When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
$$ \phi \xrightarrow{{K}_{r}} [mRNA] $$
$$ [mRNA] \xrightarrow{{K}_{p_1}} [mRNA] + [{Protein}_{1}] + [{Protein}_{2}] $$
$$ [mRNA] \xrightarrow{{K}_{d_1}} [{mRNA}_{1}] + [{mRNA}_{2}] $$
$$ [{mRNA}_{1}] \xrightarrow{{K}_{{p}_{11}}} [{mRNA}_{1}] + [{Protein}_{1}] $$
$$ [{mRNA}_{2}] \xrightarrow{{K}_{{p}_{12}}} [{mRNA}_{2}] + [{Protein}_{2}] $$
$$ [mRNA] \xrightarrow{{K}_{d_0}} \phi $$
$$ [{mRNA}_{1}] \xrightarrow{{K}_{{d}_{11}}} \phi $$
$$ [{mRNA}_{2}] \xrightarrow{{K}_{{d}_{12}}} \phi $$
$$ [{Protein}_{1}] \xrightarrow{{K}_{{d}_{p_1}}} \phi $$
$$ [{Protein}_{2}] \xrightarrow{{K}_{{d}_{p_2}}} \phi $$
| Symbol |
Definition |
Units |
| [mRNA] |
|
|
| $[{mRNA}_{1}]$ |
|
|
| $[{mRNA}_{2}]$ |
|
|
| $[{Protein}_{1}]$ |
|
|
| $[{Protein}_{2}]$ |
|
|
| $K_r$ |
|
|
| ${K}_{p_1}$ |
|
|
| ${K}_{d_1}$ |
|
|
| ${K}_{{p}_{11}}$ |
|
|
| ${K}_{{p}_{12}}$ |
|
|
| ${K}_{{d}_{0}}$ |
|
|
| ${K}_{{d}_{11}}$ |
|
|
| ${K}_{{d}_{12}}$ |
|
|
| ${K}_{{d}_{p_1}}$ |
|
|
| ${K}_{{d}_{p_2}}$ |
|
|
| Symbol |
Definition |
Units |
| [AraC] |
The concentration of dissociative repressor protein - AraC |
|
| [Arab] |
The concentration of arabinose |
|
| [AraC·Arab] |
The concentration of complex - [AraC·Arab] |
|
| ${[Arac]_T}$ |
The sum of the concentration of both dissociative repressor protein - Arac and complex - AraC·Arab |
|
| $K_i$,i = 1, 2, 3 |
reaction rate constant |
|
| $K_m$ |
Michaelis constant |
|
| v |
transcription rate |
|
$$Arac + Arab \overset{k_1}{\underset{k_2}{\rightleftharpoons}} AraC·Arab \xrightarrow{{K}_{3}} mRNA + Arac$$
$$ { {d[Arac·Arab]} \over {dt} } = {{k_1}·{({{[AraC]}_T}-{[AraC·Arab]})} - {k_2}·{[AraC·Arab]} - {k_3}·{[AraC·Arab]} }$$
$$ { {d[Arac·Arab]} \over {dt} } = 0 $$
$$ { { {k_2} + {k_3}} \over {k1} } = { { {({{[AraC]}_T}-{[AraC·Arab]})}·[Arab] } \over {[AraC·Arab]} } $$
$$ {k_m} = { { {k_2} + {k_3}} \over {k1} } $$
$$ { [AraC·Arab] } = { { { { [AraC] }_T } · [Arab] } \over { {k_m} + [Arab] } } $$
$$ {v} = {k_3}·{ [AraC·Arab] } = {k_3} · { {{[AraC]}_T} } · { {[Arab]} \over { {k_m} + [Arab] } } $$
$$ {K_r} = { v \over { [AraC·Arab] } }$$
| Symbol |
Definition |
Units |
| $P_{bound}$ |
Probability of ribosome binding to RBS |
/ |
| $P$ |
Effective number of ribosome available for binding to RBS |
|
| $N_{NS}$ |
The number of nonspecific site of mRNA |
|
| $K^{S}_{pd}$ |
Dissociation constants for specific binding |
nM |
| $K^{NS}_{pd}$ |
Dissociation constants for non-specific binding |
nM |
| ${\epsilon }^{S}_{pd}$ |
Binding energy for ribosome on the RBS |
J |
| ${\epsilon }^{NS}_{pd}$ |
Average binding energy of ribosome to the genomic background |
J |
| $k_B$ |
Boltzmann constants |
/ |
| T |
Temperature |
K |
| Rate |
Rate of reaction |
|
| Volume |
Volume |
L |
| Avogadro |
Avogadro constants |
/ |
| ${[mRNA]}_0$ |
Initial concentration of mRNA |
|
$$ { P_{bound} } = { 1 \over { 1 + { { N_{NS} } \over P }exp({ {{{\epsilon }^{S}_{pd}}-{{\epsilon }^{NS}_{pd}}} \over {{k_B}T} }) } } $$
$$ { {{\epsilon }^{S}_{pd}} - {{\epsilon }^{NS}_{pd}} } \approx { {{k_B}T}ln({ {K^{S}_{pd}} \over {K^{NS}_{pd}} }) } $$
$$ { P_{bound} } = { 1 \over { 1 + { { {N_{NS}} \over {P} } · { {{K^{S}_{pd}}} \over {{K^{NS}_{pd}}} } } } } $$
$$ Rate = {1000*{ P_{bound} }} \over {Volume*Avogadro} $$
$$ k = { {1000*{ P_{bound} }} \over {Volume*Avogadro*{{[mRNA]}_0}} } $$
$$ {K_{d_1}} = { {{[H^+]}{K_{E1}}{k_0}} \over { {{K_{E1}}·{K_{E2}}} + { {[H^+]}{K_{E1}} } + {{[H^+]}^2} } } $$
$$ {lg{1 \over C}} = { {lgA} - {0.434{ {\Delta G} \over {RT} }} } $$
| Symbol |
Definition |
| C |
|
| A |
|
| $\Delta G$ |
|
| R |
|
| T |
|
$$ {lg{1 \over C}} = { {-a{\pi}^2} + {b \pi} + {\rho \pi} + {\delta E_S} + c} $$
$$ lgP = { {lgP_H} + {\Sigma (\pi x_i)} } $$
$$ lg{ {k_X} \over {k_H} } = \rho {\sigma}_X $$
$$ MR = { {({n^2}-1){M_W}} \over {({n^2}+2)d} } $$
$$ r = \sqrt{ 1 - { { \Sigma {( {Y_{cal}} - {Y_{exp}} )}^2 } \over { \Sigma {( {Y_{exp}} - {{\tilde{Y}}_{exp}} )}^2 } } } $$
$$ s = \sqrt{ { \Sigma {( {Y_{cal}} - {Y_{exp}} )}^2 } \over {n-k-1} } $$
$$ F = \sqrt{ {{r^2}(n-k-1)} \over {k{(1-r)}^2} } $$
