Mass Action Reactions

Reaction	Description or Expansion
${\tt\emptyset \to X}$	Creation (introduction)
${\tt X \to \emptyset}$	Annihilation (removal)
${\tt p_1 A_1 + p_2 A_2 + \cdots \to}$ ${\tt q_1 B_1 + q_2 B_2 + \cdots }$	Standard mass action with stoichiometry
${\tt A\rightleftarrows B}$	$\left\{\begin{matrix}{\tt A\to B} {\tt B\to A}\end{matrix}\right.$
${\tt p_1 A_1 + p_2 A_2 + \cdots \rightleftarrows}$ ${\tt q_1 B_1 + q_2 B_2 + \cdots }$	$\left\{\begin{matrix} {\tt p_1 A_1 + \cdots \to q_1 B_1 + \cdots}\\ {\tt q_1B_1+ \cdots \to p_1 A_q + \cdots} \end{matrix}\right.$
Representation of ${\tt A+X\rightleftarrows A\_X \rightleftarrows B+X}$
${\tt A {\overset {X}{\rightleftarrows}}B}$	$\left\{\begin{matrix} {\tt A+X\to A\_X}\\ {\tt A\_X \to A+X}\\ {\tt A\_X \to B+X}\\ {\tt B+X \to A\_X } \end{matrix}\right.$
Representation of ${\tt A+X\rightleftarrows A\_X \rightleftarrows B\_X \rightleftarrows B+X}$
${\tt A{\overset{X}{\rightleftharpoons}} B}$	$\left\{\begin{matrix} {\tt A+X\to A\_X}\\ {\tt A\_X \to A+X}\\ {\tt A\_X \to... ...B\_X \to A\_X}\\ {\tt B\_X \to B+X}\\ {\tt B+X \to B\_X } \end{matrix}\right.$
Typical Cascades
${\tt A_1 \to A_2 \to A_3 \cdots }$	$\left\{\begin{matrix} {\tt A_1 \to A_2}\\ {\tt A_2 \to A_3} \vdots \end{matrix}\right.$
${\tt A_1 \rightleftarrows A_2 \rightleftarrows A_3 \cdots }$	$\left\{\begin{matrix} {\tt A_1 \rightleftarrows A_2}\\ {\tt A_2 \rightleftarrows A_3} \vdots \end{matrix}\right.$
$\overset{\tt X}{\tt A_1 \rightleftarrows A_2 \rightleftarrows A_3 \cdots }$	$\left\{\begin{matrix} {\tt A_1 \overset{X}\rightleftarrows A_2}\\ {\tt A_2 \overset{X}\rightleftarrows A_3} \vdots \end{matrix}\right.$
$\overset{\tt\{X_1,X_2\dots\}}{\tt A_1 \rightleftarrows A_2 \rightleftarrows A_3 \cdots }$	$\left\{\begin{matrix} {\tt A_1 \overset{X_1}\rightleftarrows A_2}\\ {\tt A_2 \overset{X_2}\rightleftarrows A_3} \vdots \end{matrix}\right.$

