Question

**SOLVE THIS USING MATHLAB**

NOTE: Set the radioactive decay constant 'k' equal to 0.000100128

DIFFERENTIAL EQUATIONS 2. One of the applications of Differential Equations is Growth and Decay. Radioactive decay is an exponential process. If xo is the initial quantity of a radioactive substance at time t= 0, then the amount of that substance that will be present at any time t in the future is given by x(t) = Where 'k' is the radioactive decay constant. Create a script file that reads the half-life of the radioactive element remaining in a sample, calculates the age of the sample from it, and prints out the result with proper units, Xekt

Answer 1

Assume a reaction X k→ ∅ as the receptor disappears from the membrane, which leads to an equation   

dX dt = −kX

The solution to this equation is X(t) = X0e −kt where X0 is the initial concentration.

(This is easily checked by taking the time derivative of X(t).)

The kinetic parameter can be estimated by

e −kt = X(t) X0 = 0.05

k = − 1 t ln X(t) X0 = − 1 15 ln 0.05 = 0.2 min−1

This estimate could be improved further by fitting a curve X = X0e −kt to a dynamical measurment of the labeled ALK1 receptors.

Matlab Code :

yBase           = y_equil;

y               = y - y_equil;

fh_objective    = @(param) norm(param(2)+(param(3)-param(2))*exp(-param(1)*(x-x(1))) - y, 2);

initGuess(1)    = -(y(2)-y(1))/(x(2)-x(1))/(y(1)-y(end));

initGuess(2)    = y(end);

initGuess(3)    = y(1);

param           = fminsearch(fh_objective,initGuess);

k2              = param(1);

yInf            = param(2) + yBase;

y0              = param(3) + yBase;

yFit2           = yInf + (y0-yInf) * exp(-k2*(x-x(1)));

Output:

0.16 Data k = 0.030448 Initial Curve k = 0.019749 Adjusted Curve 0.155 0.15 0.145 0.14 Concentration 0.135 0.13 0.125 0.12 0.115 0.11 4900 5000 5100 5200 5300 5400 5500 5600 Time (s)