Team:OUC-China/Model

When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ $$ \phi \xrightarrow{{K}_{r}} [mRNA] $$ $$ [mRNA] \xrightarrow{{K}_{p_1}} [mRNA] + [{Protein}_{1}] + [{Protein}_{2}] $$ $$ [mRNA] \xrightarrow{{K}_{d_1}} [{mRNA}_{1}] + [{mRNA}_{2}] $$ $$ [{mRNA}_{1}] \xrightarrow{{K}_{{p}_{11}}} [{mRNA}_{1}] + [{Protein}_{1}] $$ $$ [{mRNA}_{2}] \xrightarrow{{K}_{{p}_{12}}} [{mRNA}_{2}] + [{Protein}_{2}] $$ $$ [mRNA] \xrightarrow{{K}_{d_0}} \phi $$ $$ [{mRNA}_{1}] \xrightarrow{{K}_{{d}_{11}}} \phi $$ $$ [{mRNA}_{2}] \xrightarrow{{K}_{{d}_{12}}} \phi $$ $$ [{Protein}_{1}] \xrightarrow{{K}_{{d}_{p_1}}} \phi $$ $$ [{Protein}_{2}] \xrightarrow{{K}_{{d}_{p_2}}} \phi $$

两行三列的表格

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

$$ {lg{1 \over C}} = { {-a{\pi}^2} + {b \pi} + {\rho \pi} + {\delta E_S} + c} $$ $$ lgP = { {lgP_H} + {\Sigma (\pi x_i)} } $$ $$ lg{ {k_X} \over {k_H} } = \rho {\sigma}_X $$ $$ MR = { {({n^2}-1){M_W}} \over {({n^2}+2)d} } $$ $$ r = \sqrt{ 1 - { { \Sigma {( {Y_{cal}} - {Y_{exp}} )}^2 } \over { \Sigma {( {Y_{exp}} - {{\tilde{Y}}_{exp}} )}^2 } } } $$ $$ s = \sqrt{ { \Sigma {( {Y_{cal}} - {Y_{exp}} )}^2 } \over {n-k-1} } $$ $$ F = \sqrt{ {{r^2}(n-k-1)} \over {k{(1-r)}^2} } $$

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