Michaelis-Menten-Henri Style Reactions

Syntax ${\tt\{A\implies B, MM[K,v]\}}$ or ${\tt\{A\overset{X}\implies B, MM[K,v]\}}$
${\tt A\implies B}$	${\tt [B]'=\dfrac{v[A]}{K+[A]}=-[A]' }$
${\tt A\overset{X}\implies B}$	${\tt [B]'=\dfrac{v[A][X]}{K+[A]}=-[A]' }$
${\tt A\Longleftrightarrow B}$	${\tt [B]'=\dfrac{v_A[A]}{K_A+[A]}-\dfrac{v_B[B]}{K_B+[B]}=-[A]' }$
${\tt A\underset{Y}{\overset{X}{\Longleftrightarrow}} B}$	${\tt [B]'=\dfrac{v_A[A][X]}{K_A+[A]}-\dfrac{v_B[B][Y]}{K_B+[B]}=-[A]' }$
Syntax ${\tt\{A\implies B, MM[k_1,k_2,k_3]\}}$ or ${\tt\{A\overset{X}\implies B, MM[k_1,k_2,k_3]\}}$
${\tt A\implies B}$	${\tt [B]'=\dfrac{k_1[A]}{\frac{k_2+k_3}{k_1}+[A]}=-[A]' }$
${\tt A\overset{X}\implies B}$	${\tt [B]'=\dfrac{k_1 [A][X]}{\frac{k_2+k_3}{k_1}+[A]}=-[A]' }$
Cascades
${\tt A_1 \implies A_2 \implies A_3 \implies \cdots}$	$\left\{\begin{matrix} {\tt A_1 \implies A_2}\\ {\tt A_2 \implies A_3} \vdots \end{matrix}\right.$
$\overset{\tt X}{{\tt A_1 \implies A_2 \implies A_3 \implies \cdots}}$	$\left\{\begin{matrix} {\tt A_1 \overset{X}\implies A_2}\\ {\tt A_2 \overset{X}\implies A_3} \vdots \end{matrix}\right.$
$\overset{\tt\{X_1,X_2,\dots\}}{{\tt A_1 \implies A_2 \implies A_3 \implies \cdots}}$	$\left\{\begin{matrix} {\tt A_1 \overset{X_1}\implies A_2}\\ {\tt A_2 \overset{X_2}\implies A_3} \vdots \end{matrix}\right.$

Regulatory Hill Functions

Syntax ${\tt\{ A\mapsto B, hill[v,n,K,b,T]\}}$
${\tt A\mapsto B}$	${\tt [B]'=\dfrac{v(b+T[A])^n}{K^n+(b+T[A])^n}}$ and ${\tt [A]'=0}$
Syntax ${\tt \{ A_1+A_2+\cdots \mapsto B, hill[v,n,K,b,\{T_1,T_2,\dots\}] \} }$
$$\begin{array}{l} {\tt A_1 + A_2 +\cdots \mapsto B}\text{ or ... ...\ {\tt A_3\mapsto B}\\ \vdots \end{matrix}\right. \end{array}$$	$$\begin{array}{l} {\tt [B]'=\dfrac{v(b+T_1[A_1]+T_2[A_2]+\cdo... ...2]+\cdots)^n}} \\ {\tt [A_1]'=[A_2]'=\cdots = 0} \end{array}$$

Catalytic Hill Functions

Syntax ${\tt \{ A\overset{X}\mapsto B, hill[v,n,K,b,T]\}}$
${\tt A\overset{X}{\mapsto} B}$	${\tt [B]'=\dfrac{v[X](b+T[A])^n}{K^n+(b+T[A])^n} = -[A]' }$
Syntax ${\tt \{ A_1+A_2+\cdots \mapsto B, hill[v,n,K,b,\{T_1,T_2,\dots\}] \} }$
$$\begin{array}{l} {\tt A_1 + A_2 +\cdots \overset{X}\mapsto B... ...\overset{X}\mapsto B}\\ \vdots \end{matrix}\right. \end{array}$$	$$\begin{array}{l} {\tt [B]'=\dfrac{v[X](b+T_1[A_1]+T_2[A_2]+\... ...cdots)^n}} \\ {\tt [A_1]'=[A_2]'=\cdots = -[B]'} \end{array}$$

Logistic Rate Functions

Syntax $\{\tt A\mapsto B, GRN[v,\beta, n,h]\}$
${\tt A{\mapsto} B}$	$$\begin{array}{l} \\ {\tt [B]'=\dfrac{v}{1+e^{-h-\beta [A]^n}}}\\ {\tt [A]'=0 } \\ \end{array}$$
Syntax $\{\tt A_1+A_2+\cdots \mapsto B, GRN[v, \{\beta_1,\beta_2,\dots\}, \{n_1,n_2,\dots\},h]\}$
$$\begin{array}{l} {\tt A_1 + A_2 + \cdots \mapsto B }\text{ o... ...o B}\\ {\tt A_2\to B} \vdots \end{matrix}\right. \end{array}$$	$$\begin{array}{l} {\tt [B]'=\dfrac{v}{1+e^{-h-\beta_1 [A_1]^{... ... - \cdots}}} \\ {\tt [A_1]' = [A_2]'=\cdots = 0} \end{array}$$

