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. 2018 Mar 21;19(4):926. doi: 10.3390/ijms19040926

Algorithms

Define $α$ , $Δ t$ , all $φ_{i}$ , all E_i,
${\vec{ϵ}}^{m}$ is pre-defined for all time points m; most entries are 0.
Iteration:
- ○
  starting with ${\vec{V}}^{1} = V (0, x)$ , ${\vec{p i}}^{1} = p i (0, x, V (0, x))$
- ○
  m = 1, …, M − 1
  - ▪
    calculate ${\vec{b}}^{m}$ , ${\vec{a}}^{m}$ , $[A^{m}]$ , ${[A^{m}]}^{- 1}$ and $[B^{m}]$
  - ▪
    calculate ${\vec{V}}^{m + 1}$
  - ▪
    calculate ${\vec{p i}}^{m + 1}$ via separate linear equation systems of the type:
  - ▪
    n = 1…N
    
    i = 1, …, T (number of transporters involved)
    ○
    $p i_{n}^{m + 1} = p i_{n}^{m} + Δ t \cdot [a i (V_{n}^{m}) \cdot (1 - p i_{n}^{m}) - d i (V_{n}^{m}) \cdot p i_{n}^{m}]$ with the transporter-specific coefficients $d i (V_{n}^{m})$ and $a i (V_{n}^{m})$ , which depend only on $V_{n}^{m}$
    
    End i-loop
  - ▪
    End n-loop
- ○
  End m-loop
End Iteration