="td11">            Decay Rate of mGTFC mRNA                   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-7-3"
+                                                                                                                                            class="td11">   <img
+                src="https://static.igem.org/mediawiki/2018/e/eb/T--MIT--MITMorpheqs4x.png" alt="1--
+"  class="frac" align="middle">      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-7-4"
+                                                  class="td11">              transcripts per second                       </td></tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-8-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-8-1"
+                                                                    class="td11"> <span
+                class="cmmi-10">pComC </span></td> <td  style="white-space:nowrap; text-align:center;" id="TBL-2-8-2"
+                                                        class="td11"> Number of ComC genes </td> <td  style="white-space:nowrap; text-align:center;" id="TBL-2-8-3"
+                                                                                                      class="td11"> 1 </td> <td  style="white-space:nowrap; text-align:center;" id="TBL-2-8-4"
+                                                                                                                                 class="td11"> genes</td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-9-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-9-1"
+                                                                    class="td11">  <span
+                class="cmmi-10">pComD  </span></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-9-2"
+                                                        class="td11">               Number of ComD genes                        </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-9-3"
+                                                                                                                                          class="td11">    1      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-9-4"
+                                                                                                                                                                            class="td11">                     genes                                  </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-10-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-10-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">pComE   </span></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-10-2"
+                                                         class="td11">               Number of ComE genes                        </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-10-3"
+                                                                                                                                           class="td11">    1      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-10-4"
+                                                                                                                                                                             class="td11">                     genes                                  </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-11-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-11-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">pGTFC  </span></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-11-2"
+                                                        class="td11">               Number of GTFC genes                        </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-11-3"
+                                                                                                                                          class="td11">    1      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-11-4"
+                                                                                                                                                                            class="td11">                     genes                                  </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-12-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-12-1"
+                                                                     class="td11">   <span
+                class="cmmi-10">&#x03B1;</span><sub><span
+                class="cmmi-7">CSP</span></sub>    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-12-2"
+                                                             class="td11">               Translation Rate of CSP                       </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-12-3"
+                                                                                                                                                 class="td11">  <img
+                src="https://static.igem.org/mediawiki/2018/0/06/T--MIT--MITMorpheqs5x.png" alt="-22-
+"  class="frac" align="middle">    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-12-4"
+                                                  class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-13-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-13-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B1;</span><sub><span
+                class="cmmi-7">ComD</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-13-2"
+                                                             class="td11">              Translation Rate of ComD                      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-13-3"
+                                                                                                                                                class="td11">  <img
+                src="https://static.igem.org/mediawiki/2018/0/0d/T--MIT--MITMorpheqs6x.png" alt="-22-
+"  class="frac" align="middle">    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-13-4"
+                                                  class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-14-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-14-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B1;</span><sub><span
+                class="cmmi-7">ComE</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-14-2"
+                                                             class="td11">              Translation Rate of ComE                      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-14-3"
+                                                                                                                                                class="td11">  <img
+                src="https://static.igem.org/mediawiki/2018/9/9c/T--MIT--MITMorpheqs7x.png" alt="-22-
+"  class="frac" align="middle">    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-14-4"
+                                                  class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-15-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-15-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B1;</span><sub><span
+                class="cmmi-7">GTFC</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-15-2"
+                                                             class="td11">              Translation Rate of GTFC                      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-15-3"
+                                                                                                                                                class="td11">  <img
+                src="https://static.igem.org/mediawiki/2018/3/3b/T--MIT--MITMorpheqs8x.png" alt="-22-
+"  class="frac" align="middle">    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-15-4"
+                                                  class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-16-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-16-1"
+                                                                     class="td11">   <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">CSP</span></sub>     </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-16-2"
+                                                              class="td11">                 Decay Rate of CSP                           </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-16-3"
+                                                                                                                                                  class="td11"> <span
+                class="cmsy-10">&#x2223;</span><img
+                src="https://static.igem.org/mediawiki/2018/e/e1/T--MIT--MITMorpheqs9x.png" alt="ln(1&#x2215;2)-
+"  class="frac" align="middle"><span
+                class="cmsy-10">&#x2223;</span> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-16-4"
+                                                          class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-17-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-17-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">ComE</span></sub>    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-17-2"
+                                                              class="td11">                Decay Rate of ComE                          </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-17-3"
+                                                                                                                                                 class="td11"> <span
+                class="cmsy-10">&#x2223;</span><img
+                src="https://static.igem.org/mediawiki/2018/7/7a/T--MIT--MITMorpheqs10x.png" alt=" ln(1&#x2215;2)
+"  class="frac" align="middle"><span
+                class="cmsy-10">&#x2223;</span> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-17-4"
+                                                          class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-18-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-18-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">ComD</span></sub>    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-18-2"
+                                                              class="td11">                Decay Rate of ComD                         </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-18-3"
+                                                                                                                                                class="td11"><span
+                class="cmr-10">1</span><span
+                class="cmmi-10">.</span><span class="overline"><span
+                class="cmr-10">5</span></span> <span
+                class="cmsy-10">&#x00D7; </span><span
+                class="cmr-10">10</span><sup><span
+                class="cmsy-7">-</span><span
+                class="cmr-7">5</span></sup></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-18-4"
