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<h1> Modeling</h1>
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<p>Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.</p>
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<h3> Gold Medal Criterion #3</h3>
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To complete for the gold medal criterion #3, please describe your work on this page and fill out the description on your <a href="https://2017.igem.org/Judging/Judging_Form">judging form</a>. To achieve this medal criterion, you must convince the judges that your team has gained insight into your project from modeling. You may not convince the judges if your model does not have an effect on your project design or implementation.
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Please see the <a href="https://2017.igem.org/Judging/Medals"> 2017 Medals Page</a> for more information.
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<h3>Best Model Special Prize</h3>
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To compete for the <a href="https://2017.igem.org/Judging/Awards">Best Model prize</a>, please describe your work on this page  and also fill out the description on the <a href="https://2017.igem.org/Judging/Judging_Form">judging form</a>. Please note you can compete for both the gold medal criterion #3 and the best model prize with this page.
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<br><br>
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You must also delete the message box on the top of this page to be eligible for the Best Model Prize.
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<h5> Inspiration </h5>
 
<p>
 
Here are a few examples from previous teams:
 
</p>
 
<ul>
 
<li><a href="https://2016.igem.org/Team:Manchester/Model">Manchester 2016</a></li>
 
<li><a href="https://2016.igem.org/Team:TU_Delft/Model">TU Delft 2016  </li>
 
<li><a href="https://2014.igem.org/Team:ETH_Zurich/modeling/overview">ETH Zurich 2014</a></li>
 
<li><a href="https://2014.igem.org/Team:Waterloo/Math_Book">Waterloo 2014</a></li>
 
</ul>
 
  
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    <div class="display-t" style="background-image:url(https://static.igem.org/mediawiki/2017/c/c4/Sfls-title-modeling.svg); background-repeat:no-repeat; background-attachment:fixed">
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      <div class="display-tc">
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        <h1>Modeling</h1>
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      </div>
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    </div>
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  </div>
  
</div>
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          <li class="taoyu" href="#index_0" id="BigTitle">
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            <div class="taoyu_title"> <a href="#index_0" title="0">Off-target Prediction</a></div>
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          </li>
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          <li class="taoyu" href="#index_1">
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            <div class="taoyu_title"> <a href="#index_1" title="1">Introduction</a></div>
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          </li>
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          <li class="taoyu" href="#index_2">
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            <div class="taoyu_title"> <a href="#index_2" title="2">Methods</a></div>
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          </li>
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          <li class="taoyu" href="#index_3">
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            <div class="taoyu_title"> <a href="#index_3" title="3">Results</a></div>
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          </li>
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          <li class="taoyu" href="#index_4">
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            <div class="taoyu_title"> <a href="#index_4" title="4">Conclusion</a></div>
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          </li>
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          <li class="taoyu" href="#index_5" id="BigTitle">
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            <div class="taoyu_title"> <a href="#index_5" title="5">Drug Designing</a></div>
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          </li>
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          <li class="taoyu" href="#index_6">
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            <div class="taoyu_title"> <a href="#index_6" title="6">Introduction</a></div>
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          </li>
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          <li class="taoyu" href="#index_7">
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            <div class="taoyu_title"> <a href="#index_7" title="7">Diffusion part</a></div>
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          </li>
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          <li class="taoyu" href="#index_8">
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            <div class="taoyu_title"> <a href="#index_8" title="8">Reaction part</a></div>
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          </li>
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          <li class="taoyu" href="#index_9">
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            <div class="taoyu_title"> <a href="#index_9" title="9">Future work</a></div>
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            <h3 id="index_0">Prediction of Off-target Effect</h3>
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            <h5 id="index_1">INTRODUCTION</h5>
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            <p>Since the CRISPR/Cas9 system was first used in genetic engineering, the researches on its off-target effects have never stopped. The methods of Hsu-Zhang scoring (1) and CCTop (2) are two widely used algorithms for designing a single guide
 +
              RNA (sgRNA) sequence and finding potential off-target locus. Last year, a new algorithm named CFD (Cutting Frequency Determination) scoring method was developed to evaluate potential off-target sits with 240 parameters (Fig. 1) (3). All
 +
              of these three methods (Hsu-Zhang scoring, CCTop, and CFD) take  different weight coefficients of different mismatch position into consideration, however, only CFD scoring method considers mismatch types as a factor as well.</p>
 +
            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/5/5b/SFLS_2017_Modeling_CFDscore.png" width="500" alt="demonstrate_fig.1">
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            </div>
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            <br/><br/>
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            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/b/b7/SFLS_2017_Modeling_Mismatch_types.png" width="700" alt="demonstrate_fig.2">
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            </div>
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            <br/><br/>
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          </div>
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          <p>Fig.1 The values of CFD scores change over mismatch positions and types (Data derived from Doench 2016). Mismatch position is counted from 5’ end of gene, position 20 represents the nucleotide nearest to protospacer adjacent motif (PAM), and
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            position 1 represent the nucleotide farthest from PAM.</p>
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          <div class="mainpage mainpage2">
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            <h5 id="index_2">Methods</h5>
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            <p>We chose to use CFD scoring method instead of Hsu-Zhang scoring or CCTop for the following reasons: First, it is reported that the CFD method has higher Pearson correlation (3), compared with Hsu-Zhang scoring method and CCTop, especially
 +
              when the number of mismatched bases is large; Second, Computing the scores by using CFD method is much easier than the other two methods. </p>
 +
            <p>In order to obtain the CFD score of a certain DNA locus, we multiply all the scores of single base mismatch together. If the DNA loci and sgRNA have mismatched bases at position α, β, γ… with mismatch types rA-dC, rC-dC, rU-dT…(Fig. 1B), its
 +
              CFD score is calculated as:</p>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/a/ad/SFLS_2017_Modeling_CFD_score_calculation.png" width="700" alt="demonstrate-sequencing">
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            </div>
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            <br/><br/>
 +
            <p>It is reported that about 60% of melanomas contain a mutation in the v-raf murine sarcoma viral oncogene homolog B (BRAF), and V600E (1799T> A) variation (Fig. 2) in BRAF is the main type of mutations in the cancer tissues, which plays a critical
 +
              role in carcinogenesis of melanoma.</p>
 +
 
