Team:Northwestern/Project/Modeling
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Introduction / ObjectiveWe 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:
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:
ResultsThe plots of the non-induced and the induced system are as follows:
Future ConcernsOverall, 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.
Future WorkAside from being generally accurate, the model should perform the following functions:
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 |