+                                                      class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-19-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-19-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">GTFC</span></sub>    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-19-2"
+                                                              class="td11">                Decay Rate of GTFC                         </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-19-3"
+                                                                                                                                                class="td11"> <span
+                class="cmsy-10">&#x2223;</span><img
+                src="https://static.igem.org/mediawiki/2018/1/1d/T--MIT--MITMorpheqs11x.png" alt="ln(1&#x2215;2)
+-"  class="frac" align="middle"><span
+                class="cmsy-10">&#x2223;</span> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-19-4"
+                                                          class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-20-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-20-1"
+                                                                     class="td11"><span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">CSPComDP</span></sub></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-20-2"
+                                                              class="td11">              Decay Rate of CSPComDP                     </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-20-3"
+                                                                                                                                              class="td11"><span
+                class="cmr-10">1</span><span
+                class="cmmi-10">.</span><span class="overline"><span
+                class="cmr-10">5</span></span> <span
+                class="cmsy-10">&#x00D7; </span><span
+                class="cmr-10">10</span><sup><span
+                class="cmsy-7">-</span><span
+                class="cmr-7">5</span></sup></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-20-4"
+                                                      class="td11">               complexes per second                       </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-21-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-21-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">ComEP</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-21-2"
+                                                              class="td11">         Decay Rate of Phosphorylated ComE               </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-21-3"
+                                                                                                                                              class="td11">  <img
+                src="https://static.igem.org/mediawiki/2018/c/c3/T--MIT--MITMorpheqs12x.png" alt="--1--
+"  class="frac" align="middle">    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-21-4"
+                                                   class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-22-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-22-1"
+                                                                     class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">b</span></sub>       </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-22-2"
+                                                              class="td11">           Binding Activity of CSP to ComD                 </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-22-3"
+                                                                                                                                                class="td11"> <img
+                src="https://static.igem.org/mediawiki/2018/0/0e/T--MIT--MITMorpheqs13x.png" alt="--2----
+10000000"  class="frac" align="middle">  </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-22-4"
+                                                   class="td11">      complexes per CSP per ComD per second          </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-23-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-23-1"
+                                                                     class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">ub</span></sub>      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-23-2"
+                                                              class="td11">Unbinding Activity of CSP:Phosphorylated ComD Complex</td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-23-3"
+                                                                                                                                          class="td11"> <img
+                src="https://static.igem.org/mediawiki/2018/9/9c/T--MIT--MITMorpheqs14x.png" alt="---5---
+100000000"  class="frac" align="middle"> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-23-4"
+                                                   class="td11">      dissasociations per CSPComDP per second         </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-24-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-24-1"
+                                                                     class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">k</span></sub>       </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-24-2"
+                                                              class="td11">        Kinase Activity of Phosphorylated ComD            </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-24-3"
+                                                                                                                                               class="td11"> <img
+                src="https://static.igem.org/mediawiki/2018/9/98/T--MIT--MITMorpheqs15x.png" alt="---5---
+100000000"  class="frac" align="middle"> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-24-4"
+                                                   class="td11">phosphorylations per CSPComDP per ComE per second</td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-25-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-25-1"
+                                                                     class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">e</span></sub>       </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-25-2"
+                                                              class="td11">              Enzyme Activity of GTFC                      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-25-3"
+                                                                                                                                                class="td11"> <img
+                src="https://static.igem.org/mediawiki/2018/7/7c/T--MIT--MITMorpheqs16x.png" alt="--1----
+10000000"  class="frac" align="middle">  </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-25-4"
+                                                   class="td11">    glucans formed per GTFC per Sugar per second      </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-26-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-26-1"
+                                                                     class="td11">   <span
+                class="cmmi-10">&#x03D5;</span><sub><span
+                class="cmmi-7">CSP</span></sub>    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-26-2"
+                                                             class="td11">            Export Rate of CSP from a Cell                  </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-26-3"
+                                                                                                                                                class="td11">   <img
+                src="https://static.igem.org/mediawiki/2018/7/7c/T--MIT--MITMorpheqs17x.png" alt="1--
+"  class="frac" align="middle">      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-26-4"
+                                                  class="td11">                 <sup class="nicefrac"><span
+                class="cmr-10">1</span></sup><span
+                class="cmmi-10">&#x2215;</span><sub class="nicefrac"><span
+                class="cmmi-10">CSP </span><span
+                class="cmsy-10">&#x00D7; </span><span
+                class="cmmi-10">second</span></sub>                      </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-27-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-27-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03D5;</span><sub><span
+                class="cmmi-7">Glucan</span></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-27-2"
+                                                               class="td11">          Export Rate of Glucans from a Cell                </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-27-3"
+                                                                                                                                                  class="td11">    <img
+                src="https://static.igem.org/mediawiki/2018/3/30/T--MIT--MITMorpheqs18x.png" alt="1
+"  class="frac" align="middle">       </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-27-4"
+                                                 class="td11">                <sup class="nicefrac"><span
+                class="cmr-10">1</span></sup><span
+                class="cmmi-10">&#x2215;</span><sub class="nicefrac"><span
+                class="cmmi-10">Glucan </span><span
+                class="cmsy-10">&#x00D7; </span><span
+                class="cmmi-10">second</span></sub>                     </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-28-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-28-1"
+                                                                     class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">ab</span></sub>      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-28-2"
+                                                              class="td11">           Binding Activity of ScFv to GTFC                 </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-28-3"
+                                                                                                                                                 class="td11">  <img
+                src="https://static.igem.org/mediawiki/2018/b/b4/T--MIT--MITMorpheqs19x.png" alt="-4--
+"  class="frac" align="middle">    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-28-4"
+                                                  class="td11">      Complexes per ScFv per GTFC per second         </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-29-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-29-1"