 +
            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/5/58/SFLS_2017_Modeling_V600E.png" width="500" alt="demonstrate-sequencing-ATCG">
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            </div>
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 +
            <p>Fig. 2 The graphic of BRAF 1799T>A (V600E) mutation</p>
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 +
 
 +
 
 +
            <br/><br/>
 +
            <p>In our project, we try to disrupt the mutant BRAF in the two melanoma cell lines (A375 and G361) by CRISPR/Cas9 technology. A typical PAM is ‘NGG’. However, we didn’t find it. It has been reported that alternative PAM sequence ‘NAG’ has rather
 +
              high cleavage efficiency while ‘NTG’ shows no tendency of cutting (3). To meet the goal of specific cleavage, we arranged the mutant base on the three PAM bases as shown in Fig 3. </p>
 +
            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/7/7e/SFLS_2017_Modeling_target_region_%26_sgRNA.png" width="700" alt="demonstrate-sequencing-ATCG">
 +
            </div>
 +
 
 +
            <p>Fig.3 The sgRNA for targeting mutant BRAf in melanoma cells</p>
 +
 
 +
            <br/><br/>
 +
            <p>After setting the sgRNA sequence, we searched for the potential off-target locus. The potential off-target locus must meet the following conditions: 1) Having a PAM sequence (‘NGG’, ‘NAG’, ‘NCG’, or ‘NGA’) at 3’ end; 2) More than 13 base identities are
 +
              identified by <a href="https://blast.ncbi.nlm.nih.gov/Blast.cgi?PROGRAM =blastn&PAGE_TYPE=BlastSearch&LINK_LOC=blasthome">MegaBLAST</a>; 3) CFD score is greater than 5%.</p>
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          <div class="mainpage mainpage3">
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            <h5 id="index_3">RESULTS</h5>
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            <p>Using Megablast, we find that over 500 alignments have potential off-target effects, and 62 of them have a PAM sequence (‘NGG’, ‘NAG’, ‘NCG’, or ‘NGA’). Seven of them are scored higher than 5% (Fig. 4). Since ‘NGG’ PAM has much higher efficiency
 +
              of cleavage than ‘NAG’ (Fig.5)(3), the off-target probability of Seq 2, 3, 4, 5, 6 and 7 may be higher than its scores.</p>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/d/df/SFLS_2017_Modeling_potential_off_target_region.png" width="700" alt="demonstrate-sequencing-ATCG">
 +
            </div>
 +
 
 +
            <p>Fig.4 Potential off-target sites of our sgRNA. PAM is marked in red. Compared with the sgRNA, the different bases are marked in yellow.</p>
 +
            <br/><br/>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/d/d2/SFLS_2017_Modeling_PAM.png" width="400" alt="demonstrate-sequencing-ATCG">
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            </div>
 +
            <br/><br/>
 +
 
