Introduction / Objective
We constructed a mathematical model to explore and characterize the operation as well as the modulation of our experimental system.
In particular, we wanted to investigate the effect of varying parameters that we could experimentally modulate on the system. Those parameters include:
 SpatialGradient of Inducer Concentration
 Repressor Concentration
 Ribosome Binding Site (Rate of Translation) for Productproducing Protein
In terms of our iGEM project, this model was employed to explore the effect of IPTG concentration and diffusion, lacI concentration (determined by the combination part of constitutive promoter, ribosome binding site, lacI gene, double terminator, lac promoter/operon), and the ribosome binding site (of the final productproducing enzyme) on the concentrations of all species involved  described in the following sections  and especially on Chitin Synthase and Chitin concentration and the corresponding rates.
Modeling
Overall Model
Using enzyme kinetics equations, we elected to mathematically simulate the following model:
Variables
 Iex: External Inducer, determined by diffusion through Fick's law (IPTG in our experiment)
 Iin: Internal Inducer (IPTG)
 Ii: Inducer bound to Repressor (IPTG bound to lacI)
 i: Repressor (lacI)
 Db: Repressorbound DNA (lacIbound DNA(CHS3) region in plasmid)
 Dunb: transcribeable or Repressorunbound DNA (lacIunbound DNA(CHS3))
 Re: mRNA for Enzyme (CHS3 mRNA)
 E: Enzyme (CHS3)
 S: Substrate (NAcetyl Glucosamine)
 C: Enzyme Substrate Complex (CHS3(NAcetylGlucosamine)Chitin or (NAG)n Complex)
 P: Protein Product (Chitin or (NAG)n+1)
Equations
The differential of the variables were found as follows:
 dIin = kIin*Iex  kIex*Iin + kIir*Ii  kIif*Iin*i
 dIi = kIif*Iin*i  kIir*Ii
 di = kIir*Ii  kIif*Iin*i + kDbr*Db  kDbf*i*Dunb
 dDb = kDbf*i*Dunb  kDbr*Db
 dDunb = kDbr*Db  kDbf*i*Dunb
 dRe = ktscribe*Dunb  kRdeg*Re
 dE = ktslate*Re  kEdeg*E + kCr*CC + kP*CC  kCf*E*Si
 dSi = ksin*So  kCf*E*Si + kCr*CC
 dCC = kCf*E*Si  kCr*CC  kP*CC
 dP = kP*CC  kPdeg*P
IPTG Diffusion
First, Fick's Law of Diffusion was modeled through MATLAB. The diffusion constant used was 220um^2/s.[4]
IPTG was sprayed at the top of the colony, which then diffuses as according to Fick's law.
The spatially different local IPTG concentration will then differentially induce downstream processes.
This distinction was necessary in our project in order to establish a Chitin layer on the top of the biofilm.
SemiEmpirical Variable/Constant Determination
Status: Under Development
The initial plan was to use lacIconstitutive expression / lacoperon (CPLacpI) part with Green Fluorescent Protein to acquire empirical data.
By testing various combinations of CP/LacpI, RBS, and IPTG concentrations, the acquisition of a broad range of expression level (GFP fluorescence) over time could be acquired through a plate reader.
This data would be used to determine many of the rate constants as well as initial concentration values, thus generating a more accurate semiempirical kinetics model.
However, at the time of the wikifreeze, data acquisition is incomplete.
Current Model
The plots of the noninduced and the induced system are as follows:
 Not Induced  Induced