+                                                                     class="td11"> <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">GTFCScFv</span></sub> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-29-2"
+                                                               class="td11">          Decay Rate of GTFC:ScFv Complex                </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-29-3"
+                                                                                                                                               class="td11"> <span
+                class="cmsy-10">&#x2223;</span><img
+                src="https://static.igem.org/mediawiki/2018/3/37/T--MIT--MITMorpheqs20x.png" alt="ln(1&#x2215;2)-
+"  class="frac" align="middle"><span
+                class="cmsy-10">&#x2223;</span> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-29-4"
+                                                          class="td11">               complexes per second                       </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-30-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-30-1"
+                                                                     class="td11">  <span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">ScFv</span><sub><span
+                class="cmmi-5">cell</span></sub></sub>   </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-30-2"
+                                                                   class="td11">                 Decay Rate of ScFv                           </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-30-3"
+                                                                                                                                                        class="td11"> <span
+                class="cmsy-10">&#x2223;</span><img
+                src="https://static.igem.org/mediawiki/2018/9/97/T--MIT--MITMorpheqs21x.png" alt="ln(1&#x2215;2)
+--"  class="frac" align="middle"><span
+                class="cmsy-10">&#x2223;</span> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-30-4"
+                                                          class="td11">                proteins per second                        </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-31-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-31-1"
+                                                                     class="td11">    <span
+                class="cmmi-10">k</span><sub><span
+                class="cmmi-7">kc</span></sub>      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-31-2"
+                                                              class="td11">           Binding Activity of Kappa-Casein                 </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-31-3"
+                                                                                                                                                 class="td11">    1      </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-31-4"
+                                                                                                                                                                                   class="td11">          <sup class="nicefrac"><span
+                class="cmr-10">1</span></sup><span
+                class="cmmi-10">&#x2215;</span><sub class="nicefrac"><span
+                class="cmmi-10">KCasein </span><span
+                class="cmsy-10">&#x00D7; </span><span
+                class="cmmi-10">Glucan </span><span
+                class="cmsy-10">&#x00D7; </span><span
+                class="cmmi-10">second</span></sub>              </td>
+        </tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-32-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-32-1"
+                                                                     class="td11"><span
+                class="cmmi-10">&#x03B4;</span><sub><span
+                class="cmmi-7">KCasein</span><sub><span
+                class="cmmi-5">cell</span></sub></sub></td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-32-2"
+                                                                class="td11">             Decay Rate of Kappa-Casein                    </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-32-3"
+                                                                                                                                                  class="td11"> <span
+                class="cmsy-10">&#x2223;</span><img
+                src="https://static.igem.org/mediawiki/2018/4/48/T--MIT--MITMorpheqs22x.png" alt=" ln(1&#x2215;2)
+"  class="frac" align="middle"><span
+                class="cmsy-10">&#x2223;</span> </td><td  style="white-space:nowrap; text-align:center;" id="TBL-2-32-4"
+                                                          class="td11">                proteins per second                        </td></tr><tr
+                class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr
+                style="vertical-align:baseline;" id="TBL-2-33-"><td  style="white-space:nowrap; text-align:center;" id="TBL-2-33-1"
+                                                                     class="td11"> </td> </tr></table>
+        </div>
+    </div>
+    <button class="accordion">2. Morpheus and the Cellular Potts Model</button>
+    <div class="panel">
+        <p class="noindent" >
+        <!--l. 217--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">2.1   </span> <a
+                id="x1-30002.1"></a>Cell Energy and Migration</h4>
+        <!--l. 219--><p class="noindent" ><div class="eqnarray">
+        <center class="math-display" >
+            <img
+                    src="https://static.igem.org/mediawiki/2018/e/ea/T--MIT--MITMorpheqs23x.png" alt="P (&#x03C3; &#x2032;  &#x2192;   &#x03C3; ) = { 1-i(f&#x0394; &#x0394;HH+Y)+ Y &#x003E; 0              (1)
+    x       x     e---T---otherwise
+" class="math-display" ></center>
+    </div>
+        <table
+                class="equation"><tr><td><a
+                id="x1-3002r2"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/1/1b/T--MIT--MITMorpheqs24x.png" alt="     &#x2211;
+H  =    &#x03BB;V(&#x03BD;&#x03C3; - Vt)2
+     &#x03C3;&#x003E;0
+" class="math-display" ></center></td><td class="equation-label">(2)</td></tr></table>
+        <!--l. 228--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-3003r3"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/4/4d/T--MIT--MITMorpheqs25x.png" alt="     &#x2211;
+H  =    [&#x03BB;V (&#x03BD;&#x03C3; - Vt)2 + &#x03BB;P (&#x03C1;&#x03C3; - Pt)2]
+     &#x03C3;&#x003E;0
+" class="math-display" ></center></td><td class="equation-label">(3)</td></tr></table>
+        <!--l. 232--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-3004r4"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/b/b4/T--MIT--MITMorpheqs26x.png" alt="    &#x2211;
+H =    J[&#x03C4;(&#x03C3;i),&#x03C4;(&#x03C3;j)](1- &#x03B4;&#x03C3;i&#x03C3;j)
+    i,j
+" class="math-display" ></center></td><td class="equation-label">(4)</td></tr></table>
+        <!--l. 236--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-3005r5"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/7/70/T--MIT--MITMorpheqs27x.png" alt="&#x03B4;&#x03C3;i&#x03C3;j = {1,&#x03C3;i = &#x03C3;j;0,&#x03C3;i &#x2044;= &#x03C3;j}
+" class="math-display" ></center></td><td class="equation-label">(5)</td></tr></table>
+        <!--l. 240--><p class="nopar" >
+        <!--l. 242--><p> </p>Morpheus is based on the Cellular Potts Model, which defines cells as spaces in a two
+        or three dimensional lattice (our model used a hexagonal lattice in 2d) and
+        determines changes to cell shape and size by using the Hamiltonian (Equations
+-4). Equation 2 determines changes to the volume of a cell <span
+                class="grmn-1000">&#x03C3; </span>with current
+        volume v<span
+                class="grmn-1000">&#x03C3; </span>in lattice sites and intended volume <span
+                class="cmmi-10">V</span> <sub><span
+                class="cmmi-7">t</span></sub> where <span
+                class="cmmi-10">&#x03BB;</span><sub><span
+                class="cmmi-7">V</span> </sub> is a constant
+        parameter of elasticity governing the extent to which the difference between
+        the cell&#8217;s immediate and intended volume contributes to a rise in its free
+        energy <span
+                class="ecti-1000">H</span>. Equation 3 incorporates a similar system for cell perimeter into the
+        equation.
+        <!--l. 254--><p> </p> In addition to modelling adjustments to cells&#8217; shape and size as they migrate, the
+        Cellular Potts Model also uses the Hamiltonian to model the interactions between
+        cells. Equation 4 determines the interaction energies between different cell types
+        where <span
+                class="cmmi-10">&#x03C4;</span><span
+                class="cmr-10">(</span><span
+                class="cmmi-10">&#x03C3;</span><sub><span
+                class="cmmi-7">i</span></sub><span
+                class="cmmi-10">,&#x03C3;</span><sub><span
+                class="cmmi-7">j</span></sub><span
+                class="cmr-10">) </span>represents the cell types of two cells <span
+                class="cmmi-10">&#x03C3;</span><sub><span
+                class="cmmi-7">i</span></sub> and <span
+                class="cmmi-10">&#x03C3;</span><sub><span
+                class="cmmi-7">j</span></sub> and <span
+                class="ecti-1000">J </span>specifies
+        said energies in matrix form. In order to prevent cells from returning values
+        for interactions with themselves, the term known as the Kronecker Delta
+        defined in Equation 5 is included. Each update to the configuration of cells in
+        the lattice as a result of equations 2-4 only occurs with a certain likelihood
+        governed by the Boltzmann probability in Equation 1. This equation states
+        that the cell&#8217;s chance of changing state is 100% if its change in energy <span
+                class="grmn-1000">&#x0394;</span><span
+                class="ecti-1000">H</span>
+        added to its resistance to change <span
+                class="ecti-1000">Y </span>is favorable (<span
+                class="grmn-1000">&#x0394;</span><span
+                class="ecti-1000">H</span>+<span
+                class="ecti-1000">Y </span>&#x003C; 0) and decreases
+        exponentially with a rate of <img
+                src="https://static.igem.org/mediawiki/2018/9/9d/T--MIT--MITMorpheqs28x.png" alt="1T-"  class="frac" align="middle"> where <span
+                class="ecti-1000">T </span>represents the amount of unfavorable
+        updates to the cell lattice, defined as modifications to the cell&#8217;s perimeter or
+        volume.