S-System Rate Functions
(Synergistic Systems)

Syntax $\{\tt A\mapsto B, SSystem[\tau,K_{+},K_{-},c_{+},c_{-}]\}$
${\tt A\mapsto B}$	$\begin{array}{l} \\ {\tt [B]'=\dfrac{1}{\tau}\left( K_{+}[A]^{C_{+}} - K_{-}[A]^{C_{-}} \right)} \\ {\tt [A]'=0} \\ \end{array}$
Syntax $\{\tt A_1+A_2+\cdots\mapsto B, SSystem[\tau,K_{+},K_{-},\{p_1,p_2,\dots\},\{m_1,m_2,\dots\}]\}$
$$\begin{array}{l} {\tt A_1 + A_2 + \cdots \mapsto B }\text{ o... ...o B}\\ {\tt A_2\to B} \vdots \end{matrix}\right. \end{array}$$	$$\begin{array}{l} {\tt [B]'=\dfrac{ K_{+}[A_1]^{p_1}[A_2]^{p_... ...2}\cdots}{\tau}} \\ {\tt [A_1]'=[A_2]'=\cdots=0} \end{array}$$

NHCA Rate Function
(Non-Hierarchical Cooperative Activation)

Syntax $\{\tt A\mapsto B, NHCA[v, \{\alpha, \beta\}, n, m, k]\}$
${\tt A\mapsto B}$	$$\begin{array}{l} {\tt [B]'=\dfrac{v(1+\alpha [A]^n)^m}{(1+\alpha [A]^n)^m + k (1+\beta [A]^n)^m}}\\ {\tt [A]'=0} \end{array}$$
Syntax $\{\tt A_1 + A_2 + \cdots \mapsto B, NHCA[v, \{\{\alpha_1,\beta_1\}, \{\alpha_2,\beta_2\},\dots\}, \{n_1,n_2,\dots\}, m, k]\}$
Syntax $\{\tt A_1 + A_2 + \cdots \mapsto B, NHCA[v, \{\{\alpha_1,\beta_1\}, \{\alpha_2,\beta_2\},\dots\}, n, m, k]\}$
$$\begin{array}{l} {\tt A_1+A_2+\cdots \mapsto B} \text{ or }... ..._2 \mapsto B}\\ {\tt\vdots} \end{matrix} \right. \end{array}$$	$$\begin{array}{l} {\tt [B]'=\dfrac{v\prod_j(1+\alpha_j [A_j]^... ...{n_j})^m} } \\ {\tt [A_1]'=[A_2]' = \cdots = 0} \end{array}$$
Syntax $\{\tt A \mapsto B, NHCA[v,T, n, m, k]\}$
Syntax $\{\tt A_1+A_2+\cdots \mapsto B, NHCA[v,\{T_1,T_2,\dots\}, \{n_1,n_2,\dots\}, m, k]\}$
${\tt A\mapsto B}$	$$\begin{array}{l} {\tt [B]'=\dfrac{v(1+T\mathcal{U}(T)[A]^n)^m... ... + k (1+T\mathcal{U}(-T) [A]^n)^m}}\\ {\tt [A]'=0} \end{array}$$
$$\begin{array}{l} {\tt A_1 + A_2 + \cdots \mapsto B} \text{ ... ...tt A_2 \mapsto B}\\ \vdots \end{matrix} \right . \end{array}$$	$$\begin{array}{l} {\tt [B]' = \dfrac{v\prod_j (1+T_j\mathcal... ...]^{n_j})^m} } \\ {\tt [A_1]'=[A_2]'=\cdots = 0} \end{array}$$