External IPTG  

Internal IPTG  

IPTG bound Lac Repressor  

Free Lac Repressor  

Repressor Bound DNA  

Free DNA  

mRNA  

Chitin Synthase 3  

NAcetylGlucosamine  

CHS3NAG/Chitin Complex  

Chitin  

Future Concerns
Overall, the model, though logical, is not yet fit for application, partially due to the lack of empirical data to which the model could be fit and/or tested. Once fit, the rate constant values and initial concentration values will be much more accurate. The following concerns deal primarily with problems that could possibly not be corrected even with empirical data.
 Currently the model predicts the final product expression level between the top layer and bottom layer of a gradientinduced system as only about 0.00001%. Determine if this is primarily due to model inaccuracy or the inefficacy of diffusionbased layer differential induction method.
 Currently the model only predicts only a 10% increase in product when induced by what is suspected to be a significant concentration of inducer. This could be due to a model design problem.
 The model assumes constant substrate; this assumption may or may not be accurate.
 The model assumes all components are in first order; this assumption may or may not be sufficient.
Future Work
Aside from being generally accurate, the model should perform the following functions:
 Differentiate protein production between the top and bottom layer of the biofilm based on a diffusiongradient inducer.
 Differentiate protein production between expression vectors with different repressor production levels.
 Differentiate protein production between expression vectors with different IPTG Induction levels.
 Differentiate protein production between expression vectors with different Promoters (ktranscribe).
 Differentiate protein production between expression vectors with different Ribosome Binding Sites (ktranslate).
 Differentiate protein production between expression vectors with different Copy Number Plasmids.
References
1. A novel structured kinetic modeling approach for the analysis of plasmid instability in recombinant bacterial cultures
William E. Bentley, Dhinakar S. Kompala
Article first published online: 18 FEB 2004
DOI: 10.1002/bit.260330108
http://onlinelibrary.wiley.com/doi/10.1002/bit.260330108/pdf
2. Mathematical modeling of induced foreign protein production by recombinant bacteria
Jongdae Lee, W. Fred Ramirez
Article first published online: 19 FEB 2004
DOI: 10.1002/bit.260390608
http://onlinelibrary.wiley.com/doi/10.1002/bit.260390608/pdf
3. Pool Levels of UDP NAcetylglucosamine and UDP NAcetylglucosamineEnolpyruvate in Escherichia coli and Correlation with Peptidoglycan Synthesis
DOMINIQUE MENGINLECREULX, BERNARD FLOURET, AND JEAN VAN HEIJENOORT*
E.R. 245 du C.N.R.S., Institut de Biochimie, Universit' ParisSud, Orsay, 91405, France
Received 9 February 1983/Accepted 15 March 1983
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC217602/pdf/jbacter002470262.pdf
4. Diffusion in Bioﬁlms
Philip S. Stewart
Center for Bioﬁlm Engineering and Department of Chemical Engineering, Montana
State University–Bozeman, Bozeman, Montana, 597173980
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC148055/pdf/0965.pdf
5. Regulation of the Synthesis of the Lactose Repressor
PATRICIA L. EDELMANN' AND GORDON EDLIN
Department of Genetics, University of California, Davis, California 95616
Received for publication 21 March 1974
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC245824/pdf/jbacter003350105.pdf
MATLAB mfile
%IPTG PREDETERMINATION
%finite difference method
%diffusion equation (Fick's 2nd Law)
%c=c0*(erfc*x/sqrt(2*D*t))
%D*(Ci+12C+Ci1)/dx^2 = Ci/dt
%x goes from 0 (top) to 100 micrometers
%D=IPTG is a modified monosaccharide, so we can estimate from Table 1 that
%its diffusion coefficient in water at 25°C will be ca. 6.5 × 10?6 cm2 s?1.
%Scaling to 37°C and taking De/Daq to be 0.25, De is found to be 2.2 × 10?6 cm2 s?1
%http://www.ncbi.nlm.nih.gov/pmc/articles/PMC148055/
%%
clear
clc
%INITIALIZE
D=220; %um^2/s
c0=14.298; %mg so 3ml spray of 20mM iptg
%in comparison 23.83mg of iptg is in 5ml 20mM
dx=1; %um