+        <p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">2.2   </span> <a
+                id="x1-40002.2"></a>Diffusion and the Cell Lattice Space</h4>
+        <table
+                class="equation"><tr><td><a
+                id="x1-4001r6"></a>
+            <center class="math-display" >
+                <img
+src="https://static.igem.org/mediawiki/2018/f/f4/T--MIT--MITMorpheqs29x.png" alt="-&#x2202;c    -&#x2202;2c
+&#x2202;T = D &#x2202;X2
+" class="math-display" ></center></td><td class="equation-label">(6)</td></tr></table>
+        <!--l. 277--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4002r7"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/b/b9/T--MIT--MITMorpheqs30x.png" alt="    X-    T-
+x = L ,t = &#x03C4;
+" class="math-display" ></center></td><td class="equation-label">(7)</td></tr></table>
+        <!--l. 281--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4003r8"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/4/4d/T--MIT--MITMorpheqs31x.png" alt="&#x2202;c   &#x03C4;D-&#x2202;2c-
+&#x2202;t = L2 &#x2202;x2
+" class="math-display" ></center></td><td class="equation-label">(8)</td></tr></table>
+        <!--l. 285--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4004r9"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/fb/T--MIT--MITMorpheqs32x.png" alt="c(x,t) &#xE306; ci,j &#x2261; c(xi,tj),xi &#x2261; i&#x03B4;x,tj &#x2261; j&#x03B4;t
+" class="math-display" ></center></td><td class="equation-label">(9)</td></tr></table>
+        <!--l. 289--><p class="nopar" >
+        <!--l. 291--><p class="noindent" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4005r10"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/2/21/T--MIT--MITMorpheqs33x.png" alt="&#x2202;c-|xi,tj&#xE306; ci+1,j --ci--1,j= ci+1,j---ci-1,j-
+&#x2202;x         xi - xi-1        2&#x03B4;x
+" class="math-display" ></center></td><td class="equation-label">(10)</td></tr></table>
+        <!--l. 295--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4006r11"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/b/b4/T--MIT--MITMorpheqs34x.png" alt="         &#x2202;c       &#x2202;c
+&#x2202;2c-| &#xE306; -&#x2202;x |xi+-12 --&#x2202;x |xi--12= ci+1,j---2ci,j-+ci-1,j
+&#x2202;x2  xi     xi+12 - xi- 12            (&#x03B4;x)2
+" class="math-display" ></center></td><td class="equation-label">(11)</td></tr></table>
+        <!--l. 299--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4007r12"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/f8/T--MIT--MITMorpheqs35x.png" alt="c    = c  + -&#x03B4;t--(c    - 2c  + c    )
+ i,j+1    i,j  (&#x03B4;x)2 i+1,j    i,j   i-1,j
+" class="math-display" ></center></td><td class="equation-label">(12)</td></tr></table>
+        <!--l. 303--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4008r13"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/a/af/T--MIT--MITMorpheqs36x.png" alt="(  2     2 )
+  &#x2202;-c+ &#x2202;-c   &#xE306; ci+1,j --2ci,j-+-ci-1,j-+ ck+1,j --2ck,j +-ck-1,j
+  &#x2202;x2  &#x2202;y2           (&#x03B4;x)2                (&#x03B4;y)2
+" class="math-display" ></center></td><td class="equation-label">(13)</td></tr></table>
+        <!--l. 307--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4009r14"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/6/63/T--MIT--MITMorpheqs37x.png" alt="ci,k,j+1 = ci,j +-&#x03B4;t2(ci+1,j - 2ci,j +ci-1,j)+ --&#x03B4;t2(ck+1,j - 2ck,j + ck-1,j)
+             (&#x03B4;x)                      (&#x03B4;y)
+" class="math-display" ></center></td><td class="equation-label">(14)</td></tr></table>
+        <!--l. 311--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-4010r15"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/0/04/T--MIT--MITMorpheqs38x.png" alt="        4  &#x2211;&#x221E;    sin[(n-+-1)&#x03B8;]sin[(m-+-1)&#x03B8;]--sin-(n&#x03B8;)sin[(m---1)&#x03B8;]
+h2(2s) = 3                  [(n+ 12m )2 + 3(12m)2]s
+         m,n=-&#x221E;
+" class="math-display" ></center></td><td class="equation-label">(15)</td></tr></table>
+        <!--l. 315--><p class="nopar" >
+        <!--l. 317--><p> </p>   Another important component of our model was treating sugar, CSP, and our
+        potential outputs not simply as cell-associated values but as scalar fields representing
+        the concentration of said molecules. Fortunately Morpheus also incorporates the
+        ability to overlay fields like these on simulated cell populations and allows the field to
+        be updated locally based on cell activity at a specific lattice site (cells can also report
+        on fields or other cells surrounding them, a function which is explained in the
+        next section). Diffusion Fields were evaluated using the Central Difference
+        Method to solve the 2-D Diffusion Equation (Equations 6-14). Equation 6 is a
+        differential equation modelling diffusion in one dimension where the concentration
+        c is a function of the X-coordinate and time T. D signifies the diffusion
+        coefficient and L represents the length between nodes in (X, T) space where
+        the domain of solutions is 0<span
+                class="cmsy-10">&#x2264;</span>X<span
+                class="cmsy-10">&#x2264;</span>L. By making the change of variables in
+        Equation 7, the Diffusion Equation can be rewritten in the form of Equation
+.
+        <!--l. 334--><p class="indent" >
+        <video src="https://static.igem.org/mediawiki/2018/7/7c/T--MIT--MITvid1.mp4" controls muted replay autoplay style="width: 30vw; position: relative; left: 32vw;"></video>
+        <p> </p>Next, a set of approximations are made in order to solve for the first and second
+        derivatives of concentration with respect to x (Equation 9). The central difference
+        method is a version of the finite difference method of approximations to solve the
+        Diffusion Equation, which assumes that there is a minimum distance in both
+        dimensions, <span
+                class="cmmi-10">&#x03B4;</span>x and <span
+                class="cmmi-10">&#x03B4;</span>t, for which the concentration changes. This creates
+        a grid in (x, t) space as shown in Figure . The central difference method
+        approximates the x-derivative of concentration based on concentration values on
+        either side of a point (i, j), one for each of the smallest change in x from
+        the starting point: (i+1, j) and (i-1, j) (Equation 10). This is more precise
+        than the forwards or backwards difference methods which only take into
+        account one direction of incrementation in x. Next, the second derivative of
+        concentration with respect to x is approximated using the values of the first
+        derivative on a range half as large as in the previous approximation (Equation
+).
