Difference between revisions of "Team:Harvard BioDesign/Model"

Line 110: Line 110:
 
</ul>
 
</ul>
 
</nav>
 
</nav>
 +
<section>
 +
<h3>Modelling outline:</h3>
 +
<li>
 +
Calculating extinction coefficients</li>
 +
<li>Deciding what size of plastic to sense
 +
<li>Comparing std. enzyme kinetics to thin film kinetics
 +
<li>Concluding micro much more efficient degradation/ faster feedback to counter sensor lag--> deciding to focus on these for our project</li>
 +
</li>
 +
</li>
 +
<li>Determining how much plastic is needed to set off the sensor
 +
<li>The whole pipeline/ process: g of plastic to - tpa - electricity - sensor</li>
 +
</li>
 +
</section>
 +
 +
<section>
 +
<h3>Supporting our Experiments: Calculating Extinction Coefficients</h3>
 +
 +
<p>During our wet lab work with PETase, we <a href="https://2016.igem.org/Team:Harvard_BioDesign/Results">successfully isolated the enzyme</a>and obtained optical density measures of PETase suspended in solution. In order to calculate the concentration of enzyme present, our team needed to know PETase's extinction coefficient. The extinction coefficient of a substance describesThe paper, "Calculation of Protein Extinction Coefficients from Amino Acid Sequence Data", guided us through the process and allowed us to use the number of tryptophan, tyrosine, and cystine amino acids present in PETase to arrive at an extinction coefficient of 39170 M^-1 cm^-1. </p>
 +
 +
<p>Below are equations from the paper that enabled us to determine the extinction coefficient: </p>
 +
 +
 +
<img src="https://static.igem.org/mediawiki/2016/3/3a/T--Harvard_BioDesign--Modelling_ExtinctionEq1.png"/>
 +
</section>
 +
 +
<section>
 +
<h3>Informing our Design: Modelling Michaelis-Menten Kinetics</h3>
 +
 +
<p>As we were refining the design of our plastic-sensing device, we were having a difficult time accounting for the lag between plastic uptake and plastic degradation. Recall that plastic in our bioreactor needs to be broken down by PETase in order to produce the terephthalic acid necessary to power our sensor. Though PETase is the most efficient PET-degrading enzyme discovered to date, it by no means degrades plastic instantaneously. This means that our devices gives off a signal at a later time and later location from where plastic was originally collected. </p>
 +
 +
<p>The team decided to turn to modelling to find quantitative solutions to this design challenge. We realized that understanding degradation rates as determined by enzyme kinetics would allow us to adjust our reactor design to minimize the lag in signal. </p>
 +
 +
<p>The task required us to understand enzyme kinetics under two broad sets of circumstances: </p>
 +
<ol>
 +
<li>When the pieces of plastic were small enough in size so that they were effectively suspended in a solution. </li>
 +
<li>When the pieces of plastic were too large to be considered in suspension. This would require us to apply models geared towards heterogeneous reactions that involve interactions between a solid substrate and an enzyme in solution. </li>
 +
</ol>
 +
<p>To understand conditions involving small pieces of plastic, our team relied on Michaelis-Menten, a widely-applied model for enzyme kinetics. Prior to beginning our work, we ensured that the assumptions inherent in the model were reasonable for modelling PETase activity:  </p>
 +
<ul>
 +
<li>The model assumes that the product is not converted back into the substrate. This assumption holds when the concentration of product is very low. It is valid at the beginning of reactions and is especially compatible with our reactor design considering that the products of the reaction are funneled to another chamber where none of the original substrate or enzyme are present. The assumption also makes sense considering that our enzyme functions by breaking down polymers: after the reaction, the newly formed monomers no longer fit in the enzyme's active site. </li>
 +
