Separation of Timescales

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Separation of Timescales is a common approach to simplify systems of differential equations, in order to find an approximate solution. To illustrate the idea and it's practical use we elaborate on the most popular example known, which is the Michaelis Menten Kinetics. In the related article separation of timescales entered as the quasi steady state assumption. It was argued that in the reaction \begin{equation} E+S \underset{k_{-1}}{\overset{k_1}{\rightleftarrows}} ES \overset{k_2}{\rightarrow}E+P \tag{1} \end{equation} the first reaction steps with rate constants $k_1,k_{-1}$ occur much faster than the second (rate constant $k_2$) \begin{equation} k_1,k_{-1}\gg k_2 \end{equation} In practice this means the first reaction step reaches equilibrium faster than the second. As an approximation we can therefore assume that fast reactions are at equilibrium when we consider slower timescales. In other words, we can separate timescales and assume that fast reactions are in equilibrium while considering reactions with slower dynamics. The concept becomes more clear, if we calculate the numerical solution for (1) without making the quasi steady state assumption and comparing it with Michaelis Menten Kinetics. We will solve the system \begin{equation} \frac{dS}{dt}=-k_1E\cdot S + k_{-1}ES \end{equation} \begin{equation} \frac{dES}{dt}=k_1E\cdot S-(k_{-1}+k_{2})ES \tag{2} \end{equation} \begin{equation} \frac{dE}{dt}=-k_1E\cdot S+(k_{-1}+k_{2})ES \tag{3} \end{equation} \begin{equation} \frac{dP}{dt}=k_2 ES \tag{4} \end{equation} for parameters $k_1=10^7s^{-1}$, $k_{-1}=45.5 \cdot 10^{3} s^{-1}$ and $k_2=4.5 \cdot 10^{3}$ and compare the results to Michaelis Menten Kinetics with parameters $v_{max} = 4.5 \cdot 3 \cdot 10^{-4} / 105000$ and $K_m = 5.0 \cdot 10^{−3}$. If we plot both, the numerical solution with (blue) and without (red) steady state assumption for the time period at the very beginning (up to $100 \mu s$) we get the following graph.


You see, that in the beginning the red curve needs a little time to reach the equilibrium (=reach the same slope as the blue curve), that is why it lags a little bit behind the blue line, where we already assumed steady state $\frac{dES}{dt}=0$ for the calculation. On the time scales we are interested in however, the error due to this approximation is negligible and it is reasonable to separate time scales, to lump the dynamics for fast reactions and to assume steady state.

Further Reading:

  • Philip W. Kuchel $\&$ Peter J. Mulquiney - Modelling Metabolism with Mathematica: Analysis of Human
  • Uri Alon - An Introduction to Systems Biology: Design Principles of Biological Circuits