+        <!--l. 350--><p> </p>Using these results, an approximation of the concentration at a point 1 time step
+        in the future can be obtained based on the concentrations at the surrounding points
+        at the initial time (Equation 12), which can be appended with terms based on the
+        approximations for the first and second derivatives in Equations 10 and 11 in order to
+        obtain a more accurate approximation. Of course, in our model this was done in two
+        dimensions, but the steps for the second dimension y are the same as those above and
+        can be combined into Equations 13 and 14 in order to obtain values for a 2-D
+        diffusion model.
+        <!--l. 360--><p> </p>  In order to sense nearby concentrations or cells in Morpheus, each cell can take
+        advantage of the NeighborhoodReporter function, which maps values within a
+        specified node length from a cell to an average, variance, or sum specific to that cell.
+        In our model, cells employed the sum function of the NeighborhoodReporter to
+        interact with the various diffusion fields. Equation 15 is the equation for a lattice sum
+        in a 2-dimensional hexagonal lattice, which Morpheus approximates to write field
+        values to individual cells.
+        <!--l. 369--><p class="noindent" >
+    </div>
+    <button class="accordion">3. Model Specifics</button>
+    <div class="panel">
+        <p> </p>The first objective of accurately modelling a bacterial population was to model
+            population growth over time. We chose to define our time step in the model as one
+            second, and used the canonical doubling time of around 3800 seconds to
+            determine a probability of cell division at each time step. Next, in order to reflect
+            the constantly shifting environment of saliva on teeth, we programmed the
+            cells with random motion defined within a realistic range for cell velocity in
+            the salivary microbiome. This ensured that the cells would interact so that
+            they would be affected by the modified adhesion associated with biofilm
+            initiation.
+            <!--l. 382--><p class="noindent" >
+        <image src="https://static.igem.org/mediawiki/2018/4/4f/T--MIT--MITpic1.png" style="width: 60%; position: relative; left: 20vw;">
+        <h4 class="subsectionHead"><span class="titlemark">3.1   </span> <a
+                id="x1-60003.1"></a>Transcription and Translation of the Relevant Genes</h4>
+        <table
+                class="equation"><tr><td><a
+                id="x1-6001r16"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/c/c1/T--MIT--MITMorpheqs39x.png" alt="dmCSP-- = ktx &#x00D7; pComC + ktf &#x00D7; ComEP  &#x00D7; pComC  - &#x03B4;mCSP &#x00D7; mCSP
+   dt
+" class="math-display" ></center></td><td class="equation-label">(16)</td></tr></table>
+        <!--l. 386--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6002r17"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/b/bd/T--MIT--MITMorpheqs40x.png" alt="dmComD---= ktx &#x00D7; pComD - &#x03B4;mComD &#x00D7; mComD
+   dt
+" class="math-display" ></center></td><td class="equation-label">(17)</td></tr></table>
+        <!--l. 390--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6003r18"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/4/46/T--MIT--MITMorpheqs41x.png" alt="dmComE---= ktx &#x00D7; pComE - &#x03B4;mComE &#x00D7; mComE
+   dt
+" class="math-display" ></center></td><td class="equation-label">(18)</td></tr></table>
+        <!--l. 394--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6004r19"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/c/c5/T--MIT--MITMorpheqs42x.png" alt="dmGT--FC-= k  &#x00D7; ComEP   &#x00D7; pGTF C - &#x03B4;      &#x00D7; mGT  FC
+   dt       tf                      mGTFC
+" class="math-display" ></center></td><td class="equation-label">(19)</td></tr></table>
+        <!--l. 398--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6005r20"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/f3/T--MIT--MITMorpheqs43x.png" alt="dCSPin- = &#x03B1;    &#x00D7; mCSP  - &#x03B4;    &#x00D7; CSP
+   dt      CSP            CSP       in
+" class="math-display" ></center></td><td class="equation-label">(20)</td></tr></table>
+        <!--l. 402--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6006r21"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/b/b6/T--MIT--MITMorpheqs44x.png" alt="dComD--= &#x03B1;ComD &#x00D7;mComD   - &#x03B4;ComD&#x00D7;ComD+kub   &#x00D7;CSP ComDP   - kb&#x00D7;CSPout &#x00D7;ComD
+   dt
+" class="math-display" ></center></td><td class="equation-label">(21)</td></tr></table>
+        <!--l. 406--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6007r22"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/1/1f/T--MIT--MITMorpheqs45x.png" alt="dComE--= &#x03B1;ComE &#x00D7;mComE   - &#x03B4;ComE &#x00D7;ComE - kk&#x00D7;CSP ComDP  &#x00D7;ComE+ktf  &#x00D7;pComC  &#x00D7;ComEP   +ktf&#x00D7;pGT F C&#x00D7;ComEP
+  dt
+" class="math-display" ></center></td><td class="equation-label">(22)</td></tr></table>
+        <!--l. 410--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-6008r23"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/e/e3/T--MIT--MITMorpheqs46x.png" alt="dGT-FC- = &#x03B1;GTFC &#x00D7; mGT F C - &#x03B4;GTFC &#x00D7; GT FC
+   dt
+" class="math-display" ></center></td><td class="equation-label">(23)</td></tr></table>
+        <!--l. 414--><p class="nopar" >
+        <!--l. 416--><p> </p>The biggest hurdle in creating an accurate model of biofilm formation in S. mutans
+        was simulating the two-component signaling system known as ComCDE. We began
+        by creating differential equations for the transcription and translation of the ComC,
+        ComD, and ComE genes based on transcription rates, translation rates and decay
+        rates for mRNAs and proteins from the literature. We also researched the genome
+        size and operon structure of the bacteria in order to better reflect its production of
+        the relevant proteins in our model. Equations 16-18 represent the transcription of
+        mRNAs for the three genes we chose to study, and the first four terms in each of
+        Equations 20-22 represent the translation of those mRNAs. All terms and
+        constants are defined in the table at the beginning of this section. Now that we
+        had the cells generating values for these variables at each time step, we
+        could add terms to model the kinetics of the two-component sensing system
+        itself.