<li>Another assumption is that the enzyme must be either free or bound to the substrate. That is, PETase should not at any time be bound to a molecule that does not consist of PET. The pnpb assays we conducted during our characterization of different esterases (include hyperlink to appropriate page) demonstrate how PETase is significantly less likely that other cutinases to catalyze the breakdown of esters. This suggests that PETase has a more specific binding site and comes closer than alternative enzymes to satisfying the assumption. </li>
 +
</ul>
 +
 +
<p>An overview of Michaelis-Menten: The Michaelis-Menten constant (Km) corresponds to the substrate concentration at which the speed of the reaction is half of the maximum possible velocity: </p>
 +
 +
 +
<img src="https://static.igem.org/mediawiki/2016/3/3a/T--Harvard_BioDesign--Modelling_MichaelisGeneric.png"/>
 +
 +
<p>Our team initially intended to compare the Michaelis-Menten constant associated with small plastics suspended in solutions to that of large pieces of plastics. However, we soon realized that the Michaelis-Menten model, with its references to substrate concentration, assumes that the substrate is in solution and would therefore be unsuitable for describing the kinetics associated with large solid substrates. </p>
 +
 +
<p>As our team and mentors investigated more sophisticated variations of Michaelis-Menten, we uncovered a highly relevant paper, "Surface Enzyme Kinetics for Biopolymer Microarrays: a Combination of Langmuir and Michaelis-Menten Concepts". This model incorporates additional parameters - adsorption and desorption coefficients - to help explain surface kinetics. Using the paper as a blueprint, our team constructed a model in Matlab (include downloadable matlab file?) that enables its users to enter values for adsorption, desorption, and catalysis constants and values for initial substrate and enzyme concentrations to immediately receive values for the Michaelis-Menten constant and the percentage of substrate that is bound to the enzyme at any point in time. </p>
 +
</section>
 +
This is the code for the Matlab model: could either include in body of text in a different font/color/size, or could include as a downloadable matlab file
 +
 +
%% PET Surface Enzyme Kinetics
 +
 +
% Define variables
 +
h = 0.001; % step size
 +
t = 1:h:100; % time frame
 +
 +
E = 0*t; %concentration of enzyme
 +
 +
k_cat = 0; % enzyme catalysis
 +
k_a = 0; % enzyme adsorption
 +
k_d = 0; % enzyme desorption
 +
 +
gamma_S = 0*t; % surface coverage of substrate
 +
gamma_ES = 0*t; % surface coverage of enzyme-substrate complex
 +
gamma_P = 0*t; % surface coverage of product
 +
gamma_total = gamma_S + gamma_ES + gamma_P; % total number of surface sites
 +
 +
d_gamma_S = diff(gamma_S)/h % derivative of surface coverage of substrate wrt t
 +
d_gamma_ES = diff(gamma_ES)/h % derivative of surface coverage of complex wrt t
 +
d_gamma_P = diff(gamma_P)/h % derivative of surface coverage of product wrt t
 +
 +
theta_S = gamma_S/gamma_total; % relative surface coverage of substrate
 +
theta_ES = gamma_ES/gamma_total; % relative surface coverage of complex
 +
theta_P = gamma_P/gamma_total; % relative surface coverage of product
 +
 +
d_theta_S = diff(theta_S)/h % derivative of relative coverage of substrate wrt t
 +
d_theta_ES = diff(theta_ES)/h % derivative of relative coverage of complex wrt t
 +
d_theta_P = diff(theta_P)/h % derivative of relative coverage of product wrt t
 +
 +
% Set-up
 +
d_gamma_ES = k_a*gamma_S - k_d*gamma_ES - k_cat*gamma_ES; % rxn rate for production of complex
 +
d_gamma_P = k_cat*gamma_ES; % rxn rate for production of product
 +
% Rates of change of surface coverage
 +
d_theta_ES = k_a*theta_S - k_d*theta_ES - k_cat*theta_ES;
 +
d_theta_P = k_cat*theta_ES;
 +
 +
% Assuming k_cat << k_a, k_d
 +
theta_ES = k_a*theta_S*E/(k_d + k_cat); % equilibrium surface coverage of complex
 +
K_m = k_d + k_cat/(k_a); % Michaelis-Menten constant
 +
 +
<p>Since our assessment of enzyme kinetics was purely comparative (that is, we judged the efficiency of surface kinetics by comparing them to kinetics in solution), we focused our investigation on the parameters that are unique to surfaces: adsorption and desorption.
 +
</p>
 +
 +
<!--Will write if time allows
 +
[Validating our Vision: Estimating Amount of Plastic Required to Power a Sensor]
 +
-->
 +
 +
<section>
 +
<h3>Footnotes</h3>
 +
 +
</section>
  