MWC/GWMC Rate Function
(Generalized Monod-Wyman-Changeaux)

Syntax $\{\tt A\overset{X}\Rightarrow B, MWC[k_{cat}, n, c, \ell,K]\}$
${\tt A\overset{X}\Rightarrow B}$	$$\begin{array}{l} {\tt [B]' = k_{cat}[X]\dfrac {\alpha(1+\alph... ... where } \alpha=\dfrac{[A]}{K} \\ {\tt [A]'=-[B]'} \end{array}$$
${\tt S\underset{\{A,I\}}{\overset{X}\Rightarrow} P}$	$$\begin{array}{l} {\tt [B]' = k_{cat}[X]\dfrac {s(1+s)^{n-1}+c... ...I]}{K_I},\ a=\dfrac{[A]}{K_A} \\ {\tt [A]'=-[B]'} \end{array}$$
Syntax $\{\tt\underset{\{\{A_1,A_2,\dots\}, \{I_1,I_2,\dots\}\}}{\{S_1,S_2,\dots\}\overset{X}\Rightarrow \{ P_1,P_2,\dots \}}, MWC[\{k_1,k_2\dots \},n,c,\ell,K]\}$
$$\begin{array}{l} {\tt [P_q]'=k_q[X]\dfrac {\prod_i(1+a_i)^n ... ...ac{[A_j]}{K_{A_j}}, \ i_j = \dfrac{[I_j]}{K_{I_j}} \end{array}$$
Syntax $\{\tt\underset{ \{\{A_1,A_2,\dots\}, \{I_1,I_2,\dots\}\}, \{ \{C'_{11},C'_{12... ...overset{X}\Rightarrow \{ P_1,P_2,\dots \}}, MWC[\{k_1,k_2,\dots\},n,c,\ell,K]\}$
here $C_{j1},C_{j2},\dots$ are competitive inhibitors of $S_j$, $A'_{i1},A'_{i2},\dots$ are competitive activators of $A_i$
$$\begin{array}{l} [P_q]'=\dfrac {\prod_i(1+a_i+\overline{a_... ...verline{a_j} = \sum_q\dfrac{[A'_{jq}]}{K_{A'_{jq}}} \end{array}$$

Rational Function Rate Law

$$\begin{array}{l} {\tt\{\{\{ A_1, A_2,\dots \}, \{B_1,B_2,\do... ...ots\},\{b_0,\dots\},\{p_1,\dots\},\{q_1,\dots\}]\} }\end{array}$$

$$\begin{array}{l} {\tt [X]'=\dfrac {a_0 + a_1[A_1]^{p_1} + a... ...tt [A_1]'=[A_2]'=\cdots = [B_1]'=[B_2]'=\cdots = 0} \end{array}$$

User-Defined Stoichiometric Rate Laws

$$\begin{array}{l} \text{Syntax } {\tt\{p_1A_1 + p_2A_2 + \c... ...t f[A_1[t],A_2[t],\dots, B_1[t],B_2[t],\dots] \} } \end{array}$$

${\tt [X_i]'= (q_i-p_i)f[A_1[t],A_2[t],\dots,B_1[t],\dots] }$

User-Defined Regulatory Rate Laws

${\tt\{A\mapsto B, name[r, T, n, h, f]\} }$	${\tt [B]' = rf(h+T[A]^n)}$
$$\begin{array}{l} {\tt A_1 + A_2 + \cdots \mapsto B} \text{ ... ...tt A_2 \mapsto B}\\ \vdots \end{matrix} \right . \end{array}$$	${\tt [B]' = rf(h+T_1[A_1]^{n_1} + T_2[A_2]^{n_2} + \cdots )}$

Flux-Only Reactions

${\tt \{ p_1 X_1 + p_2 X_2 + \cdots \to q_1 Y_2 + q_2 Y_2 + \cdots, }$

${\tt Flux[} \textit{lower bounds }, \textit{variable name}, \textit{upper bounds}, \textit{value}, \textit{objective coefficient}]$

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