+        <!--l. 432--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">3.2   </span> <a
+                id="x1-70003.2"></a>Two-Component Sensing System: Ligand Binding, Response Regulator
+            Phosphorylation and DNA Binding</h4>
+        <table
+                class="equation"><tr><td><a
+                id="x1-7001r24"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/1/17/T--MIT--MITMorpheqs47x.png" alt="dCSPout
+---dt---= kub&#x00D7;CSP ComDP   - kb&#x00D7;CSPout &#x00D7;ComD+kk &#x00D7;CSP  ComDP  &#x00D7;ComE
+" class="math-display" ></center></td><td class="equation-label">(24)</td></tr></table>
+        <!--l. 436--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-7002r25"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/5/52/T--MIT--MITMorpheqs48x.png" alt="dCSP-ComD--
+     dt     = kb&#x00D7;CSPout&#x00D7;ComD   - kub&#x00D7;CSP ComDP  - kk&#x00D7;CSP ComDP  &#x00D7;ComE  - &#x03B4;CSP ComDP &#x00D7;CSP comDP
+" class="math-display" ></center></td><td class="equation-label">(25)</td></tr></table>
+        <!--l. 440--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-7003r26"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/c/c5/T--MIT--MITMorpheqs49x.png" alt="dComEP---= kk&#x00D7;CSP  ComD &#x00D7;ComEP   - ktf&#x00D7;pComC &#x00D7;ComEP  - ktf&#x00D7;pGT FC &#x00D7;ComEP  - &#x03B4;ComEP&#x00D7;ComEP
+   dt
+" class="math-display" ></center></td><td class="equation-label">(26)</td></tr></table>
+        <!--l. 444--><p class="nopar" >
+        <!--l. 446--><p> </p>We used the Law of Mass Action to create differential equations reflecting the
+        kinetics of ComD binding its ligand CSP, the autophosphorylation of ComD, the
+        phosphorylation of ComE by ComD, and finally the DNA binding and transcription
+        factor activity of phosphorylated ComE. For a full explanation of the way the
+        two-component signalling system works in live bacteria, please refer to our project
+        page, as this section assumes knowledge of the relevant components and their
+        functions. The first step in the signalling system is the binding of CSP to
+        ComD. In order to initiate this activity, we used the NeighborhoodReporter
+        function to update the variable CSPout at each time step. ComD binds
+        CSP at the rate kb, at which point the two are considered one complex,
+        CSPComDP for autophosphorylated ComD:CSP, and the amount of CSP and
+        ComD decrease accordingly (Equations 21 and 24). CSPComDP has an
+        unbinding rate kub which decreases the amount of ComD:CSP complexes while
+        increasing both CSPout and ComD by the same amount. In order to simplify
+        the system of equations, the assumption was made that once CSPComDP
+        phosphorylates its response regulator ComE, the complex dissociates and
+        returns to separate ComD and CSPout. kk represents the rate of the complex
+        phosphorylating ComE (Equation 25). Once ComE is phosphorylated, it is considered
+        a different variable, ComEP for phosphorylated ComE. Phosphorylated ComE
+        upregulates the ComC and Glycosyltransferase genes with the rate ktf in order
+        to create a positive feedback loop within the population characteristic of
+        quorum-sensing systems and produce the necessary enzymes for biofilm formation.
+        Once again, the assumption was made that after ComEP binds DNA it is
+        dephosphorylated by available phosphatases and returns to its initial state,
+        hence the amount of ComEP decreases with the rate ktf for each gene it
+        upregulates and ComE increases by the same amount (Equations 22 and
+).
+        <!--l. 475--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">3.3   </span> <a
+                id="x1-80003.3"></a>Water-Insoluble Glucan Synthesis via Glucosyltransferase Activity</h4>
+        <table
+                class="equation"><tr><td><a
+                id="x1-8001r27"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/ff/T--MIT--MITMorpheqs50x.png" alt="dGlucan
+---dt---= ke &#x00D7; GT F C &#x00D7;Sugarcell - &#x03D5;Glucan &#x00D7; Glucan
+" class="math-display" ></center></td><td class="equation-label">(27)</td></tr></table>
+        <!--l. 479--><p class="nopar" >
+        <!--l. 481--><p> </p>Biofilm formation in the model depends on two main factors: Sugar available to the
+        cell and the amount of Glycosyltransferase enzymes a cell possesses (GTFC). mRNAs
+        for GTFC (Equation 19) are only transcribed when ComEP binds the gene to
+        upregulate it, and GTFC proteins are translated according to Equation 23.
+        Sugar concentration (<span
+                class="cmmi-10">Sugar</span><sub><span
+                class="cmmi-7">field</span></sub>) is given an initial value as a diffusion field
+        in the model, which was defined as a homogeneous diffusion field due to
+        sugar&#8217;s high solubility in saliva. <span
+                class="cmmi-10">Sugar</span><sub><span
+                class="cmmi-7">cell</span></sub> is defined at each time step via the
+        NeighborhoodReporter function based on the current concentration of the
+        <span
+                class="cmmi-10">Sugar</span><sub><span
+                class="cmmi-7">cell</span></sub>.
+        <!--l. 492--><p> </p> The final differential equation based on the Law of Mass Action in the most basic
+        form of the model, Equation 27, governs the production of water-insoluble
+        glucans from available sugar. The enzymatic activity of GTFC is denoted
+        ke, and the glucans become the basis of the extracellular biofilm by being
+        incorporated into the extracellular field of the same name at the rate <span
+                class="cmmi-10">&#x03D5;</span><sub><span
+                class="cmmi-7">Glucan</span></sub>. In
+        order to simulate the transition of S. mutans from free-floating planktonic
+        cells to biofilm-bound immobilized cells which grow in localized colonies,
+        each cell was programmed to also use the NeighborhoodReporter to return
+        a constantly updated value for how concentrated biofilm-forming glucans
+        are around that cell. Once the local biofilm reaches a threshold value of
+arbitrary units, cells are considered to be adhered and their movement
+        ceases.