 
</div>
 
</div>
Line 121: Line 226:
 
<header>
 
<header>
 
<h2><a href="#">Model</a></h2>
 
<h2><a href="#">Model</a></h2>
−
<p>
 
−
Morbi convallis lectus malesuada sed fermentum dolore amet
 
−
</p>
 
 
</header>
 
</header>
−
<a href="#" class="image featured"><img src="images/bom01.jpg" alt="" /></a>
+
−
<p>
+
 
−
Commodo id natoque malesuada sollicitudin elit suscipit. Curae suspendisse mauris posuere accumsan massa
+
 
−
posuere lacus convallis tellus interdum. Amet nullam fringilla nibh nulla convallis ut venenatis purus
+
 
−
lobortis. Auctor etiam porttitor phasellus tempus cubilia ultrices tempor sagittis. Nisl fermentum
+
<!--Rebekah - START - insert contents here-->
−
consequat integer interdum integer purus sapien. Nibh eleifend nulla nascetur pharetra commodo mi augue
+
 
−
interdum tellus. Ornare cursus augue feugiat sodales velit lorem. Semper elementum ullamcorper lacinia
+
 
−
natoque aenean scelerisque vel lacinia mollis quam sodales congue.
+
 
−
</p>
+
 
−
<section>
+
<section>
−
<header>
+
<h3>Modelling outline:</h3>
−
<h3>Ultrices tempor sagittis nisl</h3>
+
<li>
−
</header>
+
Calculating extinction coefficients</li>
−
<p>
+
<li>Deciding what size of plastic to sense
−
Nascetur volutpat nibh ullamcorper vivamus at purus. Cursus ultrices porttitor sollicitudin imperdiet
+
<li>Comparing std. enzyme kinetics to thin film kinetics
−
at pretium tellus in euismod a integer sodales neque. Nibh quis dui quis mattis eget imperdiet venenatis
+
<li>Concluding micro much more efficient degradation/ faster feedback to counter sensor lag--> deciding to focus on these for our project</li>
−
feugiat. Neque primis ligula cum erat aenean tristique luctus risus ipsum praesent iaculis. Fermentum elit
+
</li>
−
fringilla consequat dis arcu. Pellentesque mus tempor vitae pretium sodales porttitor lacus. Phasellus
+
</li>
−
egestas odio nisl duis sociis purus faucibus morbi. Eget massa mus etiam sociis pharetra magna.
+
<li>Determining how much plastic is needed to set off the sensor
−
</p>
+
<li>The whole pipeline/ process: g of plastic to - tpa - electricity - sensor</li>
−
<p>
+
</li>
−
Eleifend auctor turpis magnis sed porta nisl pretium. Aenean suspendisse nulla eget sed etiam parturient
+
</section>
−
orci cursus nibh. Quisque eu nec neque felis laoreet diam morbi egestas. Dignissim cras rutrum consectetur
+
 
−
ut penatibus fermentum nibh erat malesuada varius.
+
<section>
−
</p>
+
<h3>Supporting our Experiments: Calculating Extinction Coefficients</h3>
−
</section>
+
 
−
<section>
+
<p>During our wet lab work with PETase, we <a href="https://2016.igem.org/Team:Harvard_BioDesign/Results">successfully isolated the enzyme</a>and obtained optical density measures of PETase suspended in solution. In order to calculate the concentration of enzyme present, our team needed to know PETase's extinction coefficient. The extinction coefficient of a substance describesThe paper, "Calculation of Protein Extinction Coefficients from Amino Acid Sequence Data", guided us through the process and allowed us to use the number of tryptophan, tyrosine, and cystine amino acids present in PETase to arrive at an extinction coefficient of 39170 M^-1 cm^-1. </p>
−
<header>
+
 