xmax=100; %um; 100um total
dt=.002; %s
tmax=10; %s; 1 hour total
C=zeros(xmax/dx,tmax/dt); %rows = same time %also, (x,t) x is row, t is column
C_Rate=zeros(xmax/dt,1);
C(1,1)=c0;
%Time Step
for t=1:1:tmax/dt1; %s
for x=1:1:(xmax/dx) %um
if x==1%account for x=1 boundary condition
C_Rate(x)=D*(C(x+1,t)C(x,t))/(dx^2);
elseif x==xmax/dx %account for x=xmax boundary condition
C_Rate(x)=D*(C(x1,t)C(x,t))/(dx^2);
else%BULK
C_Rate(x)=D*(C(x1,t)+C(x+1,t)2*C(x,t))/(dx^2);
end
C(x,t+1)=C(x,t)+C_Rate(x)*dt;
end
t*.002
end
%%
%Compile into 1s parts
%newC=zeros(100,tmax/dt/500);
counts=1;
for t=1:500:tmax/dt1 %.002*x=1;
newC(:,counts)=C(:,t);
counts=counts+1;
end
%%
%visualize newC
for t=1:1:tmax/dt/500
XX=0:dx:xmaxdx;
YY=newC(:,t);
plot(XX,YY,'k.');
axis([0 100 0 2])
text(50,1.8,sprintf('Time is: %g s', t*dt));
text(50,1.7,sprintf('IPTG Mass balance is: %g mg', sum(C(:,t))));
xlabel('Distance from surface (um)')
ylabel('IPTG (mg)')
Mov(t)=getframe();
%pause(1);
end
%%
%VISUALIZE
blah=1;
for t=1:7:tmax/dt
XX=xmaxdx100:dx:0100;
YY=C(:,t);
plot(YY,XX,'k.');
axis([0 2 100 0])
text(1.2,30,sprintf('Time: %g s', t*dt));
%text(50,1.7,sprintf('IPTG Mass balance is: %g mg', sum(C(:,t))));
ylabel('Distance from surface (um)')
xlabel('IPTG (mg)')
VID(blah)=getframe(gcf);
blah=blah+1;
end
% Updated kinetics model
%{
Iex  external inducer or iptg
Iin  internal inducer or iptg
Ii  lac inhibitor / iptg complex
i  lac inhibitor
Db  DNA bound to inhibitor
Dunb  DNA not bound to inhibitor
R  mRNA
E  Enzyme Chitin Synthase
So  External Substrate  NAcetyl Glucosamine
Si  Internal Substrate  NAcetyl Glucosamine
CC  Enzyme Substrate Complex
P  Protein  Chitin
%}
dt=1;
tmax=3600;
dx=1;
xmax=100;
%General Array Structure: (Depth (um), Time (s))
%Depth = 0100 micrometers
%Time = 0 3600 seconds
preset=zeros(xmax,tmax);
Iin=preset;Ii=preset;i=preset;Db=preset;Dunb=preset;Re=preset;E=preset;Si=preset;CC=preset;P=preset;
%initial concentration
a=load('newC.mat'); %Iex is diff  assume cell intake << exist
Iex=zeros(100,3600);%a.newC();%
Iin(:,1)=0; %initially IPTG inside the cell is 0
Ii(:,1)=0; %no iptg, no Ii
i(:,1)=0.9973*10^4; %10^7 to 10^8 M which is 10^4 to 10^5 mM http://www.ncbi.nlm.nih.gov/pmc/articles/PMC245824/pdf/jbacter003350105.pdf
%DNA  partsregistry  200 copy number  200/(6.23*10^23)/(6*10^16
%volume)*1000 = 5.53*10^4mM
dper=0.9008;
Db(:,1)=5.53*dper*10^4; %say 90% is bound no idea
Dunb(:,1)=5.53*(1dper)*10^4; %say 10% unbound no idea
Re(:,1)=1.82735*10^4; %initially, no RNA
E(:,1)=6.08*10^4; %initially no enzyme
So=33.9; %1 pill per plate ~ 750mg/100ml /221.21g/mol /1000*1000*1000 is 33.9mM
Si(:,1)=10^1; %http://www.ncbi.nlm.nih.gov/pmc/articles/PMC217602/pdf/jbacter002470262.pdf
CC(:,1)=3*10^5; %initially 0
P(:,1)=1*10^4; %initially 0
%rate constants
kIin=.05;%safe to say diffuses within minutes
kIex=.05;%permeability must be similar
kIif=.01;
kIir=.01;
kDbf=100;
kDbr=.0011;
ktscribe=.01;
ktslate=.01;
kRdeg=.003;
kEdeg=.003;
kPdeg=.003;
ksin=0.000000009;
kCf=.01;
kCr=.01;
kP=.01;
%Begin Loop
%FILL IN THE BLANKS
%General Array Structure: (Depth (um), Time (s))
for tt=1:1:tmax/dt1
for xx=1:1:xmax/dx
%Predetermine diffusion
%dIex=DIF(xx,tt) + kIin*Iin(xx,tt)  kIex*Iex(xx,tt);
dIin=kIin*Iex(xx,tt) kIex*Iin(xx,tt) +kIir*Ii(xx,tt) kIif*Iin(xx,tt)*i(xx,tt);
dIi=kIif*Iin(xx,tt)*i(xx,tt) kIir*Ii(xx,tt);
di=kIir*Ii(xx,tt) kIif*Iin(xx,tt)*i(xx,tt) +kDbr*Db(xx,tt) kDbf*i(xx,tt)*Dunb(xx,tt);
dDb=kDbf*i(xx,tt)*Dunb(xx,tt) kDbr*Db(xx,tt);
dDunb=kDbr*Db(xx,tt) kDbf*i(xx,tt)*Dunb(xx,tt);
dRe=ktscribe*Dunb(xx,tt) kRdeg*Re(xx,tt);
dE=ktslate*Re(xx,tt) kEdeg*E(xx,tt) +kCr*CC(xx,tt) +kP*CC(xx,tt) kCf*E(xx,tt)*Si(xx,tt);
dSi=ksin*So kCf*E(xx,tt)*Si(xx,tt) +kCr*CC(xx,tt);
dCC=kCf*E(xx,tt)*Si(xx,tt) kCr*CC(xx,tt)  kP*CC(xx,tt);
dP=kP*CC(xx,tt) kPdeg*P(xx,tt);
%Iex(xx,tt+1)=Iex(xx,tt)+dIex*dt;
Iin(xx,tt+1)=Iin(xx,tt)+dIin*dt;
Ii(xx,tt+1)=Ii(xx,tt)+dIi*dt;
i(xx,tt+1)=i(xx,tt)+di*dt;
Db(xx,tt+1)=Db(xx,tt)+dDb*dt;
Dunb(xx,tt+1)=Dunb(xx,tt)+dDunb*dt;
Re(xx,tt+1)=Re(xx,tt)+dRe*dt;
E(xx,tt+1)=E(xx,tt)+dE*dt;
Si(xx,tt+1)=Si(xx,tt)+dSi*dt;
CC(xx,tt+1)=CC(xx,tt)+dCC*dt;
P(xx,tt+1)=P(xx,tt)+dP*dt;
end
tt*dt/tmax*100
end