+        <!--l. 505--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">3.4   </span> <a
+                id="x1-90003.4"></a>Extracellular Diffusion Fields</h4>
+        <table
+                class="equation"><tr><td><a
+                id="x1-9001r28"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/c/cb/T--MIT--MITMorpheqs51x.png" alt="dCSP
+--dt-- = &#x03D5;CSP &#x00D7;CSPin
+" class="math-display" ></center></td><td class="equation-label">(28)</td></tr></table>
+        <!--l. 509--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-9002r29"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/1/1a/T--MIT--MITMorpheqs52x.png" alt="dBiofilm
+---------= &#x03D5;Glucan &#x00D7;Glucan
+   dt
+" class="math-display" ></center></td><td class="equation-label">(29)</td></tr></table>
+        <!--l. 513--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-9003r30"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/4/4c/T--MIT--MITMorpheqs53x.png" alt="dSugarfield
+----dt----= - ke &#x00D7; GT FC &#x00D7; Sugarcell
+" class="math-display" ></center></td><td class="equation-label">(30)</td></tr></table>
+        <!--l. 517--><p class="nopar" >
+        <!--l. 519--><p> </p>Equations 28-30 are evaluated at each time step at each point in the lattice in order
+        to affect the changes individual cells make to field values. Both the biofilm and CSP
+        field equations increase at an export rate times the number of proteins a cell at a
+        given point has, and the Sugar decreases as GTFC transforms it into glucans. Sugar
+        undergoes homogeneous or well-mixed diffusion, CSP was given a diffusion
+        constant of <span
+                class="ecti-1000">D </span>= 1, and the Biofilm was given the diffusion constant <span
+                class="ecti-1000">D </span>=
+.001, both in as employed in Equation 6 and its subsequent approximate
+        solutions.
+        <!--l. 529--><p class="noindent" >
+       </div>
+    <button class="accordion">4. Results</button>
+    <div class="panel">
+        <p> </p>Our model proved quite successful in qualitatively reproducing multiple aspects of
+        the formation of live <span
+                class="ecti-1000">S. mutans </span>biofilms. The simulated cells form clear microcolonies
+        in the locations where glucan concentration is the highest which expand
+        from there. CSP increases everywhere throughout the simulation as the
+        time steps modelled only encompass the phase of growth when that is the
+        case.
+        <!--l. 537--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">4.1   </span> <a
+                id="x1-110004.1"></a>Effect of Sugar Concentration on Biofilm Formation</h4>
+        <img src="https://static.igem.org/mediawiki/2018/0/07/T--MIT--MITpic2.png">
+        <!--l. 539--><p> </p>The first parameter we varied was sugar by modifying the initial value of the
+        diffusion field. After fifteen thousand time steps, there is a clear difference in biofilm
+        density in plots of the cells themselves. Graphs of data collected from a
+        representative cell in each condition show a clear distinction in glucan production
+        between sugar concentrations. For the time period simulated, limiting sugar
+        concentration was between 10 and 25 arbitrary units, at which point the cells were
+        unable to generate enough glucans to adhere and create microcolonies in a
+        biofilm. These results reflect our experiments with live <span
+                class="ecti-1000">S. mutans </span>in which
+        we varied sucrose concentration and measured its effect on biofilm growth
+        through image analysis of of colony-forming units as well as crystal violet
+        staining. As expected based on our differential equations, the concentration
+        of sugar decreases exponentially over time, whereas the total amount of
+        glucans produced increases exponentially over time. The rates of exponential
+        growth and decay also depend on the sugar available, with the difference
+        in glucans produced by the end of the simulation being around one order
+        of magnitude less than the difference in sugar available in all cases. This
+        data was useful in helping us match the arbitrary units in the model to real
+        experiments.
+        <!--l. 559--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">4.2   </span> <a
+                id="x1-120004.2"></a>Inhibition of GTFC Activity by Single-Chain Variable Fragments</h4>
+        <!--l. 561--><p class="noindent" >
+        <h5 class="subsubsectionHead"><span class="titlemark">4.2.1   </span> <a
+                id="x1-130004.2.1"></a>Additional Cellular Equations Governing ScFv Activity</h5>
+        <table
+                class="equation"><tr><td><a
+                id="x1-13001r31"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/8/89/T--MIT--MITMorpheqs54x.png" alt="dGT-F-CScF-v = kab &#x00D7; ScFvcell &#x00D7; GT FC - &#x03B4;GTFCScFv &#x00D7; GT FCScF v
+     dt
+" class="math-display" ></center></td><td class="equation-label">(31)</td></tr></table>
+        <!--l. 565--><p class="nopar" >
+        <table
+                class="equation"><tr><td><a
+                id="x1-13002r32"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/f0/T--MIT--MITMorpheqs55x.png" alt="dScF-vcell= - k  &#x00D7;ScF v   &#x00D7; GT FC - &#x03B4;      &#x00D7; ScF v
+   dt         ab       cell           ScFvcell       cell
+" class="math-display" ></center></td><td class="equation-label">(32)</td></tr></table>
+        <!--l. 569--><p class="nopar" >
+        <!--l. 572--><p class="noindent" >
+        <h5 class="subsubsectionHead"><span class="titlemark">4.2.2   </span> <a
+                id="x1-140004.2.2"></a>Additional Field Equations Governing ScFv Activity</h5>
+        <table
+                class="equation"><tr><td><a
+                id="x1-14001r33"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/9/93/T--MIT--MITMorpheqs56x.png" alt="dScF-vfield-= (- k &#x00D7; ScF v   &#x00D7;GT F C - &#x03B4;      &#x00D7; ScFv   ) ÷100
+    dt         ab       cell          ScFvcell      cell
+" class="math-display" ></center></td><td class="equation-label">(33)</td></tr></table>
+        <!--l. 576--><p class="nopar" >
+        <!--l. 579--><p class="noindent" >
+        <h5 class="subsubsectionHead"><span class="titlemark">4.2.3   </span> <a
+                id="x1-150004.2.3"></a>Modified Equations Integating ScFv Activity</h5>
+        <table
+                class="equation"><tr><td><a
+                id="x1-15001r34"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/5/5a/T--MIT--MITMorpheqs57x.png" alt="dGT FC
+--dt---= &#x03B1;GT FC &#x00D7; mGT F C - &#x03B4;GT FC &#x00D7;GT F C - kab &#x00D7; ScFvcell &#x00D7; GT FC
+" class="math-display" ></center></td><td class="equation-label">(34)</td></tr></table>
+        <!--l. 583--><p class="nopar" >
+        <img src="https://static.igem.org/mediawiki/2018/1/1a/T--MIT--MITpic3.png">
+        <!--l. 585--><p> </p>The primary purpose of our model was to preemptively compare the effectiveness of
+        our potential outputs by modelling their inhibition of different parts of the pathway
+        and its effects on overall biofilm growth. We began by implementing the above
+        differential equations (Equations 31-33), and modifying Equation 23 into equation 34.
