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Michaelis and Menten considered a quasi equilibrium between E and the ES-Complex to simplify the system. | Michaelis and Menten considered a quasi equilibrium between E and the ES-Complex to simplify the system. | ||
</p> | </p> | ||
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[S] \over dt} = -k_1[E] \cdot S + k_{-1}ES}$$</p> |
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[ES]\over dt} = k_1[E] \cdot S - (k_{-1} + k_2)ES}$$</p> |
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[E] \over dt} = k_1[E] \cdot S - (k_{-1} + k_2)ES}$$</p> |
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[P] \over dt} = k_2[ES]}$$</p> |
<p class="pcontent">After derivation $${v = {-ds \over dt} = {dP \over dt}}$$</p> | <p class="pcontent">After derivation $${v = {-ds \over dt} = {dP \over dt}}$$</p> | ||
<p class="pcontent">where V<sub>m</sub> represents the maximum rate achieved by the system, | <p class="pcontent">where V<sub>m</sub> represents the maximum rate achieved by the system, | ||
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it is thus called turnover number. | it is thus called turnover number. | ||
</p> | </p> | ||
− | <p class="pcontent">$${Vmax = k_2 \cdot E_{total} = k_{cat} \cdot E_{total}}$$</p> | + | <p class="pcontent">$${Vmax = k_2 \cdot [E_{total}] = k_{cat} \cdot [E_{total}]}$$</p> |
<p class="pcontent">B. A more realistic description of a enzymatic reactions than pure Michaelis Menten Kinetics | <p class="pcontent">B. A more realistic description of a enzymatic reactions than pure Michaelis Menten Kinetics | ||
is given by considering the product forming reaction step as reversible. | is given by considering the product forming reaction step as reversible. | ||
</p> | </p> | ||
<p class="pcontent">$${E + S \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} ES \xrightarrow{k_2} E + P}$$</p> | <p class="pcontent">$${E + S \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} ES \xrightarrow{k_2} E + P}$$</p> | ||
− | <p class="pcontent">The Enzyme(E) forms a Complex (ES) with the Substrate (S) and modifies it to the Product (P). | + | <p class="pcontent">The Enzyme (E) forms a Complex (ES) with the Substrate (S) and modifies it to the Product (P). |
In the reversible Michaelis-Menten reaction the Product can bind the enzyme again and react back to the substrate. | In the reversible Michaelis-Menten reaction the Product can bind the enzyme again and react back to the substrate. | ||
This process is described by the ODE system. | This process is described by the ODE system. | ||
</p> | </p> | ||
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[S] \over dt} = -k_1[E] \cdot [S] + k_{-1}[ES]}$$</p> |
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[ES] \over dt} = k_1[E] \cdot [S] + k_{-2}[E] \cdot P - (k_{-1} + k_2)[ES]}$$</p> |
− | <p class="pcontent">$${{ | + | <p class="pcontent">$${{d[E] \over dt} = k_1[E] \cdot [S] - k_{-2}[E] \cdot P - (k_{-1} + k_2)[ES]}$$</p> |
− | <p class="pcontent">$${v = {dP \over dt} = | + | <p class="pcontent">$${v = {dP \over dt} = k_2[ES] - k_{-2}E \cdot P = v_f - v_b}$$</p> |
<p class="pcontent">After derivation $${v = {-ds \over dt} = {dP \over dt}}$$</p> | <p class="pcontent">After derivation $${v = {-ds \over dt} = {dP \over dt}}$$</p> | ||
<p class="pcontent">Finally we use V<sub>fmax</sub> = k2⋅Etotal and V<sub>bmax</sub> = k<sub>−1</sub>⋅Etotal to get the common form for the reversible Michaelis Menten equation</p> | <p class="pcontent">Finally we use V<sub>fmax</sub> = k2⋅Etotal and V<sub>bmax</sub> = k<sub>−1</sub>⋅Etotal to get the common form for the reversible Michaelis Menten equation</p> |
Revision as of 19:44, 17 October 2018