−
<h3>Augue euismod feugiat tempus</h3>
+
<p>Below are equations from the paper that enabled us to determine the extinction coefficient: </p>
−
</header>
+
 
−
<p>
+
 
−
Pretium tellus in euismod a integer sodales neque. Nibh quis dui quis mattis eget imperdiet venenatis
+
<img src="https://static.igem.org/mediawiki/2016/3/3a/T--Harvard_BioDesign--Modelling_ExtinctionEq1.png"/>
−
feugiat. Neque primis ligula cum erat aenean tristique luctus risus ipsum praesent iaculis. Fermentum elit
+
</section>
−
ut nunc urna volutpat donec cubilia commodo risus morbi. Lobortis vestibulum velit malesuada ante
+
 
−
egestas odio nisl duis sociis purus faucibus morbi. Eget massa mus etiam sociis pharetra magna.
+
<section>
−
</p>
+
<h3>Informing our Design: Modelling Michaelis-Menten Kinetics</h3>
−
</section>
+
 
 +
<p>As we were refining the design of our plastic-sensing device, we were having a difficult time accounting for the lag between plastic uptake and plastic degradation. Recall that plastic in our bioreactor needs to be broken down by PETase in order to produce the terephthalic acid necessary to power our sensor. Though PETase is the most efficient PET-degrading enzyme discovered to date, it by no means degrades plastic instantaneously. This means that our devices gives off a signal at a later time and later location from where plastic was originally collected. </p>
 +
 
 +
<p>The team decided to turn to modelling to find quantitative solutions to this design challenge. We realized that understanding degradation rates as determined by enzyme kinetics would allow us to adjust our reactor design to minimize the lag in signal. </p>
 +
 
 +
<p>The task required us to understand enzyme kinetics under two broad sets of circumstances: </p>
 +
<ol>
 +
<li>When the pieces of plastic were small enough in size so that they were effectively suspended in a solution. </li>
 +
<li>When the pieces of plastic were too large to be considered in suspension. This would require us to apply models geared towards heterogeneous reactions that involve interactions between a solid substrate and an enzyme in solution. </li>
 +
</ol>
 +
<p>To understand conditions involving small pieces of plastic, our team relied on Michaelis-Menten, a widely-applied model for enzyme kinetics. Prior to beginning our work, we ensured that the assumptions inherent in the model were reasonable for modelling PETase activity:  </p>
 +
<ul>
 +
<li>The model assumes that the product is not converted back into the substrate. This assumption holds when the concentration of product is very low. It is valid at the beginning of reactions and is especially compatible with our reactor design considering that the products of the reaction are funneled to another chamber where none of the original substrate or enzyme are present. The assumption also makes sense considering that our enzyme functions by breaking down polymers: after the reaction, the newly formed monomers no longer fit in the enzyme's active site. </li>
 +
<li>Another assumption is that the enzyme must be either free or bound to the substrate. That is, PETase should not at any time be bound to a molecule that does not consist of PET. The pnpb assays we conducted during our characterization of different esterases (include hyperlink to appropriate page) demonstrate how PETase is significantly less likely that other cutinases to catalyze the breakdown of esters. This suggests that PETase has a more specific binding site and comes closer than alternative enzymes to satisfying the assumption. </li>
 +
</ul>
 +
 
 +
<p>An overview of Michaelis-Menten: The Michaelis-Menten constant (Km) corresponds to the substrate concentration at which the speed of the reaction is half of the maximum possible velocity: </p>
 +
 
 +
 
 +
<img src="https://static.igem.org/mediawiki/2016/3/3a/T--Harvard_BioDesign--Modelling_MichaelisGeneric.png"/>
 +
 
 +
<p>Our team initially intended to compare the Michaelis-Menten constant associated with small plastics suspended in solutions to that of large pieces of plastics. However, we soon realized that the Michaelis-Menten model, with its references to substrate concentration, assumes that the substrate is in solution and would therefore be unsuitable for describing the kinetics associated with large solid substrates. </p>
 +
 