 +
          </div>
 +
 
 +
          <p>Fig.5 Proportion of active sgRNAs with different PAM on targeting region(Data derived from Doench 2016)</p>
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        <div class="row">
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          <div class="mainpage mainpage2" id="reference">
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            <h5 id="index_4">Conclusion</h5>
 +
            <p>Our sgRNA sequence has high cleavage efficiency on the mutated BRAF gene, as well as a high risk of off-target effect. To avoid the off-target effect, we designed an artificial microRNA complementary to SAMMSON gene, which is specifically
 +
              expressed in human melanomas. The CRISPR/CAS9 system is only activated in cancer cells, having no any effects on normal cells.</p>
 +
            <p>View our regulatory system at <a href="https://2017.igem.org/Team:Shenzhen_SFLS/Design#index_3">https://2017.igem.org/Team:Shenzhen_SFLS/Design</a></p>
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          </div>
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<div class="row">
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          <div class="mainpage mainpage2" id="reference">
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            <h5>References</h5>
 +
            <li>http://crispr.mit.edu/</li>
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            <li> https://crispr.cos.uni-heidelberg.de/ </li>
 +
            <li> Doench, J. G., Fusi, N., Sullender, M., Hegde, M., Vaimberg, E. W., Donovan, K. F., ... & Virgin, H. W. (2016). Optimized sgRNA design to maximize activity and minimize off-target effects of CRISPR-Cas9. Nature biotechnology, 34(2), 184-191.</li>
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            <li> Kleinstiver, B. P., Prew, M. S., Tsai, S. Q., Topkar, V. V., Nguyen, N. T., Zheng, Z., ... & Aryee, M. J. (2015). Engineered CRISPR-Cas9 nucleases with altered PAM specificities. Nature, 523(7561), 481-485.</li>
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        <!-- Part 2 -->
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          <div class="mainpage mainpage5">
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            <h3 id="index_5">Drug Designing</h3>
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          </div>
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        </div>
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        <div class="row">
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          <div class="mainpage mainpage6">
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            <h5 id="index_6">Introduction</h5>
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            <p>At first, we intended to use liposome as genetic vector. However, experiment data shows that liposome (has blank plasmid in it) has a high cytotoxic effect. When the liposome concentration is 4μg/mL, the number of cells died from cytotoxic
 +
              effect is even greater than the number of survival cells (G361). Because the malignant degree of A375 is higher than G361, cytotoxic effect seems much weaker in A375.</p>
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            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/8/82/SFLS_2017_Modeling_A375_Cytotoxic_effect.png" width="400px">
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              <img src="https://static.igem.org/mediawiki/2017/b/bd/SFLS_2017_Modeling_G361_Cytotoxic_effect.png" width="400px">
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            </div>
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            <p>Fig.1 Growth curves of the two cell lines: A375 and G361 (Stannard fitting)</p>
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            <br/>
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            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/9/94/SFLS_2017_Modeling_comparison_of_growth_curves.png" width="700">
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            </div>
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            <p>Fig.2 A comparison of R-Squares of exponential curve and different sigmoid curves. “⭐” marks the best fitting equation.</p>
 +
            <br/><br/>
 +
 
