Team:Northwestern/Project/Modeling
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=='''Introduction / Objective'''== | =='''Introduction / Objective'''== | ||
- | + | The general objective of our experimental system is to perform the following functions: | |
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- | The general objective of our system is to perform the following functions: | + | |
*Differentiate protein production between the top and bottom layer of the biofilm based on a diffusion-gradient inducer. | *Differentiate protein production between the top and bottom layer of the biofilm based on a diffusion-gradient inducer. | ||
- | * | + | *Respond to varying Repressor production levels |
- | * | + | *Respond to varying IPTG Induction levels |
- | * | + | *Respond to varying Promoters |
- | * | + | *Respond to varying Ribosome Binding Sites |
- | * | + | *Respond to varying Copy Number Plasmids |
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+ | The objective of our model is to explore and characterize the system, and the effect of parameter modulation on the system. | ||
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 product-producing 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. | 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 product-producing 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. | ||
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=='''Model Development'''== | =='''Model Development'''== |
Revision as of 02:13, 28 October 2010
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Introduction / ObjectiveThe general objective of our experimental system is to perform the following functions:
The objective of our model is to explore and characterize the system, and the effect of parameter modulation on the system. 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 product-producing 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. Model DevelopmentThe following schematic summarizes the mathematical model we formulated to describe our system.
Variables
ConstantsRate Constants:
EquationsThe differential of the variables were found as follows:
AssumptionsIn order to determine the initial or steady state concentrations of the involved species and to determine the rate constants, the following assumptions were made:
First, Fick's Law of Diffusion was modeled through MATLAB. The diffusion constant used was 220um^2/s.[4] It was assumed that IPTG was not consumed nor degraded We also assumed that IPTG uptake was minor was compared to the concentration in the biofilm, and so that the external IPTG was determined solely by Fick's Law, not by internalization.
Initially, the following initial concentration values were assumed:
The rate constant values were assumed to be the following:
Results
Future ConsiderationsAlthough the this model, as shown, captures the topology of our engineered network, its predictive prowess can be improved by obtaining constants and parameters from empirical observations. One possible method is to characterize the system using GFP instead of CHS3 and fluorescence as an indicator of product generation. Possible future work:
References1. 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
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
DOMINIQUE MENGIN-LECREULX, BERNARD FLOURET, AND JEAN VAN HEIJENOORT* E.R. 245 du C.N.R.S., Institut de Biochimie, Universit' Paris-Sud, Orsay, 91405, France Received 9 February 1983/Accepted 15 March 1983 http://www.ncbi.nlm.nih.gov/pmc/articles/PMC217602/pdf/jbacter00247-0262.pdf
Philip S. Stewart Center for Biofilm Engineering and Department of Chemical Engineering, Montana State University–Bozeman, Bozeman, Montana, 59717-3980 http://www.ncbi.nlm.nih.gov/pmc/articles/PMC148055/pdf/0965.pdf
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/jbacter00335-0105.pdf
MATLAB m-file%IPTG PREDETERMINATION %finite difference method %diffusion equation (Fick's 2nd Law) %c=c0*(erfc*x/sqrt(2*D*t)) %D*(Ci+1-2C+Ci-1)/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/dt-1; %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(x-1,t)-C(x,t))/(dx^2); else%BULK C_Rate(x)=D*(C(x-1,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/dt-1 %.002*x=1; newC(:,counts)=C(:,t); counts=counts+1; end %% %visualize newC for t=1:1:tmax/dt/500 XX=0:dx:xmax-dx; 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=xmax-dx-100:-dx:0-100; 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 - N-Acetyl Glucosamine Si - Internal Substrate - N-Acetyl Glucosamine CC - Enzyme Substrate Complex P - Protein - Chitin %} dt=1; tmax=3600; dx=1; xmax=100; %General Array Structure: (Depth (um), Time (s)) %Depth = 0-100 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/jbacter00335-0105.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*(1-dper)*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/jbacter00247-0262.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/dt-1 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 |