 +
<p>As our team and mentors investigated more sophisticated variations of Michaelis-Menten, we uncovered a highly relevant paper, "Surface Enzyme Kinetics for Biopolymer Microarrays: a Combination of Langmuir and Michaelis-Menten Concepts". This model incorporates additional parameters - adsorption and desorption coefficients - to help explain surface kinetics. Using the paper as a blueprint, our team constructed a model in Matlab (include downloadable matlab file?) that enables its users to enter values for adsorption, desorption, and catalysis constants and values for initial substrate and enzyme concentrations to immediately receive values for the Michaelis-Menten constant and the percentage of substrate that is bound to the enzyme at any point in time. </p>
 +
</section>
 +
This is the code for the Matlab model: could either include in body of text in a different font/color/size, or could include as a downloadable matlab file
 +
 
 +
%% PET Surface Enzyme Kinetics
 +
 +
% Define variables
 +
h = 0.001; % step size
 +
t = 1:h:100; % time frame
 +
 +
E = 0*t; %concentration of enzyme
 +
 +
k_cat = 0; % enzyme catalysis
 +
k_a = 0; % enzyme adsorption
 +
k_d = 0; % enzyme desorption
 +
 +
gamma_S = 0*t; % surface coverage of substrate
 +
gamma_ES = 0*t; % surface coverage of enzyme-substrate complex
 +
gamma_P = 0*t; % surface coverage of product
 +
gamma_total = gamma_S + gamma_ES + gamma_P; % total number of surface sites
 +
 +
d_gamma_S = diff(gamma_S)/h % derivative of surface coverage of substrate wrt t
 +
d_gamma_ES = diff(gamma_ES)/h % derivative of surface coverage of complex wrt t
 +
d_gamma_P = diff(gamma_P)/h % derivative of surface coverage of product wrt t
 +
 +
theta_S = gamma_S/gamma_total; % relative surface coverage of substrate
 +
theta_ES = gamma_ES/gamma_total; % relative surface coverage of complex
 +
theta_P = gamma_P/gamma_total; % relative surface coverage of product
 +
 +
d_theta_S = diff(theta_S)/h % derivative of relative coverage of substrate wrt t
 +
d_theta_ES = diff(theta_ES)/h % derivative of relative coverage of complex wrt t
 +
d_theta_P = diff(theta_P)/h % derivative of relative coverage of product wrt t
 +
 +
% Set-up
 +
d_gamma_ES = k_a*gamma_S - k_d*gamma_ES - k_cat*gamma_ES; % rxn rate for production of complex
 +
d_gamma_P = k_cat*gamma_ES; % rxn rate for production of product
 +
% Rates of change of surface coverage
 +
d_theta_ES = k_a*theta_S - k_d*theta_ES - k_cat*theta_ES;
 +
d_theta_P = k_cat*theta_ES;
 +
 +
% Assuming k_cat << k_a, k_d
 +
theta_ES = k_a*theta_S*E/(k_d + k_cat); % equilibrium surface coverage of complex
 +
K_m = k_d + k_cat/(k_a); % Michaelis-Menten constant
 +
 
 +
<p>Since our assessment of enzyme kinetics was purely comparative (that is, we judged the efficiency of surface kinetics by comparing them to kinetics in solution), we focused our investigation on the parameters that are unique to surfaces: adsorption and desorption.
 +
</p>
 +
 
 +
<!--Will write if time allows
 +
[Validating our Vision: Estimating Amount of Plastic Required to Power a Sensor]
 +
-->
 +
 
 +
<section>
 +
<h3>Footnotes</h3>
 +
 
 +
</section>
 +
 
 +
 
 +
 
 +
 
 +
 
 +
<!-- Rebekah - STOP -->
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 
</article>
 
</article>
−
<hr />
 
−
<div class="row">
 
−
<article class="4u 12u(mobile) special">
 
−
<a href="#" class="image featured"><img src="images/pic07.jpg" alt="" /></a>
 
−
<header>
 
−
<h3><a href="#">Gravida aliquam penatibus</a></h3>
 
−
</header>
 
−
<p>
 
−
Amet nullam fringilla nibh nulla convallis tique ante proin sociis accumsan lobortis. Auctor etiam
 
−
porttitor phasellus tempus cubilia ultrices tempor sagittis. Nisl fermentum consequat integer interdum.
 
