Biomod/2013/NanoUANL/Enzyme: Difference between revisions

Revision as of 15:36, 26 October 2013

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<html>  <script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\$','\$']]}}); MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script> <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <head> <title></title> </head> <body>    ENZYME   <img alt="" src="http://openwetware.org/images/e/e4/UANLEnzyme1.jpg" style="width: 410px; height: 385px;" /> Enzyme In biological systems, chemical transformations are typically accelerated by enzymes, macromolecules capable of turning one or more compounds into others (substrates and products). The activity is determined greatly by their three-dimensional structure. Most enzymes are proteins, although several catalytic RNA molecules have been identified. They may also need to employ organic and inorganic cofactors for the reaction to occur. The process is based upon the diminishment of the activation energy needed for a reaction, greatly increasing its rate of reaction. The rate enhancement provided by these proteins can be as high as 10^19, while maintaining high substrate specificity. Because of this, reaction rates are millions of times faster than un-catalyzed reactions. Enzymes are not consumed by the reactions that they take part in, and they do not alter the equilibrium. Enzyme activity can be affected by a wide variety of factors. Inhibitors and activators intervene directly in the reaction rate, environmental factors like temperature, pressure, pH and substrate concentration also play a part in these kinetics. For temperature and pH, usually exist a range of values for which the enzyme works better (optimal conditions). The enzyme activity lowers dramatically as you get farther away from this range of values. As for concentration, other kind of relationship is observed. With increasing concentration, enzyme activity increases, until we reach the most optimal performance. Further increase of concentration generally won’t have an impact on the enzyme activity. Being able to determine these conditions allow us to manipulate the enzyme activity, thus achieving greater control over the reaction. Michaelis-Menten kinetics Michaelis-Menten kinetics is one of the oldest models for describing the catalytic activity of enzymes. The reaction cycle is divided into two basic steps: the reversible binding between the enzyme and substrate to form an intermediate complex, and the irreversible catalytic step to generate the product and release the enzyme; in which the first step is affected by the constants k1 and k-1, whereas the irreversible step only takes into account  k2. \begin{equation} E + S \leftrightarrow ES \rightarrow E^0 + P \end{equation} The rate of consumption can be expressed by the formation of the ES complex in the following equation: \begin{equation} \frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES] \end{equation} Using a steady-approximation and rearranging eq. 2 we obtain: \begin{equation} [ES]= \frac{[E][S]}{K_M+[S]} \end{equation} where ''KM'' is the Michaelis constant defined as \begin{equation} K_M= \frac{k_{-1} + k_2}{k_1} \end{equation} As it was mentioned in the introduction, single-enzyme studies have proven that the "traditional" enzyme kinetics do not apply, and a new approach is needed. Enzyme concentration is meaningless in a single-molecule level, so it is more appropriate to consider the probability ''PE(t)'' for the enzyme to find a catalytically active enzyme in a time ''t'' in the process. This is because the reaction is a stochastic event. Therefore, the rate equations of each species are: \begin{equation} \frac{d[E]}{dt}=-k_1[E][S]+k_{-1}[ES] \end{equation} \begin{equation} \frac{d[ES]}{dt}=k_1[E][S]-(k_{-1}+k_2)[ES] \end{equation} \begin{equation} \frac{d[E^0]}{dt}=\frac{d[P]}{dt}=k_2[ES] \end{equation} </body> </html>

Biomod/2013/NanoUANL/Enzyme: Difference between revisions

Revision as of 15:36, 26 October 2013

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@@ Line 40: / Line 40: @@
 E + S \leftrightarrow ES \rightarrow E^0 + P
 \end{equation}
+<p>
+			<span style="font-family:trebuchet ms,helvetica,sans-serif;"><span style="font-size: 12px;">The rate of consumption can be expressed by the formation of the ES complex in the following equation:</span></span></p>
+\begin{equation}
+\frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES]
+\end{equation}
+<p>
+			<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Using a steady-approximation and rearranging eq. 2 we obtain:</span></span></p>
+\begin{equation}
+[ES]= \frac{[E][S]}{K_M+[S]}
+\end{equation}
+<p>
+			<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">where &#39;&#39;K<sub>M</sub>&#39;&#39; is the Michaelis constant defined as </span></span></p>
+\begin{equation}
+K_M= \frac{k_{-1} + k_2}{k_1}
+\end{equation}
+<p>
+			<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">As it was mentioned in the introduction, single-enzyme studies have proven that the &quot;traditional&quot; enzyme kinetics do not apply, and a new approach is needed. Enzyme concentration is meaningless in a single-molecule level, so it is more appropriate to consider the probability &#39;&#39;P<sub>E</sub>(t)&#39;&#39; for the enzyme to find a catalytically active enzyme in a time &#39;&#39;t&#39;&#39; in the process. This is because the reaction is a stochastic event.</span></span></p>
+<p>
+			<span style="font-size:12px;"><span style="font-family: trebuchet ms,helvetica,sans-serif;">Therefore, the rate equations of each species are:</span></span></p>
+\begin{equation}
+\frac{d[E]}{dt}=-k_1[E][S]+k_{-1}[ES]
+\end{equation}
+\begin{equation}
+\frac{d[ES]}{dt}=k_1[E][S]-(k_{-1}+k_2)[ES]
+\end{equation}
+\begin{equation}
+\frac{d[E^0]}{dt}=\frac{d[P]}{dt}=k_2[ES]
+\end{equation}
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