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 +
            <p>Therefore, we planned to use lentivirus as gene vector, which is reported to have lower cytotoxic effect and higher transduction efficiency than liposome vector. Lentiviruses enter into cells while most kinds of viruses bind to cellular membrane and inject their genetic materials into cells when transducing.</p>
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            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/c/cd/SFLS_2017_Modeling_illustration_of_diffusion_and_reaction.png" width="800">
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            </div>
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            <p>Fig.3 Lentiviruses diffusing and entering cells in vivo (Illustration by <i>Tumblr: rainalv </i>)</P>
 +
            <br/>
 +
            <p>As the picture shows, the lentiviruses behave two ways in vivo: Diffusion and reaction (entering cells). The concentration of drug is a function of time (since injection of drug) and position (the distance to drug injection site).</p>
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            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/d/d5/SFLS_2017_Modeling_C_x_t.png" width="100">
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          <div class="mainpage mainpage7">
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            <h5 id="index_7">Diffusion part</h5>
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            <p>We assume that lentiviruses' diffusion in extracellular substance is unsteady-state diffusion, which follows Fick’s Second Law:</p>
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            <div class="picture">
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              <img src="https://static.igem.org/mediawiki/2017/8/8c/SFLS_2017_Modeling_Fick_s_second_law.png" width="200" alt="demonstrate_fig.1">
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            </div>
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            <p>C<sub>diff</sub> is the concentration of drug participating diffusion, C stands for the concentration of drug at certain time and position. t stands for time, D is the diffusion coefficient of lentivirus diffusing in melanoma tissue, &nabla;
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              is the Nabla operator.
 +
              <br/><br/>
 +
              <p>We assume the diffusion coefficient D is a constant, which does not change over time or space. Thus, the equation can be written as:</p>
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              <div class="picture">
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                <img src="https://static.igem.org/mediawiki/2017/b/b9/SFLS_2017_Modeling_FLS_short_one.png" width="150" alt="demonstrate_fig.2">
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              </div>
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              <p>&Delta; is the Laplace operator.</p>
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            <h5 id="index_8">Reaction part</h5>
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            <p>It is obviously that at each moment, the concentration of lentiviruses entering cells is proportional to the current concentration of lentivirus.</p>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/4/4c/SFLS_2017_Modeling_reaction_term.png" width="150">
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            </div>
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            <p>C<sub>reac</sub> stands for the concentration of drug participating reaction,C stands the concentration of drug, k is a constant.</p>
 +
            <br/><br/>
 +
            <p>Combining the two terms together, we have:</p>
 +
 
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/c/cd/SFLS_2017_Modeling_sum_of_diffusion_and_reaction.png" width="200">
 +
            </div>
 +
            <br/>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/2/2b/SFLS_2017_Modeling_PDE_1.png" width="150">
 +
            </div>
 +
            <br/>
 +
            <p>We assume that the volume of each dose is V, and the drug present a spherical shape at the moment of its injection into skin. R is the radius of the spherical drug system.</p>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/2/2b/SFLS_2017_Modeling_volume_of_sphere.png" width="100">
 +
            </div>
 +
            <br/><br/>
 +
            <p>The partial differential equation has the following boundary conditions:</p>
 +
 
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/b/bf/SFLS_2017_Modeling_boundary_conditions.png" width="300">
 +
            </div>
 +
            <br/><br/>
 +
 
 +
 
 +
            <p>I At the moment of injection, the drug has not diffused or entered into cells, so the concentration inside the spherical drug system is the initial drug concentration C<sub>0</sub>.</p>
 +
            <p>II&III The concentration is approaching zero when time or distance is long enough.</p>
 +
            <br/><br/>
 +
            <div class="picture">
 +
              <img src="https://static.igem.org/mediawiki/2017/a/a5/SFLS_2017_Modeling_PDE_the_blue_one.png" width="700">
 +
            </div>
 +
            <P>Fig.5 Forecast results of spherical drug diffusion and reaction in vivo</p>
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          </div>
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        </div>
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        <br/><br/><br/>
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          <div class="mainpage mainpage9">
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            <h5 id="index_9">Future work</h5>
 +
            <p>Since the time is limited, we cannot put our genetic circuit into drugs. We also failed to find data from the existing articles with which the values of parameters can be perfectly determined.</p>
 +
            <p>The parameters D and k can be determined using the method mentioned in Minchinton 2006(4).</p>
 +
            <p>We can know the concentration at any time and position, when the initial concentration and the volume of each dose are set. Thus, the drug release is under control.</p>
 +
            <p>When the drug gets to the border of the dermis and subcutaneous tissue (where the distance to injection site is x<sub>s</sub> x_s), it will be taken away through the complex network of blood vessels. To avoid doing harm to normal cells,
 +
C_max is the concentration which results in the maximal tolerable cytotoxic effect., the concentration at the border should be lower than the concentration resulting in the maximal tolerable cytotoxic effect, which is to be determined using the future experiment data.</p>
 +
          </div>
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 +
 
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        </div>
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          <div class="mainpage mainpage2" id="reference">
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            <h5>References</h5>
 +
            <li>Zwietering M H, Jongenburger I, Rombouts F M, et al. Modeling of the bacterial growth curve[J]. Applied and environmental microbiology, 1990, 56(6): 1875-1881.</li>
 +
            <li>https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion</li>
 +
            <li>Siepmann J, Siepmann F. Modeling of diffusion controlled drug delivery[J]. Journal of Controlled Release, 2012, 161(2): 351-362.</li>
 +
            <li>Minchinton A I, Tannock I F. Drug penetration in solid tumours[J]. Nature Reviews Cancer, 2006, 6(8): 583-592.</li>
 +
 