−
</p>
 
−
</article>
 
−
<article class="4u 12u(mobile) special">
 
−
<a href="#" class="image featured"><img src="images/pic08.jpg" alt="" /></a>
 
−
<header>
 
−
<h3><a href="#">Sed quis rhoncus placerat</a></h3>
 
−
</header>
 
−
<p>
 
−
Amet nullam fringilla nibh nulla convallis tique ante proin sociis accumsan lobortis. Auctor etiam
 
−
porttitor phasellus tempus cubilia ultrices tempor sagittis. Nisl fermentum consequat integer interdum.
 
−
</p>
 
−
</article>
 
−
<article class="4u 12u(mobile) special">
 
−
<a href="#" class="image featured"><img src="images/pic09.jpg" alt="" /></a>
 
−
<header>
 
−
<h3><a href="#">Magna laoreet et aliquam</a></h3>
 
−
</header>
 
−
<p>
 
−
Amet nullam fringilla nibh nulla convallis tique ante proin sociis accumsan lobortis. Auctor etiam
 
−
porttitor phasellus tempus cubilia ultrices tempor sagittis. Nisl fermentum consequat integer interdum.
 
−
</p>
 
−
</article>
 
−
</div>
 
 
</div>
 
</div>
  

Revision as of 21:23, 19 October 2016

Harvard BioDesign 2016

Model

Modelling outline:

  • Calculating extinction coefficients
  • Deciding what size of plastic to sense
  • Comparing std. enzyme kinetics to thin film kinetics
  • Concluding micro much more efficient degradation/ faster feedback to counter sensor lag--> deciding to focus on these for our project
  • Determining how much plastic is needed to set off the sensor
  • The whole pipeline/ process: g of plastic to - tpa - electricity - sensor

    • Supporting our Experiments: Calculating Extinction Coefficients

    During our wet lab work with PETase, we successfully isolated the enzymeand obtained optical density measures of PETase suspended in solution. In order to calculate the concentration of enzyme present, our team needed to know PETase's extinction coefficient. The extinction coefficient of a substance describesThe paper, "Calculation of Protein Extinction Coefficients from Amino Acid Sequence Data", guided us through the process and allowed us to use the number of tryptophan, tyrosine, and cystine amino acids present in PETase to arrive at an extinction coefficient of 39170 M^-1 cm^-1.

    Below are equations from the paper that enabled us to determine the extinction coefficient:

    Informing our Design: Modelling Michaelis-Menten Kinetics

    As we were refining the design of our plastic-sensing device, we were having a difficult time accounting for the lag between plastic uptake and plastic degradation. Recall that plastic in our bioreactor needs to be broken down by PETase in order to produce the terephthalic acid necessary to power our sensor. Though PETase is the most efficient PET-degrading enzyme discovered to date, it by no means degrades plastic instantaneously. This means that our devices gives off a signal at a later time and later location from where plastic was originally collected.

    The team decided to turn to modelling to find quantitative solutions to this design challenge. We realized that understanding degradation rates as determined by enzyme kinetics would allow us to adjust our reactor design to minimize the lag in signal.

    The task required us to understand enzyme kinetics under two broad sets of circumstances:

    1. When the pieces of plastic were small enough in size so that they were effectively suspended in a solution.
    2. When the pieces of plastic were too large to be considered in suspension. This would require us to apply models geared towards heterogeneous reactions that involve interactions between a solid substrate and an enzyme in solution.

