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Thinking about pouring plates to make linear chemical gradients
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% Motivation: http://www.wormbook.org/wbg/articles/volume-18-number-3/making-linear-chemical-gradients-in-agar/ | |
% | |
% Question: how many days before can we get away with? | |
% | |
%Fick's equation is | |
% | |
% dc/dt =D d^2 C/ dx^2 where the d's are partials | |
% | |
% Since the petri dish is a solid boundary, we have neumann boundary conditions | |
% | |
% | |
% | |
% From resources like: | |
% http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_2_28_short.pdf | |
% | |
% http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf | |
% | |
% we know the general solution | |
% | |
% | |
% for the specific initial conditions f(x)=x*C0 / L, | |
% | |
% I figure that we only really care about x=0 because if we know u(x=0,t) we | |
% should know that u(x=L,t)=C0 - u(x=0,t). | |
% | |
% And I assume that the profile along x is roughly linear, although we could check. | |
L=9;%cm | |
C0=50; %mM | |
D=10^-5; %cm^2 / s | |
disp('Calculating the concentration at the far end of the plate x=0.'); | |
disp(['We assume a ' num2str(L) 'cm plate with an initial concentration gradient from 0 to ' num2str(C0) ' mM and D=' num2str(D) 'cm^2/s' ]); | |
t=0:3*60*60*24*7; %3 weeks | |
u=C0/2; | |
for n=1:2:1000 | |
u= u + ( 2*C0/(pi^2) )*(-2/(n^2))*exp(-D* t*(n*pi/L)^2 ); | |
end | |
figure; plot(t/(60*60*24),u); | |
xlabel('Days') | |
ylabel('Concentration at x=0 (mM)') | |
title('C_0=100 mM, 9 cm plate') | |
days=5; | |
disp(['After ' num2str(days) ... | |
'days the concentrations at the end point will be '... | |
num2str( u(round(days*60*60*24)) ) 'mM and ' ... | |
num2str(C0-u(round(days*60*60*24)) ) 'mM'] ) |
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