 +
 
 +
 
 +
          </div>
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Latest revision as of 03:42, 2 November 2017

Team:Shenzhen SFLS/Demonstrate - 2017.igem.org

Team:Shenzhen SFLS/Demonstrate - 2017.igem.org

Modeling

Prediction of Off-target Effect

INTRODUCTION

Since the CRISPR/Cas9 system was first used in genetic engineering, the researches on its off-target effects have never stopped. The methods of Hsu-Zhang scoring (1) and CCTop (2) are two widely used algorithms for designing a single guide RNA (sgRNA) sequence and finding potential off-target locus. Last year, a new algorithm named CFD (Cutting Frequency Determination) scoring method was developed to evaluate potential off-target sits with 240 parameters (Fig. 1) (3). All of these three methods (Hsu-Zhang scoring, CCTop, and CFD) take different weight coefficients of different mismatch position into consideration, however, only CFD scoring method considers mismatch types as a factor as well.

demonstrate_fig.1


demonstrate_fig.2


Fig.1 The values of CFD scores change over mismatch positions and types (Data derived from Doench 2016). Mismatch position is counted from 5’ end of gene, position 20 represents the nucleotide nearest to protospacer adjacent motif (PAM), and position 1 represent the nucleotide farthest from PAM.




Methods

We chose to use CFD scoring method instead of Hsu-Zhang scoring or CCTop for the following reasons: First, it is reported that the CFD method has higher Pearson correlation (3), compared with Hsu-Zhang scoring method and CCTop, especially when the number of mismatched bases is large; Second, Computing the scores by using CFD method is much easier than the other two methods.

In order to obtain the CFD score of a certain DNA locus, we multiply all the scores of single base mismatch together. If the DNA loci and sgRNA have mismatched bases at position α, β, γ… with mismatch types rA-dC, rC-dC, rU-dT…(Fig. 1B), its CFD score is calculated as:

demonstrate-sequencing


It is reported that about 60% of melanomas contain a mutation in the v-raf murine sarcoma viral oncogene homolog B (BRAF), and V600E (1799T> A) variation (Fig. 2) in BRAF is the main type of mutations in the cancer tissues, which plays a critical role in carcinogenesis of melanoma.

demonstrate-sequencing-ATCG

Fig. 2 The graphic of BRAF 1799T>A (V600E) mutation



In our project, we try to disrupt the mutant BRAF in the two melanoma cell lines (A375 and G361) by CRISPR/Cas9 technology. A typical PAM is ‘NGG’. However, we didn’t find it. It has been reported that alternative PAM sequence ‘NAG’ has rather high cleavage efficiency while ‘NTG’ shows no tendency of cutting (3). To meet the goal of specific cleavage, we arranged the mutant base on the three PAM bases as shown in Fig 3.

demonstrate-sequencing-ATCG

Fig.3 The sgRNA for targeting mutant BRAf in melanoma cells



After setting the sgRNA sequence, we searched for the potential off-target locus. The potential off-target locus must meet the following conditions: 1) Having a PAM sequence (‘NGG’, ‘NAG’, ‘NCG’, or ‘NGA’) at 3’ end; 2) More than 13 base identities are identified by MegaBLAST; 3) CFD score is greater than 5%.




RESULTS

Using Megablast, we find that over 500 alignments have potential off-target effects, and 62 of them have a PAM sequence (‘NGG’, ‘NAG’, ‘NCG’, or ‘NGA’). Seven of them are scored higher than 5% (Fig. 4). Since ‘NGG’ PAM has much higher efficiency of cleavage than ‘NAG’ (Fig.5)(3), the off-target probability of Seq 2, 3, 4, 5, 6 and 7 may be higher than its scores.

demonstrate-sequencing-ATCG

Fig.4 Potential off-target sites of our sgRNA. PAM is marked in red. Compared with the sgRNA, the different bases are marked in yellow.



demonstrate-sequencing-ATCG


Fig.5 Proportion of active sgRNAs with different PAM on targeting region(Data derived from Doench 2016)




Conclusion

Our sgRNA sequence has high cleavage efficiency on the mutated BRAF gene, as well as a high risk of off-target effect. To avoid the off-target effect, we designed an artificial microRNA complementary to SAMMSON gene, which is specifically expressed in human melanomas. The CRISPR/CAS9 system is only activated in cancer cells, having no any effects on normal cells.