    To understand conditions involving small pieces of plastic, our team relied on Michaelis-Menten, a widely-applied model for enzyme kinetics. Prior to beginning our work, we ensured that the assumptions inherent in the model were reasonable for modelling PETase activity:

    • The model assumes that the product is not converted back into the substrate. This assumption holds when the concentration of product is very low. It is valid at the beginning of reactions and is especially compatible with our reactor design considering that the products of the reaction are funneled to another chamber where none of the original substrate or enzyme are present. The assumption also makes sense considering that our enzyme functions by breaking down polymers: after the reaction, the newly formed monomers no longer fit in the enzyme's active site.
    • Another assumption is that the enzyme must be either free or bound to the substrate. That is, PETase should not at any time be bound to a molecule that does not consist of PET. The pnpb assays we conducted during our characterization of different esterases (include hyperlink to appropriate page) demonstrate how PETase is significantly less likely that other cutinases to catalyze the breakdown of esters. This suggests that PETase has a more specific binding site and comes closer than alternative enzymes to satisfying the assumption.

    An overview of Michaelis-Menten: The Michaelis-Menten constant (Km) corresponds to the substrate concentration at which the speed of the reaction is half of the maximum possible velocity:

    Our team initially intended to compare the Michaelis-Menten constant associated with small plastics suspended in solutions to that of large pieces of plastics. However, we soon realized that the Michaelis-Menten model, with its references to substrate concentration, assumes that the substrate is in solution and would therefore be unsuitable for describing the kinetics associated with large solid substrates.

    As our team and mentors investigated more sophisticated variations of Michaelis-Menten, we uncovered a highly relevant paper, "Surface Enzyme Kinetics for Biopolymer Microarrays: a Combination of Langmuir and Michaelis-Menten Concepts". This model incorporates additional parameters - adsorption and desorption coefficients - to help explain surface kinetics. Using the paper as a blueprint, our team constructed a model in Matlab (include downloadable matlab file?) that enables its users to enter values for adsorption, desorption, and catalysis constants and values for initial substrate and enzyme concentrations to immediately receive values for the Michaelis-Menten constant and the percentage of substrate that is bound to the enzyme at any point in time.

    This is the code for the Matlab model: could either include in body of text in a different font/color/size, or could include as a downloadable matlab file %% PET Surface Enzyme Kinetics % Define variables h = 0.001; % step size t = 1:h:100; % time frame E = 0*t; %concentration of enzyme k_cat = 0; % enzyme catalysis k_a = 0; % enzyme adsorption k_d = 0; % enzyme desorption gamma_S = 0*t; % surface coverage of substrate gamma_ES = 0*t; % surface coverage of enzyme-substrate complex gamma_P = 0*t; % surface coverage of product gamma_total = gamma_S + gamma_ES + gamma_P; % total number of surface sites d_gamma_S = diff(gamma_S)/h % derivative of surface coverage of substrate wrt t d_gamma_ES = diff(gamma_ES)/h % derivative of surface coverage of complex wrt t d_gamma_P = diff(gamma_P)/h % derivative of surface coverage of product wrt t theta_S = gamma_S/gamma_total; % relative surface coverage of substrate theta_ES = gamma_ES/gamma_total; % relative surface coverage of complex theta_P = gamma_P/gamma_total; % relative surface coverage of product d_theta_S = diff(theta_S)/h % derivative of relative coverage of substrate wrt t d_theta_ES = diff(theta_ES)/h % derivative of relative coverage of complex wrt t d_theta_P = diff(theta_P)/h % derivative of relative coverage of product wrt t % Set-up d_gamma_ES = k_a*gamma_S - k_d*gamma_ES - k_cat*gamma_ES; % rxn rate for production of complex d_gamma_P = k_cat*gamma_ES; % rxn rate for production of product % Rates of change of surface coverage d_theta_ES = k_a*theta_S - k_d*theta_ES - k_cat*theta_ES; d_theta_P = k_cat*theta_ES; % Assuming k_cat << k_a, k_d theta_ES = k_a*theta_S*E/(k_d + k_cat); % equilibrium surface coverage of complex K_m = k_d + k_cat/(k_a); % Michaelis-Menten constant

    Since our assessment of enzyme kinetics was purely comparative (that is, we judged the efficiency of surface kinetics by comparing them to kinetics in solution), we focused our investigation on the parameters that are unique to surfaces: adsorption and desorption.

    Footnotes