The ABSIS

// Input (X, R, ${\mathit{x}}_0$, D, θ, κ, M, λ1, λ2)

Define network $N\leftarrow (\mathcal{X},\mathcal{R})$

Initialize hash table $\mathcal{H}=\{\left({\mathit{x}}_i\right):[A_0\left({\mathit{x}}_i\right),B_0\left({\mathit{x}}_i\right),A_k\left({\mathit{x}}_i\right),B_k\left({\mathit{x}}_i\right),g_k\left({\mathit{x}}_i\right)]\}\leftarrow \varnothing$

j ← 0,   total weight wM ← 0,   weight square vM ← 0,

Number of successful paths Ns ← 0

whilej < Mdo

Path length i ← 1

Initialize path with the initial state ${\mathit{x}}_{i-1}\leftarrow {\mathit{x}}_0$

Time on current path t ← 0,   weight of current path w ← 1

whilet < θ and $d({\mathit{x}}_{i-1},\mathcal{D})>0$do

if${\mathit{x}}_{i-1}\notin \mathcal{H}$then

Calculate reaction rates $A_k\left({\mathit{x}}_{i-1}\right)$ of state ${\mathit{x}}_{i-1}$ for all reactions $R_k\in \mathcal{R}$

$A_0\left({\mathit{x}}_{i-1}\right)\leftarrow {\sum }_{k=1}^m{A}_k\left({\mathit{x}}_{i-1}\right)$

Enumerate all possible κ step paths π(i − 1, i + κ) starting from state ${\mathit{x}}_{i-1}$ using Algorithm of Ref 19.

Calculate $p_{\mathsf{F}}\left({\mathit{x}}_i\right)$ and $p_{\mathsf{B}}\left({\mathit{x}}_i\right)$ for each Rk using Eq. 9, 10

Calculate bias strength $g_k\left({\mathit{x}}_{i-1}\right)$ for each Rk according to Eq. 12

Calculate tentative reaction rate $B_k^{\prime }\left({\mathit{x}}_{i-1}\right)$ for all Rk according to Eq. 14

Calculate final biased reaction rate $B_k\left({\mathit{x}}_{i-1}\right)$ for all Rk according to Eq. 17

Calculate $B_0\left({\mathit{x}}_{i-1}\right)\leftarrow {\sum }_{k=1}^m{B}_k\left({\mathit{x}}_{i-1}\right)$

$\mathcal{H}\leftarrow \mathcal{H}\cup \{\left({\mathit{x}}_{i-1}\right):[A_0\left({\mathit{x}}_{i-1}\right),B_0\left({\mathit{x}}_{i-1}\right),A_k\left({\mathit{x}}_{i-1}\right),B_k\left({\mathit{x}}_{i-1}\right),g_k\left({\mathit{x}}_{i-1}\right)]\}$

end if

Retrieve $B_k\left({\mathit{x}}_{i-1}\right)$, $A_k\left({\mathit{x}}_{i-1}\right)$, $B_0\left({\mathit{x}}_{i-1}\right)$, $A_0\left({\mathit{x}}_{i-1}\right)$ and ${\mathit{g}}_k\left({\mathit{x}}_{i-1}\right)$ from H using key ${\mathit{x}}_{i-1}$

Generate two uniform random numbers ${\mu }_1\sim \mathcal{U}(0,1)$ and ${\mu }_2\sim \mathcal{U}(0,1)$

${\tau }_{i-1}\leftarrow -\ln \left({\mu }_1\right)/B_0\left({\mathit{x}}_{i-1}\right)$

$r\leftarrow \text{smallest}\text{integer}\text{satisfying}{\sum }_{r=1}^{k-1}{B}_r\left({\mathit{x}}_{i-1}\right)<{\mu }_2{B}_0\left({\mathit{x}}_{i-1}\right)\le {\sum }_{r=1}^k{B}_r\left({\mathit{x}}_{i-1}\right)$

t ← t + τi − 1, ${\mathit{x}}_{i-1}\leftarrow {\mathit{x}}_{i-1}+{\mathit{s}}_r$

$w\leftarrow w·\frac{A_k\left({\mathit{x}}_{i-1}\right)e^{(B_0\left({\mathit{x}}_{i-1}\right)-A_0\left({\mathit{x}}_{i-1}\right)){\tau }_{i-1}}}{B_k\left({\mathit{x}}_{i-1}\right)}$

i ← i + 1

end while

ift < θ and $d({\mathit{x}}_{i-1},\mathcal{D})=0$thenwM ← wM + w, vM ← vM + w2, Ns ← Ns + 1

end if

end while

return${\mathit{p}}_{ABSIS}({\mathit{x}}_0,\mathcal{D},\theta )=w_M/M$

return${\sigma }_{ABSIS}^2({\mathit{x}}_0,\mathcal{D},\theta )=(v_M/M)-{(w_M/M)}^2$

return${\sigma }_{ABSIS}({\mathit{x}}_0,\mathcal{D},\theta )=\sqrt{{\sigma }_{ABSIS}^2/M}$

return Success Rate: s = Ns/M