View our regulatory system at https://2017.igem.org/Team:Shenzhen_SFLS/Design




References
  • http://crispr.mit.edu/
  • https://crispr.cos.uni-heidelberg.de/
  • Doench, J. G., Fusi, N., Sullender, M., Hegde, M., Vaimberg, E. W., Donovan, K. F., ... & Virgin, H. W. (2016). Optimized sgRNA design to maximize activity and minimize off-target effects of CRISPR-Cas9. Nature biotechnology, 34(2), 184-191.
  • Kleinstiver, B. P., Prew, M. S., Tsai, S. Q., Topkar, V. V., Nguyen, N. T., Zheng, Z., ... & Aryee, M. J. (2015). Engineered CRISPR-Cas9 nucleases with altered PAM specificities. Nature, 523(7561), 481-485.






  • Drug Designing

    Introduction

    At first, we intended to use liposome as genetic vector. However, experiment data shows that liposome (has blank plasmid in it) has a high cytotoxic effect. When the liposome concentration is 4μg/mL, the number of cells died from cytotoxic effect is even greater than the number of survival cells (G361). Because the malignant degree of A375 is higher than G361, cytotoxic effect seems much weaker in A375.

    Fig.1 Growth curves of the two cell lines: A375 and G361 (Stannard fitting)


    Fig.2 A comparison of R-Squares of exponential curve and different sigmoid curves. “⭐” marks the best fitting equation.



    Therefore, we planned to use lentivirus as gene vector, which is reported to have lower cytotoxic effect and higher transduction efficiency than liposome vector. Lentiviruses enter into cells while most kinds of viruses bind to cellular membrane and inject their genetic materials into cells when transducing.

    Fig.3 Lentiviruses diffusing and entering cells in vivo (Illustration by Tumblr: rainalv )


    As the picture shows, the lentiviruses behave two ways in vivo: Diffusion and reaction (entering cells). The concentration of drug is a function of time (since injection of drug) and position (the distance to drug injection site).



    Diffusion part

    We assume that lentiviruses' diffusion in extracellular substance is unsteady-state diffusion, which follows Fick’s Second Law:

    demonstrate_fig.1

    Cdiff is the concentration of drug participating diffusion, C stands for the concentration of drug at certain time and position. t stands for time, D is the diffusion coefficient of lentivirus diffusing in melanoma tissue, ∇ is the Nabla operator.

    We assume the diffusion coefficient D is a constant, which does not change over time or space. Thus, the equation can be written as:

    demonstrate_fig.2

    Δ is the Laplace operator.




    Reaction part

    It is obviously that at each moment, the concentration of lentiviruses entering cells is proportional to the current concentration of lentivirus.

    Creac stands for the concentration of drug participating reaction,C stands the concentration of drug, k is a constant.



    Combining the two terms together, we have:



    We assume that the volume of each dose is V, and the drug present a spherical shape at the moment of its injection into skin. R is the radius of the spherical drug system.



    The partial differential equation has the following boundary conditions:



    I At the moment of injection, the drug has not diffused or entered into cells, so the concentration inside the spherical drug system is the initial drug concentration C0.

    II&III The concentration is approaching zero when time or distance is long enough.



    Fig.5 Forecast results of spherical drug diffusion and reaction in vivo




    Future work

    Since the time is limited, we cannot put our genetic circuit into drugs. We also failed to find data from the existing articles with which the values of parameters can be perfectly determined.

    The parameters D and k can be determined using the method mentioned in Minchinton 2006(4).

    We can know the concentration at any time and position, when the initial concentration and the volume of each dose are set. Thus, the drug release is under control.

    When the drug gets to the border of the dermis and subcutaneous tissue (where the distance to injection site is xs x_s), it will be taken away through the complex network of blood vessels. To avoid doing harm to normal cells, C_max is the concentration which results in the maximal tolerable cytotoxic effect., the concentration at the border should be lower than the concentration resulting in the maximal tolerable cytotoxic effect, which is to be determined using the future experiment data.




    References
  • Zwietering M H, Jongenburger I, Rombouts F M, et al. Modeling of the bacterial growth curve[J]. Applied and environmental microbiology, 1990, 56(6): 1875-1881.
  • https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion
  • Siepmann J, Siepmann F. Modeling of diffusion controlled drug delivery[J]. Journal of Controlled Release, 2012, 161(2): 351-362.
  • Minchinton A I, Tannock I F. Drug penetration in solid tumours[J]. Nature Reviews Cancer, 2006, 6(8): 583-592.
  •