Project 1_1: Own ODE Solver & Cell dilution

Part A: Practice

1. ODE solver:

Practice Finished individually, encouraging to discuss with team member, use reference material, and software (Matlab etc) to illustrate your thinking and result. You can choose question 1 and 2, or 1 and 3. 1. Try to write the matlab code to solve the enzymatic reaction kinetics without ODE solver (ode45, or any other odesolver), but do the integration yourself. You can modify the week2-3 matlab code, mm.m, mmfunc.m. The time step for the difference equation needs to be chosen carefully. You should choose two time step sizes, and compare the results. $$[E]+[S]\rightleftharpoons[ES]\overset{k_2}{\rightarrow}[E]+[P]$$ After Simplifing Equations: $$\frac{\mathrm{d} [S]}{\mathrm{d} t}=k_{-1}[ES]-k_1([E_0-[ES])$$ $$\frac{\mathrm{d} [ES]}{\mathrm{d} t}=k_{-1}[ES]+k_1[E][S]+k_2[ES]$$

Euler Method

• 1st order Euler ODE Solver (10 step, 100 step, 1000 step, 100s) $$y_{n+1}\approx y_n+(x_{n+1}-x_n)f_n$$
• 2nd order Eular ODE Solver (10 step, 100 setp, 1000 step, 100s ,1000s) $$ y_(n+1)\approx y_n+\frac{x_{n+1}-x_n}{2}(f_n+f_{n+1}) $$

Results:

• The more step and the more close to reality. By low step, the stimulation is unstable.
• The longer time and the more steps are need,
• 2nd order are moer reasonable than 1st order odesolver.

2. Dilution of protein due to cell growth

a. A single bacterial cell at time $t=0$ has volume $V_0$. After a time interval $T_d$, the doubling time, the cell grows and divides into two cells, each of volume $V_0$; after another interval $T_d$, there are four cells and so on.

b. Show that the combined volume of cells at time t may be written as $$V(t)=V_0e^{(\gamma t)}$$ Find $\gamma$ in the terms of $T_d$. The protein X is created at some rate k(t), so that the total number of molecules of X satisfies $\frac{dn}{dt}=k(t)$ . Show the concentration $[X]=\frac{n}{V}$ satisfies $$\frac{d[X]}{dt} = k(t)V−\gamma[X]$$ Discuss the origin of the decay term.

c. In addition to the term derived in (b), there should be an extra term that takes into account the degradation of protein by proteinase, which we can model by the effective reaction X→δφ . Modify the equation in part (b) in include protein degradation.

proof:

$$ V(t)=V_0 2^{\frac{t}{T_d}}=V_0e^{\frac{In2t}{T_d}}=V_0e^{\gamma t} $$ $$\Rightarrow \gamma=\frac{In2}{T_d} $$ $$\frac{d[X]}{dt}=\frac{\frac{n}{V}}{dt}=\frac{Vdn}{V^2dt}+\frac{ndV}{V^2dt}=\frac{k(t)}{V}+\frac{nV_0\gamma}{V}=\frac{k(t)}{V}+\gamma[X]$$