+        The amount of ScFvs affecting the cell is determined via the NeighborhoodReporter
+        function, and the ScFvs bind GTFC at the rate kab. Once the ScFv binds a GTFC, it
+        forms a GTFC:ScFv complex represented in Equation 31, decreasing both
+        the cell&#8217;s GTFC and ScFv count accordingly (Equations 32 and 34). The
+        complex degrades quickly based on a rate from the literature, and once it has
+        degraded both the ScFv and GTFC bound have been eliminated from the
+        simulation.
+        <!--l. 598--><p> </p> The effects of the ScFv on biofilm formation are immediately obvious from plots
+        of the cells at the end of the simulation, with only 0.075 arbitrary units of
+        concentration to start with being enough to prevent the bacteria from forming
+        adherent microcolonies throughout the time modelled. The critical point
+        at which the amount of GTFC: ScFv complexes goes from increasing to
+        decreasing determines how soon the cells are able to begin initiating biofilm
+        formation, assuming they are still in the logarithmic growth phase where CSP
+        production is increasing. Since GTFC production is decreased, everything
+        downstream of it is also affected: cells consume less sugar, produce fewer
+        glucans, and create less of a biofilm field in the presence of ScFvs, as shown
+        in the following figures. In equation 33, the decrease of the ScFv field is
+        divided by 100 due to the membrane resolution being set to 100 for the
+        simulation.
+        <!--l. 613--><p class="noindent" >
+        <h4 class="subsectionHead"><span class="titlemark">4.3   </span> <a
+                id="x1-160004.3"></a>Inhibition of Glucan Activity by Kappa-Casein</h4>
+        <!--l. 615--><p class="noindent" >
+        <h5 class="subsubsectionHead"><span class="titlemark">4.3.1   </span> <a
+                id="x1-170004.3.1"></a>Additional Cellular Equations Governing K-Casein Activity</h5>
+        <table
+                class="equation"><tr><td><a
+                id="x1-17001r35"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/f9/T--MIT--MITMorpheqs58x.png" alt="dKCaseincell
+-----dt-----= - kkc &#x00D7; KCaseincell &#x00D7; Glucan - &#x03B4;KCaseincell &#x00D7; KCaseincell
+" class="math-display" ></center></td><td class="equation-label">(35)</td></tr></table>
+        <!--l. 619--><p class="nopar" >
+        <!--l. 622--><p class="noindent" >
+        <h5 class="subsubsectionHead"><span class="titlemark">4.3.2   </span> <a
+                id="x1-180004.3.2"></a>Additional Fiels Equations Governing K-Casein Activity</h5>
+        <table
+                class="equation"><tr><td><a
+                id="x1-18001r36"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/2/2c/T--MIT--MITMorpheqs59x.png" alt="dKCaseinfield
+-----dt------= (- kkc&#x00D7;KCaseincell&#x00D7;Glucan- &#x03B4;KCaseincell&#x00D7;KCaseincell)÷100
+" class="math-display" ></center></td><td class="equation-label">(36)</td></tr></table>
+        <!--l. 626--><p class="nopar" >
+        <!--l. 629--><p class="noindent" >
+        <h5 class="subsubsectionHead"><span class="titlemark">4.3.3   </span> <a
+                id="x1-190004.3.3"></a>Modified Equations Integating K-Casein Activity</h5>
+        <table
+                class="equation"><tr><td><a
+                id="x1-19001r37"></a>
+            <center class="math-display" >
+                <img
+                        src="https://static.igem.org/mediawiki/2018/f/f7/T--MIT--MITMorpheqs60x.png" alt="dGlucan-= ke&#x00D7;GT F C&#x00D7;Sugarcell- &#x03D5;Glucan&#x00D7;Glucan - kkc&#x00D7;KCaseincell&#x00D7;Glucan
+   dt
+" class="math-display" ></center></td><td class="equation-label">(37)</td></tr></table>
+        <!--l. 633--><p class="nopar" >
+        <img src="https://static.igem.org/mediawiki/2018/0/08/T--MIT--MITpic4.png">
+        <!--l. 635--><p> </p>Next, we sought to create a similar set of equations to model the effect of
+        Kappa-Casein on biofilm formation. Equation 35 governs the effect of K-Casein on
+        an individual cell. Cells use the NeighborhoodReporter to determine the
+        amount of K-Casein molecules affecting it, and those molecules bind glucans
+        at a rate of kkc. Rather than creating a new variable for the complex of
+        K-Casein and a glucan, both are removed from the simulation immediately.
+        Due to being outside the cell, the binding rate of K-Casein was set to 1.
+        Equation 37 is a modification of equation 27 integrating the decrease in
+        glucans.
+        <!--l. 645--><p> </p> It became clear from varying the initial K-Casein concentration that the
+        equations were successful in modelling the decrease in biofilm formation due to a
+        decrease in effective adherent glucans surrounding the cells. However, despite the
+        K-Casein being given a dramatically higher binding rate than the ScFvs, a much
+        higher concentration of K-Casein was required to limit the formation of a biofilm
+        during the time simulated. In fact, ten times as high a concentration of
+        K-Casein was required to achieve the same effects as ScFvs. The K-Casein
+        also affected fewer aspects of the overall system of equations because its
+        influence was only effective on one of the most downstream components of the
+        simulation.
+        <!--l. 657--><p class="noindent" >
+        </div>
+    <button class="accordion">5. Conclusions</button>
+    <div class="panel">
+       <p> </p>The most obvious takeaway from the results of our computational model was that
+            ScFvs were more effective than K-Casein at inhibiting biofilm formation, or, more
+            broadly, inhibiting GTFC is a better method for limiting the virulence of S. mutans
+            than reducing the glucans themselves. This result had a profound effect on our
+            experimental design and our project as a whole. While Kappa-Casein can be readily
+            purchased and implemented in live experiments, the ScFvs we planned to
+            express and implement were not available for purchase anywhere, would be
+            very expensive to get synthesized, and would require additional training
+            and lab techniques to isolate from mammalian cells expressing the protein
+            itself. However, thanks to data from the model, we were able to confirm the
+            superiority of the ScFvs for our purposes. Based on these results, we decided to
+            move further ahead with characterization experiments for GTFC-inhibiting
+            ScFvs.
+            <!--l. 674--><p class="noindent" >
+    </div>
 </div>
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